V t m t m E

DIY'TIMON r Lm'

Aproved Ia ptihli'iw1vis

Distribution TUnmitl-
Acoustic Reflection

from Surfaces and Shapes

Author/Editor

Werner G. Neubauer, Ph.D.

DTIC S ELECTE

JUL 2 4 1987M

D

June 1986

~nov~DWSMIUTIONApproved toz public releosol

NAVAL RESEARCH LABORATORY

Nashlinton, D.C.
FOREWORD

This book collects, under single cover, the acoustic reflection and

scattering work, theoretical and experimental, that has been carried out

on acoustic reflection and scattering by the group that was first started

in the Sound Division at the Naval Research Laboratory by the author

under Dr. Raymond Steinberger in 1958.

The funds to produce this book were provided cooperatively by

three codes in the Office of Naval Research: Dr. Nicolas Basdekas,

Code 432, Dr. Logan Hargrove, Code 421, and Dr. Peter Rogers, Code

425UA and the Acoustics Division of the Naval Research Laboratory.

Their support and encouragement is gratefully acknowledged.

I wish to thank my co-workers, past and present, and co-authors

of the papers that are the foundation of this book for their contributions. The long association, co-authorship and many conversations

about the book and its contents with Dr. Louis R. Dragonette have

been a rewarding association for me. Also, work with Dr. Lawrence

Flax has been stimulating, provocative, and a great pleasure. In various

ways and to varying degrees the contributions of the following are

recognized and deeply appreciated:

Dr. C. M. Davis

Dr. Herbert Uberall

Richard Vogt

Dr. Susan Numrich

Dr. Peter Ugindius

Luise Schuetz

Leonard Burns

Janet Mason

Without the encouragement of the Superintendent of the Acoustics

Division, Naval Research Laboratory, Dr. John C. Munson, this

volume would probably not exist. The production of a book has editorial, mechanical, emotional, financial, and a seemingly unending list

of considerations and difficulties. It is not possible to mention all those

who had some hand in some aspect of producing these pages, but their

willing and expert help is nevertheless greatly appreciated.

iii
Typing, proofing, and coordinating many aspects of the production

of this book are the result of unceasing always cheerful efforts of Mrs.

Roberta Hopkins. I wish to thank her for her great help and patience.

The text was first typed on a word processor, transmitted by a communications link to the Technical Information Division, Computerized

Technical Composition Section at the Naval Research Laboratory. That

group, headed by Mrs. Dora Wilbanks, composed the book and

prepared it for printing. This was primarily the patient and capable

effort of Judy Kogok who made it all come together in coherent

fashion. Her contribution and the willing and cheerful assistance of

Mrs. Jean Moon is gratefully acknowledged.
PREFACE

e intent of this book is to make accessible selected reflectionand reflection-related work of a group that has been conducting such

research since approximately 1958. I have myself carried out or been

involved in or responsible for the work described in each of the

chapters in this book. In some cases, I have played the part of no more

than editor, with only minor modifications introduced from the original

publication. In all cases, when a single published paper is an entire or

major content of a chapter it is referred to at the title of the chapter.

For many chapters my role was more appropriately that of editor than

author. Time and budgetary considerations prevented extensive rewriting. I regret that, as a result, notation and mathematical conventions

are not necessarily consistent among all of the chapters. Various

degrees of new material are included in several chapters.

The material is intended to be collective rather than encyclopedic.

Therefore, regretfuily, this book does not include the work of other

groups and individuals who have contributed significantly to the field of

acoustic reflection and. scattering.

Accesion For

NTIS 

DTIG TAB

CRAM F J

Urano.:rd U_..

Justhe~at :on .................... J

By.............

a~~A [,.Ntiity ;0Olv.

Dist

v fl-( _i
CONTENTS"

Chapter Page

I APPROXIMATE FORMULATION OF REFLECTION

BY THE SUMMATION FORMULA ................. 1

2 REFLECTION FROM A FINITE PLANE

AND EXPERIMENTAL MEASUREMENTS ........... 17

3 THEORY AND DEMONSTRATION OF

CREEPING WAVES......................................... 35

4 CIRCUMFERENTIAL WAVE INTERPRETATIONS IN

CYLINDER REFLECTIONS,............................... 55

5 LAYERED ELASTIC ABSORPTIVE CYLINDERS ....... 125

6 NONABSORBING AND ABSORBING

CYLINDERS................................................. 139

7 A CYLINDRICAL CAVITY IN AN

ABSORPTIVE MEDIUM ................................... 151

8 ELASTIC SPHERES - STEADY-STATE SIGNALS...... 161

9 ELASTIC SPHERES AND RIGID SPHERES AND

SPHEROIDS - TRANSIENT SIGNALS................ 177

10 ADSORBING SPHERES........................................ 195

11 LONGITUDINAL WAVES INCIDENT ON AN

ELASTIC SPHERICAL OBSTACLE IN

ANOTHER ELASTIC MATERIAL ...................... 205

12 RUBBER CYLINDERS AND SPHERES .................... 225

13 ELASTIC CYLINDERS AND SPHERES AT HIGH ka ... 237

vii
14 REFLECTION AND VIBRATIONAL MODES

OF ELASTIC SPHERES.................................... 241

15 RESONANCE EXCITATION AND

SOUND SCATTERING .................................... 257

16 CALIBRATION OF ACOUSTIC SCATTERING

MEASUREMENTS USING SPHERE REFLECTIONS . 279

17 EVALUATION OF A REFLECTION-REDUCTION

COATING .................................................... 289

18 REFLECTION OF A BOUNDED BEAM ................... 297

19 RAYLEIGH AND SHEAR SPEED DETERMINATION

USING SCHLIEREN VISUALIZATION................. 315

20 SCHLIEREN VISUALIZATION OF *HEADWAVES!

ON PLATES.................................................. 325

21 SCHLIEREN VISUALIZATION TO DETECT FLAWS

ON PLATES.................................................. 347

22 EXPERIMENTAL FACILITIES .......................... 357

23 SPEED OF SOUND IN FRESH WATER.................... 373

INDEX ............................................................ 397

Vill
Chapter 1

APPROXIMATE FORMULATION OF REFLECTION

BY THE SUMMATION FORMULA*

INTRODUCTION

Approximate solutions to reflection and scattering problems,

beyond their usefulness in computations over limited ranges of parameters, have a value in clarifying physical understanding. This is true of

the Kirchhoff approximation. The fundamental approximation attributed to Kirchhoff has basic limitations; but, also, the description of it

leads to a basic understanding of reflection by finite bodies. There

seems ,to be no direct application of such an approximation to reflection

by Kirchhoff himself. Meecham [11, however, has considered its use

for reflection problems. The attribution of the approximation stems

from its use in a problem that is similar to the reflection problem, that

of propagation of a wave through a circular aperture in a plane screen.

In words, it demands that the wave in the aperture is everywhere what

it would be if the screen were absent. Thus, there is a restriction of the

aperture size relative to the wavelength and, also, of the accuracy of the

resultant formulation describing a field value which depends on the

location of the field point relative to the aperture. It can be shown [21

that for an aperture size greater than the wavelength a good approximation is achieved. Similarly, an angular limitation to the main lobe

between the first two nulls, caused by the diffraction, is adequately

described. The angular region for reasonable accuracy by the Kirchhoff

approximation can exceed the main lobe as will be shown in Chapter 2.

The reflection from a body, whose dimensions are large compared

to the acoustic wavelength, at a great distance from it, can be approximately described by the use of Fresnel zones on the surface of the

reflector. In a way similar to the Kirchhoff approximation, as the

reflector dimensions approach the acoustic wavelength, Fresnel zone

analysis gives poor results. The Fresnel formula can be interpreted to

describe the far field acoustic pressure as (31

PI PN

P' " T +- (I)

'Th dsvopmoMt was fkrt PrOe1d in: WMare 0. Noubuer; J. Acoust. Soc. Am. 35,

27I-285 (1963).
2 CHAPTER I

where Pri and p,. are contributions to the total reflection from the first

and nt

h quarter-wave (Fresnel) zones respectively. On most regular

curved convex bodies, such as spheres and cylinders, the last (nh) zone

is almost normal to the direction of incident wave propagation, especially if the body is much larger than an acoustic wavelength. Therefore,

the last zone contribution to total reflection is negligible and the

reflection by the body is substantially one half that from the first (central) zone. An example of the application of the Kirchhoff approximation will be given in which Fresnel zones are subdivided. It has been

called the summation formula method. By this means, the contribution

to the elements of acoustic reflection by bodies will be brought out.

It is convenient to express the solution to the wave equation in

the velocity potential 0 by

~.. exp 1w t LJ (2)

This assumes a lossless infinite homogeneous diverging acoustic wave

traveling with a speed c. The potential is at any point in the field at a

distance r from a point source. That source has a strength A exp (iwt)

at unit distance, oscillating harmonically with angular frequency (. A

smooth, rigid finite reflector insonified by such a wave results in a velocity potential in the field which depends on the reflector size, shape,

and orientation. A continuous source results in a velocity potential

which is the summation of the potential due to that source and the

potential due to the reflection by the scatterer. Here the interaction

between the source and reflected wave will be ignored and only the

velocity potential resulting from interaction of an incident wave with

the reflector will be considered. For simplicity, consider the far field

reflection in the backscattered direction, or that which is monostatic

reflector.

CONTINUOUS SURFACE

Since o - 2vf and X c/f, Eq. (2) may be written

The far field demand on the problem forces r,, the distance to the point

on the reflector nearest the source, to be considered sufficiently large that r, >> k. If, in addition, each dimension of the reflector is also

very small compared to ri, a portion of the spherical wave incident on

the reflector may be considered plane to any desired degree. We will
SUMMATION FORMULA 3

consider the space between the normal plane (Fig. 1) and the reflector

to contain a wave that is sufficiently plane so that spherical divergence

may be neglected there. Now the velocity potential at the reflector surface may be written

-B exp[ - i27" r i (4)

where ri + A r is the distance from the source to an elemental area da

on the reflector. As a consequence of our previous assumptions,

r >> Ar in Eq. (4) b is a constant factor containing the periodicity of

the source and the source strength as well as the spherical divergence

causing diminution of velocity potential over the distance ri, so B =

(Air) exp (i21rft). In the exponent of Eq. (4) the ratio rl/x may be

divided into an integral and a fractional part, since exp (-i21rn) I I

and exp (-i2ra) - a- constant where n is an integer and a is a

number less than unity. Now

-aB exp( i27r ,fJ (5)

As a part of the Kirchhoff approximation, and to satisfy the boundary

condition on particle velocity at the rigid surface, the reflecting surface

is regarded as an infinite number of elemental areas da, each of which

operates as a simple source in an infinite baffle and radiating through

half space. This construction is known to limit the applicability of the

forthcoming analysis with respect to the curvature, extent, and perimeter of the surface.

(soucr SURFACE

S RO

(RECEIVER) d

Fig. I - Geometry for monostatic reflection
4 CHAPTER 1

The incident particle velocity is

"- _w 1 + ±i ikoj I - (6)

where k - 2r/k. The imaginary term in Eq. (6) is not significant since

r 1 X, so fi ikc,.

The strength Q of the elemental area da as a source whose normal

particle velocity is equal and opposite to that of the normal component

of the incident particle velocity is Q - - fda - - iko1 cos Oda,

where 0 is the angle between the normal to da, and the direction ri of

the incident wave. See Fig. 2. The velocity potential at R, the position

of the receiver, due to the area da acting as a source is

2w,,A(r+r)

I. rr+Ar ia'k. Ar

exp - i2v r ,+ 2---riexp ia' - i2vr- cosOda. (7)

We can simplify, as we did for the incident phase, and determine that

exp (-i 2 rr/A) - a', where a' is a constant for a fixed value of r, just

as a was a constant for a fixed value of r. Substituting for the incident

field 46, from Eq. (5), Eq. (7) becomes

dOr -d,- - 2a' 21r r, exp 1 I Air - Cos Oda. (8)

The exponent -i4rAr/A may be written -i2rAr/(l/2X). The factor

Ar/(1/2A) will take values which may be expressed again as an integral

part m - 0, 1, 2, ... and b, a number less than unity, as before in

deriving the expressions in treating a and a' so that

exp(- i21r(m + b)] - exp(- i21rb),

and

d-, iia'k B exp (- i2vb) cos Oda. (9) 21r r,

The field at r resulting from the entire reflecting surface is the integral

of the contributions d4 r from the elemental areas da and is given by

, - - aaBf r cos 0 exp (- i2wrb)da, (10)

where s is the reflector surface. Geometrically, da cos 0 is the projection da' of the element da onto the normal plane (Fig. 1) and, therefore, the integral may be taken over the projection of the entire surface

iI
SUMMATION FORMULA 5

n Fig. 2 - Particle velocity at the

.9 d reflecting element

'94

onto the normal plane instead of'the surface itself. So Eq. (10) may be

written as

I f'kBf ja exp (- i21rb)da'. q = 2rr s' (11)

Now we have an expression for an acoustical quantity, the velocity potential, from which may be derived other quantities in which

one might be interested, in terms of constants derived from measurable

distances or areas on the reflector or on a projection of the reflector

onto the normal plane that can be measured. In most descriptions of

the acoustical field, quantities that are required to describe the reflected

field are acoustical quantities such as pressure or particle displacement

measured on or near the body itself. Equations (10) and (11) require

only area and distance measurements, but are encumbered by the limitations of the Kirchhoff approximation.

RIGID, FINITE PLANE

As an example of the application of the formulas which have been

derived, let us consider a rectangular reflecting plane of width d and

height h. Let the plane be located in an orthogonal coordinate frame

with the height h parallel to the Y-axis, the X-axis dividing the normal

plane in half and let Z be the direction of propagation. The positive Z

direction is the direction of the incident wave considered plane by the

time it reaches the reflector. See Fig. 3. Since b + m = Ar/(1/2X), b

(2x tan O/x) - mand da' = dxdy, Eq. (11) becomes

iaa'B Y f2h dcos* -47rXtan dx. (12) , -- r, -/2h dyJ 0 exp t
6 CHAPTER 1

REFLECTING

PLANE

Y

Xd

NORMAL

PLANE

Fig. 3 - The orientation of the plane reflector

in Cartesian coordinates

Equation (12) can be written approximately as a finite sum as

i m . Axy j exp ---L-x tan 01. (13)

j-I p-I

This equation may be considered the velocity potential from the normal

plane itself if it were composed of areas Ax. Axj qt, in number. Each

area is considered to be vibrating with phase (4ir/X) xp, tan 0 where x

- pAx.. Because of the choice of the orientation of the plane in the g

coordinate system , Ayj - h and Eq. (13) may be written

j--I

iaa' i ~xexpf A Ai X tan ej (14)

I -
SUMMATION FORMULA 7

Consider the reflecting plane divided into smaller planes spaced at

integral multiples of A/2 from the normal plane, and parallel to it as

shown in Fig. 4, where the geometry is being viewed looking down

along the Y-axis. These subplanes will intersect the reflecting surface

to define on it what may be considered half-wavelength zones. Subdivide the distances between these subplanes with even more closely

spaced planes, again parallel to the normal plane, equally subdividing

the X/2 distances into an arbitrary number P. Let the smallest planes

intersect areas having widths on the reflector surface small compared to

a wavelength. In Fig. 4 the construction is viewed in the X-Z plane

for P = 6. For any orientation parallel to the Y-axis, the projections of

the subdivided surface areas, all of length h, are proportional to the line

length Ax = Ax2 - ... Axp. These are the magnitudes of the vectors in the complex plane in Fig. 5. The constant phase increment is

the angle between successive vectors, and each time the factor

[Ax,/(A/2) tan 01 becomes an integral value i.e., when p = v, the sum

becomes zero. In Fig. 4, only two and a fraction (1/A) subzones are

taken; in the second, X/2 zone. The resultant relative velocity potential

at R is proportional to the vector A. It must be remembered that a

constant factor h, which was a coefficient for the sum, accounts for the

height of the plane and must be taken into account as a scale factor for

A.

The graphical solution of Eq. (12) can now readily be seen to be a

circle tangent to the real axis in the complex plane if Axp-- 0 as

p "". Co.

x

_. V2__

q

Z

Y

Fig. 4 - A graphical composition of the plane reflecting

surfaces as finite normally oriented areas

iI
8 CHAPTER 1

(-) IMAGINARY

AXIS

AX4

3/

A

Axe AX 2

!Ax? V/"-" 3

AXI REAL

AXIS

Fig. 5 - The plot in the complex plane of ,

approximated by the sum in Eq. 14

FINITE WEDGE

A wedge, as depicted in Fig. 6, can be considered by the previous

method if it is regarded as two finite planes rigidly fixed with respect to

each other. Under the same conditions as in the previous section, the

velocity potential at a distance R will get the summation in phase and

amplitude of the velocity potential of each plane separately. In integral

form this may be expressed as

0- 

* icia'B [h? if h/2 d fdco0 

exp - '47r!: tan 0 &C

+ h12 f d2cos exi[ l 4 x tan yjdxd (15)

which upon integration yields

aa'h I exp- [(i4ird/k)sin9] exp- [(i4vd2/) sin (0 +P)]

4wrr,[ tans tan(0+P)

- coto + cot(o +P)]. (16)

For a cube, in which case P - v/2 and dl - d2 - d, Eq. (16)

becomes

aa'h. w- exp[-(14wrdI)sin0l exp[- (4rd) cos01 c

4r rtanG + cot 02 = 0

(17)

iI
SUMMATION FORMULA 9

p t 4 /

Fig. 6 7 The wedge reflector i

t h

I

I

Taking finite elements along a plane, as shown in Fig. 5, Eq. (15) may

be expressed as a sum similar to Eq. (14) for the plane as,

i a'hB ' iw

Or X ;(x+Ax exp -- 1tntan(

Ar, - A 

Figure 7 shows two means of arriving at the complex resultant A,. In

Fig. 7(a), the resultant obtained from the separate consideration of

each face by adding A and A' associated with faces P and P' respec- tively in Fig. 6. The same resultant AR is obtained if Ax, and Ax, are

first added and then summed over p, as shown in Fig. 7(b), in which

p -,t> p' - '. Here the definition is made that for t> ', Axj, is

0. Since LJc, tan e1 -- Ix,. tan vi (Fig. 8), Eq. (18) reduces to

X (A,+ x,.ex[. p ~tan eJ. (19) Ia'B 'Ii
10 CHAPTER I

I INA Y I-) IMLGINAR

AXIS AXIS

As

A An - REAL - X X*REZ

AXIS U, + &X, AXIS

AX,

(a) (b)

Fig. 7 - Plots in the complex plane for reflection from a wedge: (a) each taken separately

and their resultants added, and (b) both planes taken as contributing to the same zone

AAXg

pp

IAxe AXI

I

Fig. I - The graphical composition of the

refloltns wedge surfaces as finite normally

oriend armew
SUMMATION FORMULA 1

RIGID, DOUBLE CURVED SURFACE

If the same construction and analysis already applied to a plane and a wedge is applied to a curved surface, the finite moduli Ax, of Eq. (19) would, in general, not be equal for equal phase increments as

p - t (see Fig. 9). In addition, a restriction must be imposed on the

principal radii of curvature of the surface. It is sufficient to demand

that at all points of the reflector the principal radii of curvature be

greater than the incident sound wavelength so that the small but finite

areas Aa, may be considered essentially plane. The general expression

for the velocity potential 0, as an integral over the illuminated surface

of the reflector is

- lekB f Iexp (- iOkAr)I cos Oda, (20)

where k2w/k. The equivalent expression over the projections da' of

the subzones onto the normal planes is

ia's exp (- i2wkAr)da. (21)

' - f2rr, ex

I-) IWAGINARY AXIS

Fig. 9 - A plot in the complex plane for

reflection from a general shape as approximated by the sum in Eq. (24)

AR

*FREAL AXIS

The distance from the normal plane, along the Z-axis, is considered to be I and is divided into p equal parts, the integrals in Eqs.

(20) and (21) respectively can be approximated by sums

iaa'kB AaP cos 0 exp - k (22)

2 r, A I- (

2wr
12 CHAPTE I

As shown in Fig. 10, Aa,. is the projection onto the normal plane of

that portion of the reflector surface between the Mh and the (p + l)th

plane subdividing L The complex plane is shown for such a sum.

Equation (22) can be evaluated by measuring the areas on the reflecting

body between successive cutting planes or alternatively the areas of

their projections onto the normal plane. The associated value of p//gs

to the area measured on the surface of the reflector or on the normal

plane must, of course, also be known. The magnitude of the velocity

potential at R then for a given orientation of the reflector may be expressed approximately as

I10- kfBI Aa,. sin 2k + Aa. cos 2kj (24)

Application of this formula result in formulas for the reflection from

rigid quadric surfaces which have been published [51 and a table for

various spheroidal shapes is reproduced here in Table I.

It would be useful to know how much of an approximation such

an approximate formula employing the Kirchhoff approximation and

then an approximate summation to an integral will cause. For this, let

us consider the reflection from the sphere. Figure II shows the solution for the reflection from a sphere using the Kirchhoff approximation

given by Eq. (24) which results from the integration of Eq. (11) over

the projection onto the normal plane of a sphere radius a. The result is

2r, cos 2ka ka

where a" - exp(-i2ka). The magnitude of Eq. (25) is

auBI _ sin 2ka + sin2 ka 11/2 (26)

2rI ka (ka)J 

The square-root expression is what is plotted against ka in Fig. I1. The

exact solution, which will be given later, is also shown. It can be seen

that the periodicity of oscillation of the solution with increasing ka is

significantly different and the magnitude in the region of 2 can be very

much different. However, as /ra increases, these oscillations in both

cases decrease and the difference in the solutions tend to be no more

than plus or minus 10% or approximately plus or minus I dB. In addition, an approximation is introduced by approximating the integral by a

finite sum. The integral is approached to a desired degree by the choice

of the ratio I/ka or the choice of ,&/p for a given ka. For ka greater

than 6, the difference between the two curves is never greater than

16%, or approximately 1.5 dB. I
SUMMATION FORMULA 13

C-v

1/41

I

44%

Fig. 10 The geometrical quantities involved in the derivation or the tflection summation formula For clarity, the contours defined on the reflecting objec are not proweced

onto the normal plane, but onto a plane parallel to the normal plane at a constant dastance from it. The family of projected contours on this paralel plane is identical to the

frmily or contours which would be obtained if the projection were made onto the normal

plane.
14 CHAPTER I

Table I - Reflection Characteristics of the Rigid Ellipsoid

and Its Special Cases

Objec dOtct A Aspect c Conitoe Coe"Iom em e- A reea MagnAtdre of the Reflected Velocity Potential

A.,, I.

EHIpSmd PtgaS ENwd Pn*W il0 , 1 2 ill 1 L1) slin' W.) uhsl si (21,1.1 + ' Ig#I)at i -iii I -

AIm 11', I 2 (hi s)3 his

(It6 IA . 1, 1

Sphere Any 1I$ 1 -3. (2t- 2p11 [ n m!! s ) ul s

in Ois) +

sa, o m , w e n 1 s 1 t1, - 2 p .+ 1 11¥ 1 sin' i *,,) 1 k s Spers _______ ____________ _______ I"( al S Ps) his

Oblate E d O $I 1, 4

2

0 2P + 1 1 /11 1L. 1 sI n' (kis) sun (2 1ls) +

Spimeroid 11 2~ ou (hs) W hS

seam ill ' 2 + 1) .-,ll±-,I n' (is) sn (2, s +,

Ipe r i " I 1? 2 ( ho s ) i s

14

INTEGRAL (KIRCHI4OFF AIPIROX)

1d2 EXACT

10-

08-

06

04

02

oil 2 4 6 a 10 12 14 16 isl 20

Fig I I - The reflio from a rqigi sphere

Lt
SUMMATION FORMULA 1

The error introduced by use of the summation formula related to

the choice of 1A for a sphere is given in Table 2. These error values

were obtained by evaluating Aa, for a sphere and algebraically summing the series thus obtained. Passing to the limit by allowingu -- c

as A,- 0 results in Eq. (26). Of course, this error is not directly

applicable to what one would expect for an arbitrary shape, since the

error introduced would definitely be a function of the local curvature of

the reflector. Evaluations such as this can however serve as a guide in

the application of the formula summation.

Table 2 - The Error in

Approximation of the Integral by a

Finite Sum for a Chosen ~

14Error(%

8 11

16 2

24 1

48 0.3

CONCLUSION

The summation formula can be employed to utilize measured

areas on the target to directly arrive at an acoustical field quantity

representative of the reflection, say the velocity potential. The formula

is only useful in those cases where an insignificant amount of energy

can be considered to penetrate the target and cause elastic vibration of

the target itself. As will be seen later, there are regions even for very

practical shapes and dimensions that will permit application of the

assumption of very little penetration into the target and, therefore, reasonable approximation by a formulation assuming a rigid boundary condition.

REFERENCES

I. W.C. Meecham, "On the Use of Kirchhoff Approximation for the

Solution of Reflection Problems," J. Rational Mech. Anal. 2, 323

(1956).

2. S. Hanish, A Treadis on Acowatc Radiation, book published by the

Naval Research Laboratory (1981).
16 CHAPTER 1

3. F.A. Jenkins and H.E. White, Fundamentals of Optics, McGrawHill Book Company, Inc. (1950).

4. A.J. Rudgers, "Calculation of the Monostatically Reflected Velocity Potential in the Far Field of Certain Finite Rigid Bodies,"

NRL Report 6285, December 1965.
Chapter 2

REFLECTION FROM A FINITE PLANE AND

EXPERIMENTAL MEASUREMENTS*

INTRODUCTION

A further application of the Kirchhoff approximation to the problem of the reflection from a rectangular plane considered to be rigid will

allow a comparison with experimental results. Theoretical description

will be accomplished by approximate evaluation of the retarded potential in a rectangular coordinate system.

THEORY

The velocity potential 0 at a distance ra from an elemental

reflecting area da is

dOa, - (dQ/2w ra) exp- (ikra), (1)

where dQ is the strength of the elemental source ua on the reflector,

radiating into half space. This essentially constitutes the Kirchhoff

approximation. In terms of the incident particle velocity f j,

dQ - - f

4 cos * cos Oda - -ik40cos 0 cos Oda, (2)

where 0; is the incident velocity potential in the absence of the

reflecting plane, k is the wave number of the incident wave and the

angles q and 0 are those defined by Fig. 1. For a simple harmonic

spherically diverging wave emitted from a source at a distance ri

0'- (Airi) exp [ik(ct - r,)]. (3)

At a sufficiently large ri , i.e., r, >> X, where x is the wavelength, a

plane wave approximation over a limited region may be achieved.

Assuming the reflecting plane to have a wave incident on it under these

conditions,

d#, - - (i,/r, ) cos # cos 0 exp [- i2wr(r, + r,')/klda, (4)

MThe theory and experiments described here first approved in: Werner G. Neubauer and

Louis R. Dragonette, 1. Acoust. Soc. Am. 41, 656-661 (1967)

17
18 CHAPTER 2

A £

1

rw~

Fig, i - The geometry which defines r, and r.

where Oi represents the incident velocity potential on the normal plane

which just touches the corner of the reflecting plane nearest the source.

The plane wave approximation permits 0' a Oi,. The quantity

(a + r1')/X represents the phase retardation that occurs as the wave

transits from the normal plane to the reflector and back to the field

point p. The distance r , is the distance from the normal plane to the

element da on the reflecting plane. Expressed as an integral over the

reflecting plane

€ i -r cos , cos 0 I7 er. x I- I2, + XT r Ida, 1o.1 (5)

where r, has been approximated by r, which means that the divergence

from each elemental area da to a reflected point p is the same for all

elements on the reflecting plane. From the geometry in Fig. 1, da

equals dC d,9 and it is possible to derive the relationship

r2 -_ (x-C)2 + (y-)2+z 2 -r 2 +C 2 +iq _ 2xC- 2y.". (6)

It can also be shown that

rl'- [(d/2n) - ] sin 0 cos 0 + [(d/2) - ,q] sin 9, (7)
FINITE PLANE 19

where d and n are defined in Fig. I. The quantity r. can be taken to be

the first two terms of the binomial expansion

r, - r + ( - 2x4 +77 2 - 2y)/2r, (8)

and the exponential in Eq. (5) can be cast into the form of the Fresnel

integral so

27r (ra + r')/, (-r/2) [4(r, + r,')/X I 7r/ 2 (at' + u'2 v'), (9)

where

a'- (4r/k) + (2d/nk) sin A cos 0 + (2d/A) sin 0

(9a)

u'2 - (2/kr)C2 - (4/Xl[(x/r) + sin sp cos 01J,

and

V2 - (2/Xr)q2 - 14/AII(y/r) + sin O),q. (9b)

Solving Eq. (9a) for and substituting

u '2 = u 2 - (2/xr) (2x) 2,

it follows that from Eq. (9a)

u - (2/Xr)1/ 2(C -- 2x).

Similarly, Eq. (9b) results in

v - (2/Ar)1/2(q - 2y),

where x r sin qi cos 0 and y - r sin 0. Now Eq. (5) becomes

WCos 4t cos J exp{a iwJ

(10)

exp I exp r2 dv

xjex1 2# d zr 2

where

a 2r + (xdlnr) + (yd/r) - 4(x2/r) - 4(y2/r)

and

u,2 (2/,kr)/ 2 [F(d/2) - 2x,

vi. 2 " (2/ Ar)"/2 lT(d/2n) - 2yj.
20 CHAFTER 2

Also,

P o Cos-Cos s f2-exp (iv U du

X If' 2 exp I iV2JIdvJ, (11)

where Po and p, are the incident and reflected pressures, respectively.

Since this result was obtained by expanding r, in a binomial series and

retaining the first two terms, it is subject to the limitation that

[(-2 - 2x4 + q2 - 2yq)r 2

I 2 , 1.

In geometric terms this means that the region inside of a hemispherical

volume containing the reflector is to be excluded from consideration.

The integrals in Eq. (10) can be expressed in terms of the complex

Fresnel integral F(u1 ) given by

F(ul) =,U1 exp 2 1 du, (12)

which by application of "Euler's" formula, can be expressed as

F(u1 ) = f0u cos 2_Idu - i fousin 2ff-'--du. (3

The conventional definitions are

C(u) f cs 2 du

and (14)

S(Ul) f0" si (- ±du- 1 

A similar definition can be made for a second integral F(u2) and by

manipulation of the integrals it may be shown that

F(ul, U2) F(U2 )- Fu 1 ) = exp du 1 2 (15)

= [C(u 2) - C(ud)I - i[S(u 2) - S(u1 )1

whose magnitude is

IF(u, u2)1 = [C(u 2) - C(u1 )12 + [S(u 2) - S(u,)]I / " 2. (16)

i
FINITE PLANE 21

This magnitude is the quantity required to compute the reflected pressure and is the line length from U1 to U2 in the complex C(u), S(u)

plane. The associated phase for this magnitude may be taken at the

angle that the directed line U1 U2 makes with the positive C(u) axis.

The functions S(u) and C(u) have been tabulated [11 and their

appropriate expansions have been evaluated by computer [2].

EXPERIMENTS

The pressure reflected monostatically from the faces of solid

metal blocks with parallel sides was experimentally measured. The purpose of experiments was to determine the degree of agreement that

could be achieved between an experiment using a real elastic reflecting

material in water and the theory which assumes the reflector to be rigid

and is based on the Kirchhoff approximation. The reflectors were

suspended in the field so that no reflections could be measured from

suspension wires. The blocks were placed in the far field of sphericalwave sources that subtended an aperture over which the incident wave

was essentially plane. Sufficiently long pulses were used so that their

flat portions could be assumed to be steady-state insonification of the

block, which justified comparison of the results with the steady-state

theory. Figure 2 gives examples of two reflected pulses indicating how

nearly flat the envelopes of the pulses were. The slight decrease in

amplitude toward the end of the pulse in Fig. 2b would indicate a

degree of penetration of acoustic energy into the block and subsequent

interaction at the front face to cause a reduction in amplitude. An

experimental determination of dilatational wave speed, density, and

acoustic impedance are tabulated in Table 1 for three materials that

were used.

Reflected pressures were referred to the reference pressure pe

measured at the position later occupied by the reflecting plane before

measuring the monostatic reflection by the plane. The receiver was a

probe hydrophone which occupied a small portion of the field. In all

cases, even in the near field measurements, the variation of the

incident pressure over the area of the reflecting plane was never greater

than 3%. A listing of all reflectors is given in Table 2 along with the

differences from theory for a rigid reflector at normal incidence. Figure

3 shows the reflected axial pressure amplitude for brass blocks of

different thicknesses. Since the pressure can be either higher or lower

than what one would expect from a rigid plane, the indication is that

* internal energy is a significant contributor to that degree. These

differences from rigid reflection were not possible to account for by

* considering a simple three-medium problem i.e., water, brass, water
22 CHAPTER 2

(a)

Fig. 2 -Reflected pulses from 4.5k square faces

of (a) K-8, and (b) brass, at normal incidence

Table 1 - Materials and Their Properties

c P PC

Material Dilatational Density Acoustic

sound speed (gc) impedance

____________ (in/sec) _________ (kg/sec inm2

)

K-8 6901 1.49 x 103 102.8 x 106

Brass 4445 8.41 x 103 37.38 x 106

Aluminum oxide 10696 3 .91 x 103 41.82 x 106

Table 2 - Reflection from Rectangular Areas.

(Dimensions are given in units of wavelength in water.)

Difference from

Face size in Depth in Material rigid theory at

wavelengths wavelength normal incidence

______W%

2.8 x 2.8 2.1 K-8 1.2

2.8 x 2.8 2.1 Brass 17

2.9 x 3.8 3.8 K-8 2

2.9 x 3.8 3.8 Brass 9

2.9 x 3.9 3.9 K-8 0.8

3.6 x 3.6 2.7 K-8 1.9

4.5 x 4.5 3.4 K-8 < 2a

4.5 x 4.5 3.4 A1202 10

4.5 x 4.5 3.4 Brass 12
FINITE PLANE 23

1.0 -- -- RIGID -

2 . Qo

5w

0 0.5 1.0 1.5 2O

THICKNESS (Xb)

Fig. 3 - The monostatically reflected axial pressure from

3)J . square face for different thicknesses of brass blocks

(k., wavelength in water; '\, wavelength in brass). The

reflecting plane is shaded.

and a subsequent reflection to the first water medium. The way axial

pressure can be expected to change is shown in Fig. 4 for a Kennametal

[31 K-8 block. It can be seen that the Fresnel theory continues to give

reasonable quantitative results, even in the near field when r - 15A.

Where Fresnel theory is only 6% higher than the actual experimental

d -4.5X

nl.[0

2.5 X .560eCm

/

/

2.0 /

/

Fig. 4 - The monostatically reflected axial /

presure from the 4.55. square face of a K-8 1.5

block vs I/r for I/r extrapolation of the far Pr/p.

field (---); Fresnel theory (-); experimental o

measurement (0). d 4,55.; n - I; 1.0

A - 0.580 cm.

0.5

0 0.05 0.10 0.15

I/r(cm-')

050 20 10 6.6

r (cm)
24 CHAPTER 2

value, while the l/r extrapolation is 20% too high; and at r - 6.6k the

Fresnel theory gives a value 10% above the experiment, while the lir

extrapolation is 65% too high. In Fig. 5 there is a comparison of experimental results with Fresnel theory for far-field monostatic reflection

from otherwise identical K-8, brass, and aluminum oxide blocks whose

faces are rotated about an angle 0 that includes the first side lobe. All

measurements were taken for # - 0. (See Fig. 1.) In this experiment

a wavelength of incident sound was 0.584 cm. Similar comparisons for

identical brass and K-8 blocks are shown in Figs. 6 and 7. In these

experiments, the wavelengths of sound were 0.952 and 0.692 cm,

respectively. Noise limited measurement at angles larger than those

shown. Near-field measurements using a 4.55k square-face K-8 block

at a distance 15k from the receiver are shown in Fig. 8. A similar comparison for a block 6.6A from the receiver is shown in Fig. 9.

0.35

0. 0

0.25

0.30-

0./p1O,5 0~

0.10 -

005-

0 2 4 6 t0 10 12 14 16 to

9 (')

Fit. 5 - The monoetatically rellected axial pressure vs rellecting plane rotation an&ie (0) for Fremnel theory (-); and experimental measurement

of faces of blocks of K-8 (0); bran (0); and aluminum oxide (A)

d - 4.3.; x - l; A - 0.5 cm; r- 68.5k.
FINITE PLANE 25

0V 0

a,.

7U

10

ly - - q

E

cyoc

o~~~0 N 0 6V ~ (M 

o o 0 0 0E E E

0 g -- ~~

SOS

4A g
26 CHAPTER 2

I.4

02,

0.4

o. \

04

0.2 .

0

00

0 2 4 6 6 10 12 14 I6

0(0)

Fig 3 - The monostatically reflected axial pressure vs reflecting

plane rotation angle (0) for Fresnel theory (-). and experimental measurement of the (ace of a K-8 block (-0-) d - 4 55k,.

x - 1. A - 0.580- r - 15,

It

0 

O

1 4 0

12

P .

08

0\

06

0

0.4

02

0 2 4 6 a 10 12 14 16

Fig. 9 - The monostatically reflected axial pressure vs reflecting

plane rotation angle (0) for Fresnel theory (-), and experimental measurement of the face of a K-8 block (-0-) d - 4.55k,

a 1. - 0.580- r - 66A I
FINITE PLANE 27

CONCLUSION

The comparison is favorable between the Kirchhoff theory and

experimental measurements using real materials for relatively small

finite planes whose edge length is approximately four wavelengths.

With only slight reduced accuracy, the theory even allows description of

the near field in the region between one and two edge lengths distant

from the plane. It appears that the differences found between approximate reflection theory for a reflecting plane and measurements of

reflection from planes of relatively high-impedance materials can be

similar to those imposed by the approximate theory itself.

For high acoustic impedance materials, quantitative agreement at

normal incidence if found to be within 2% for reflecting planes of real

material and for lower specific acoustic impedance material agreement is

somewhat poorer. For instance, for brass it was still found to be within

approximately 25% or 2.5 dB.
r

Appendix

RADIATED FIELD OF A RECTANGULAR PISTON*

By procedures similar to those in this chapter an acoustic approximation to the acoustic field radiated by a rectangular piston can be calculated. A rectangular-plane piston with dimensions d by din is located

in a plane infinite, rigid baffle. Assume a rectangular-coordinate system

with the origin at the center of the piston, as shown in Fig. A-i. The

radiation into half-space, in terms of the velocity potential resulting

from the simple-harmonic motion normal to the surface of the piston,

may be expressed [I] as

4 - -(A/2w)exp~ w0 ff (l/r.)exp(-ikr.da, (A-1)

where A is the uniform piston-velocity amplitude, r. is the radial distance from the contributing-piston element, k is the wavenumber 2w/k,

where A is the wavelength, and the integral is taken over the radiating

area.

Approximating r. in the phase, by the first two terms of an

expansion about r. - r,

r. -= r + (C2 - 2xJ + 7 2 - 2yiq)/r 2 ]. (A-2)

which is subject to the condition that

((C2 - 2xC + .92 - 2yq)/r 2 1

2 < I. (A-3)

This limitation is satisfied for all field points on or outside of the hemisphere that circumscribes the piston.

A similar approximation was previously given by Freedman 121 in

which a more limiting phase expansion was made about (r - z), resulting in a greatly restricted region of validity for the solution.

*Thi development first appeared in Werner 6 Neubauer, J Acoust Som Am 3B. 671

672 (1975)

28
FINITE PLANE 29

Fill A-1 Geometrical orientation for the radiation from a rectangular piston

Manipulation and transformation of Eq. (A-I), yields 131

0 - (A/2k)exP(fUwJ - (uw/A)112r - (x 2/r) - (y 2 /r)JJ

~L~ex~i~u/2df exp(-iwv/2)dv, (A)

where u (2/0/2q( _ X), V _ (2/Ar) 11 2

(C - y), and

U1/2 ~ (2~)1 ~ (d/2 W I. v1/2 -- (21Ar)1 2

1y ± (d/2)i

(A-5)

The integrals in Eq. (A-4) can be expressed in terms of the complex

Fresnel integral (41.

The scalar hield described by evaluating Eq. (A-4) is shown in

Figs. (A-AS). Comparison is made with a numerical integration of Eq. (A-I), a well a recomputed results, using the formulation of

Freedman in Figs. Ar-A4 . Esaential agreement between the present

approximation and the numerical integration is seen to exist even relatively clos to the piston face and far from the piston axis. In addition, expected agreement is found with the results after Freedman on and

ner the central axis.
30 CHAPTER 2

2.2

2.0

z= l.5X

1.8

- THIS WORK

1.6 - HUYGENS CONSTRUCTION

1.4

1.2

1.0

0.8

0.6

0.4

0.2-

2 4 6 8 10 y/X

Fig. A-2 - Plot of 12k4/AI computed in a rectangularcoordinate system over a range of y/A for x - 0.OSk and

z - 1.5, for a piston for which d - 2h and n - 2 (i.e.. a

A x 2A piston). Similar results of a numerical integration

(Huygens construction) are plotted for comparison.

This work. 0: Huygens construction.
FINITE PLANE 31

7W (a)

0~ (b

0

(c) -. 2

Fig. A-3 - Plots of 12kW/A I computed 0

in a rectangular-coordinate system over 0 W

a range of y/A for x - 0.05X and (a)

z - 2.5, (b) z - 5A, (c) z - 7.5A,(d) 0. 5

z - 10), (e) z - 20X, for a piston for -

which d - 2X and n - 2 (i.e., a X x 2x

piston). Similar results of an analysis ' 04

following Freedman and a numerical

integration (Huygens construction) are 02

plotted for comparison. - : This

work. -0-: After Freedman. 0:

Huygens construction. 0

(e)

1.4

1. ,2.5X

1 -1 THIS WORK

ID 0 AFTER FREEDMAN

* HUYGENS CONSTRUCTION

0.8

0.6

0.4-

0 0

2 4 6 8 y/X
32 CHAPTER 2

08

- THIS WORK

0.6 -0- AFTER FREEDMAN

" . HUYGENS CONSTRUCTION

it 04

02

0 2 4 6 8 10 12 14 16 18

(a)

2.4

2.2

2.0

.8

1.6

,d 1.4

- THIS WORK

1.2 - AFTER FREEDMAN

.0- HUYGENS CONSTRUCTION

1.0-

0.8-

0.6-

0.4-

0.2

0 2 4 6 8 10 12

y/X

(b)

Fig. A-4 - Plot of 12k#/A I computed in a rectangular-coordinate system over a range of

y/k for x - 0.05X and (a) z - 5? and (b) z - 20)t for a piston for which d - 4A and

n - 2, (i.e., a 4A x 2A piston). Similar results of an analysis following Freedman and a

numerical integration (Huygens construction) are plotted for comparison. -: This

work. -0-: After Freedman. 0: Huygens construction.
FINITE PLANE 33

(a) (b)

Fig. A-5 - Isometric plots of one quadrant of the field in a plane parallel to the piston

face at a distance z - 7.5x for (a) a 2x x X piston and (b) a 4 x 2A piston.

REFERENCES

1. J. W, Strutt Lord Rayleigh, Theory of Sound (MacMillian Co. Ltd.,

London, 1940), Vol. 2, p. 107.

2. A. Freedman, "Sound Field of a Rectangular Piston," J. Acoust.

Soc. Am. 32, 197-209 (1960).

3. A means of obtaining graphical results as well as a more detailed

discussion may be found in W. G. Neubauer, NRL Rept. 6286

(1965).

4. A treatment of Fresnel integrals is given in S. A. Schelkunoff,

Applied Mathematics for Engineers and Science (D. Van Nostrand

Co., Inc., Princeton, N.J., 1965), p. 385.
Chapter 3

THEORY AND DEMONSTRATION OF

CREEPING WAVES*

INTRODUCTION

Among the multitude of classical problems which A. Sommerfeld

[1] considered is the problem of propagation of radio waves around the

earth. He solved it by using a mathematical method which is now

known as the "Sommerfeld-Watson transformation." Also van der Pol

and Bremmer [21 addressed themselves to the same problem, and

Franz (3-71 advanced the theory considerably by applying it to diffraction of electromagnetic waves around conducting cylinders and spheres.

It was he who coined the term "creeping wave," (Kriechwelle), to

describe the phenomenon of circumferentially propagating waves.

The same creeping-wave theory furnishes a most useful description for analyzing acoustic scattering as well. The impetus for this was

given by the experimental observation by Barnard and McKinney [81

that a single underwater sound pulse which is incident on a scatterer

will produce a series of echo returns. A number of experiments [91

have corroborated the basic idea of these results: that the physical

mechanism for acoustic scattering consists of a superposition of continuously radiating circumferential waves. The creeping-wave theory

furnishes this physical picture in a most natural way, via the

Sommerfeld-Watson transformation, whereas in the classical "normal

mode solution" (an infinite-series expansion in terms of separable

eigenfunctions of the wave equation) this interpretation is hidden. (See

Ref. 10.) The creeping-wave solution is much more rapidly convergent

than the normal-mode solution. Acoustic scattering is particularly

suited for the experimental study of creeping waves, (better than electromagnetic scattering), because of the slow propagation speeds of

acoustical waves.

"The theoretical summary presented here first appeared in W. G. Neubauer, P. Uginc'ius,

and 11. Uberall. Z. Naturforsch. 24s, 691 (1969), written in memory of Arnold

Sommerfeld's hundredth birthday, December 5, 1968.

35
36 CHAPTER 3

The theory of creeping waves has been applied to various problems in acoustic scattering from rigid, soft, or elastic cylinders or

cylindrical shells using both continuous and pulsed waves [11-171.

Here we consider the creeping-wave analysis of acoustic scattering by

an infinite elastic cylinder immersed in a fluid. Two types of circumferential waves emerge: (1) highly attenuated "Franz-type" waves which

are slower than c - the speed of free acoustic waves in the liquid; and

(2) very slightly attenuated "Rayleigh-type" waves which are faster than

c. The former are the original creeping waves first identified by Franz,

and depend mainly on the geometry of the scatterer, whereas the latter

owe their existence to the scatterer's elastic properties.

THEORY

Figure 1 shows the geometry of the problem: a plane wave exp

i(klx - wt) is incident from the negative x-axis on an elastic cylinder

with radius a, density P2, and Lam6 constants X, p. The surrounding

medium is an infinite homogeneous fluid with density Pl, in which the

speed of propagation for the acoustic wave is

= lkw ; (1)

whereas inside the cylinder the longitudinal (compressional) and

transverse (shear) waves have the respective speeds

CI - I + 2P)/p2]1 /

2; C, - ("/p2)1/2. (2)

expressed in terms of the properties of the material. The total acoustic

pressure at the general observation point P(r, 0) is obtained in the

usual way, by subjecting the general separable solution in terms of the

cylinder eigenfunctions to the elastic boundary conditions at r - a.

After suppressing the time dependence exp (-iwt) the result is (see

Refs. 11,17):

P - Pinc + P'c, (3a)

where the incident and scattered pressures are given respectively by

Pinc i(2 - 8,o)J (kIr) cos (nO), (3b) n-0

B

Ps, - M2-(8,,0) B Hn (kir) cos (no), (3c)

where Bessel and Hankel functions are represented by J, and H,,()

respectively. The expansion coefficients in Eq. (3c) are found to be

Bn" - 2 a 22 a 23 ; D- a21 22 a 23 (4)

0 C 3 2 C0 3 0 C3 2 C 3 3
CREEPING WAVE 37

PP

Fig. I - Scattering geometry.

With the introduction of the dimensionless parameters

x, - ak, - awic,, (i - 1, 1, ) (5)

the elements of these determinants are given by [201 (primes indicate

difrerentiation with respect to the argument):

01 - (oPP)X,2J(x,

02- x 1Jn(x); (6a)

all"X,)j (O (XI),tt. 

a 12 - - 2xjJ(x) + (2n 2 - xt2)Jn(x 1 ) , (6b)

a 3- 2n[J. (x,) - xJ. (X,)];

a 21 - - x,(')'(x 1).

a 22 - x1Jn(x 1 ), (6C)

a23- nJ (x,);

a 32 - 2n[x1.J1(x) - J.(x)]

a 33 - 2xJ.(x,) + (x,2 - 2n 2)Jn(xt). (6d)

A quantity of experimental interest is the differential scattering cross

section defined by

do/dO - lim rlp./pincI2. (7)

By using Hankel's asymptotic expansion for large argument

H.lm(p) (2lrp)l2 exp i (lr/2) n + (8)
38 CHAPTER 3

for the scattered pressure in Eq. (3c), we arrive at the

do 2 B 2 do(2 - 8.) cos (no) . (9)

In principle this equation could be summed to arrive at a value for

cross section which is accurate to any desired degree. However, for

large values of ka(xi > 1) Eq. (9) is limited by its very slow convergence. For example, Sommerfeld ([1], p. 282), in a similar problem

finds that a normal-mode series like that of Eq. (9) would require more

than 1000 terms before it would start to converge.

THE CREEPING-WAVE SOLUTION

The normal-mode solution can be reformulated via the

Sommerfeld-Watson transformation. [1,21] This will not only improve the convergence of the solution but, more importantly, furnishes the

physical picture of the circumferential creeping waves. The

Sommerfeld-Watson transformation may be written in the form

~(2 -8'f. iPf idru' e_,rf(v), (10)

where C is a contour which passes through the origin , - 0, surrounding the positive real axis in clockwise sense, and excludes all poles of

the function f(C). The Neumann factor 2 - 8, requires that the principal value be taken for the integration through the origin. The next

step is to transform the contour C (which encloses all the zeros of sin

wv and no poles of f ()) into a different contour which surrounds all

poles of f(p,). To do this one must have an a priori knowledge of the

complex zeros of the determinant D- D (), because applying the

transformation (10) to Eq. (3c) we see that the only poles of the

function f(v) are those zeros, all other cylinder functions which appear

in B. and H,') being entire functions of their order in the complex iplane.

The determinant DG') is too complicated to yield any information

about its roots by analytical methods. For special cases, however, it

degenerates into simpler forms for which the asymptotic zeros are well

known. Thus, for example, for rigid (u -c) and soft (u -. 0)

cylinders it is found fl1] that they are the roots of H.,(' (xj) and of

H.") (x1 ) respectively. For large x, these roots are known from the

work of Sommerfeld [l] and Franz [41. There are infinitely many of

them and they all lie in the first quadrant of the v-plane on a line which

when extrapolated intersects the real axis at v, x1 .
CREEPING WAVE 39

The zeros of the determinant D(v), Eq. (4), can be found

numerically (see Ref. 17). A procedure that is essentially the NewtonRaphson method, generalized to the complex plane, is used and converges quite rapidly if the initial estimate is fairly good. To make sure

that no zeros are overlooked we also made use of the "Principle of the

Argument"

2wr(Z - P) - Ac Arg D(,), (11)

which states that the number of zeros Z minus the number of poles P

(with their multiplicities taken into account) inside any closed contour

C is equal to the net change (divided by 2 i') of the argument of D ()

in a complete traversal of C. With P - 0 Eq. (11) enables one to

determine unambiguously the number of zeros in any given region of

the complex plane. The results for an aluminum cylinder (X - 6.1 x

loll dyrn/cm 2, A - 2.5 x 1011 dyn/cm 2

, P2 - 2.7 gm/cm3 ) at x, -

kla - 5 are shown in Fig. 2. Two sets of zeros were found: (1) the set

labelled F which lies entirely in in the first quadrant, and which differs

numerically very little from the "rigid" Franz-type zeros - the roots of

H.'O (x ) - 0; and (2) the set labelled R which starts out with two

zeros close to the positive real axis, and continues into the second

quadrant approaching asymptotically the negative real axis. They seem

to coalesce pairwise into the negative integers. This second set of

Rayleigh-ype zeros is absent from either rigid or soft cylinders, and

their existence, therefore, must be attributed to the elastic properties of

the scatterer. We also find that there may be zeros in the third

quadrant; however, they were difficult to locate with the available

numerical program. Fortunately, as will be seen below, only firstquadrant zeros close to the real axis can contribute in the theory. All

these zeros are functions of xl. The behavior of the Franz-type zeros

vs x, is the same as that deduced from their asymptotic formulas (see

Ref. 1 or Ref. 4). The line of Rayleigh-type zeros, however seems to

be "pulled' into the first quadrant with increasing xl, so that it appears

that for x, -* co there may be an infinite number of them in the first

quadrant (see Fig. 5 in Ref. 15). For a cylindrical shell (15,17) the

analogous function D() is a six-by-six determinant. There we find the

zeros are qualitatively the same as in Fig. 2 with the exception of an

additional set in the fourth quadrant.

teFigure 2 also shows the tranformation of the original contour C in

the Sommerfeld-Watson integral (Eq. 10). This is done with the additional contours CO and C' closed up at oo in such a way that the new

closed contour C'+ C., + Co + C., - C + C.o does not include any poles, so that the integral in Eq. (10) over this contour must vanish.

clsdcnorC . 0+C,.-C+C.de o nld n

Hi
40 CHAPTER 3

,/® - -F

C -,

CS m

\

Fig. 2 - Contours for the Sommerfeld-Watson

transformation. The cirles show the Franz (F) and

Rayleigh-type (R) zeros of D() for an aluminum

cylinder with ka 5.

The peculiar shape of C' is dictated by the requirement (see the discussion after Eq. (22) that no second-quadrant poles should be included,

and by the fact, which Franz [71 has shown, that the integral over C.

to the left of the dashed line in the fourth quadrant (which is the

reflection of the F-line) would not converge. Vanishing of the integral

over the portions C.. shown in Fig. 2 can be readily established. The

transformation (10) applied on Eqs. (3) then yields for the total pressure

P " PA + Pill (12a)

Piv 0 e- ifve/2 cos (me)H,,'~ (k 1r) B (p) 02b P, " i 0 sin ry kl)D B(v,) D-": W~~OHI" (12b) --

t d e - vr12 cos(v) Pit - iPfc' sin vs

.X(klr)D(v) + H,(') (kr)B(v)

D (2c)

The "background" integral Pn of Eq. (12c) is absent for either soft or

rigid cylinders. For an elastic (aluminum) cylinder it has been shown

113,141 that it is negligible (except maybe at critical angles) by comparing results evaluated without that integral with the exact normal-mode

calculations of Faran. [221 We shall therefore neglect it. Note that the

incident pressure which is proportional to J(k 1 r) seems to have

dropped out of Eq. 12b. We shall see below that this is not really so,

but that it can be recovered by a saddle-point integration.
CREEPING WAVE 41

The integral Pi could be evaluated by the residue method, but

Franz 171 has shown that the resulting residue series would not converge in general, the reason being that Pt also contains the geometrically reflected wave. That wave can be separated out of Eq. 12b by

writing for cos(ve) the identity

cos(v0) - ell- cos P(ir - 9) - iei

Y(w- 0 sin(vrv). (13)

The last term cancels the sin (r,) in the denominator of Eq. (12b).

This results in an integral which has been shown 1131 to represent the

geometrically reflected wave. It is evaluated for large x, by the saddlepoint method. Using Hankel's asymptotic expansion, Eq. (8), for H."

in Eq. (12b), and Debye's asymptotic expansion

H/ 2 (x) z (2/rx sin x)l/2e±ilx(sina - a c

os

a)- w/41

, (14)

, X Cos X.

for the first-column elements of B(,) and D(,) we have shown [13,161

that the saddle point is located at a1 - 0/2,

iv, - x, cos(O/2). (15)

Figure 2 reveals that at the "critical angles"

O:- 2 cos-(Re Vk/XI), (16)

where Pk is any first-quadrant Rayleigh pole, the saddle point (Eq. 15) will lie directly under a pole. This would mean that the resulting

saddle-point evaluation for the geometrically reflected wave would be

very inaccurate, because we would be unable to distort the contour Co

in order to make it pass through the saddle point without coming too

close to a pole of the integrand. (We have shown that the path of

steepest descent must go through the saddle point at an angle of 31r/4

with the real axis). To overcome this difficulty we break up the contour CO of Fig. 2 into the two separate contours C, and C2, as shown in

Fig. 3, before the separation of the geometric term by Eq. (13). We

then evaluate the integral (Eq. (12b)) by the Residue Theorem in its

"unseparated form" (leaving cos aO unaltered) over the contour C1, and

in the "separated form" (rewriting cos ve according to Eq. (13)) over

the contour C2. These two residue series will yield the creeping waves.

The geometrically reflected wave can be evaluated by the saddle-point

method over the contour C, as outlined above. (Another saddle point is present which is shown schematically in Fig. 3 on the real axis to the

right of xj. Franz [5,71 has shown that this restores the contribution of

the incident wave, but since we are only interested in the scattered

wave, we shall ignore it here.)
42 CHAPTER 3

Fig. 3 - Contours for separation of the

geometrically reflected wave.

The results are:

P - P, + Pg; (]7a)

H,,(k')lr)B(Vk) [cOs(VkO )e - T

PC-- 21r[ 0 7b)

k-I sin (7rvk)Db(vk) COS k(Or - O)ek' 2 (b

Pg - - [a sin (0/2)/2r]RPvs)e iklr2asin/ 21 . (17c)

In the summation of Eq. (17b) D6(Ik) - aD/Ov evaluated at V - Vk,

the forms (1) and (2) must be used when

Re(k) < x, cos(0/2): (I) Unseparated Form; (18)

Re(vk) > x! cos(0/2): (2) Separated Form.

The function R (P) in Eq. (17c) was assumed to be constant in the

saddle-point integration and is given explicitly by

(PI/P2)x/2D(P) + ix, sin aD 2(V) R (P) (pl/P2)X, 2Dl(i) - ixl sin aD 2(1') ' (19)

where D1, D2 are the respective 2 x 2 minor determinants obtained by

expanding D(P) by its first column.

By the use of Eq. (8), Eq. (17b) can be written for r -- oo (the

asymptotic approximation for r- oo has already been made in Eq.

(17c)) as

I:
CREEPING WAVE 43

-PC Si" 11/2 e(kIr- /4 ) ki- (20)

B() Icos(,,kO)eI

k. I Sfcost(o-9) sin "k, D(~k)I k (-

At this point it is possible to demonstrate explicitly the circumferential

character of the creeping waves. For the unseparated form (I) in Eq.

(20) we rewrite the trigonometric terms

COS(VkO) , (AO+ 2mw,) s ,k) "- - , e , (21)

sinl in-O

which is uniformly convergent for lm(/ k) > 0. Inserting the time

dependence exp(-iwt) we see that the residue series (20) is made up

of terms having the form

el( ± #- ) -e-Im( )(± ) e ±Re(.,k)@-wtl (22)

which are clearly circumferential waves traveling in the ±O directions,

and are damped with an attenuation factor proportional to lm(v,).

Their phase velocities are given by

Cph - wa X1 (23) Re(G,) Re(,k) C.(

For pulses we can also define the group velocities

dx1

c1'- d Re(i,) c1. (24)

The additional summation over m in Eq. (21) represents waves which

have circumnavigated the cylinder m times. Equation (22) also shows

that no second or fourth quadrant zeros P. are allowed since these

would lead to physically unacceptable exponentially increasing waves.

This was one of the main factors for determining the contours in Fig. 2.

The differential scattering cross section, Eq. (7), is now given by

(approximately, because we have neglected the background integral Eq.

(12c))

(dar/d9) lim rip1 + p. 2, (25)
44 CHAPTER 3

which by using Eqs. (17c) and (20) can be put in the form

2 do'a /2R(&) + 4(w/kia )/2 e'2astn(/2)w/4j (26)

a [9sin(0/2] (26)

I2

k I B (I'k) cos(&,kO)e"1rI

This is much more complicated than the relatively simple expression in

Eq. (9). However, whereas the latter may need on the order of 1000

terms before starting to converge, we found that for k1a > 5 we need

sum only three or four terms (the first-quadrant Rayleigh-type poles

plus at most two Franz-type poles) in Eq. (26) to obtain four-digit accuracy. A numerical evaluation of Eq. (26) for an aluminum cylinder is

shown in Fig. 2. More extensive computations for aluminum shells can

be found in Refs. 15 and 17.

Finally we consider the paths of the creeping waves around the

cylinder, which determines the critical angles at which a creeping wave

may be launched on the cylinder and radiate off to an observer after m

circumnavigations. These critical angles have been derived intuitively

in Keller's 123) *geometrical theory of diffraction" by noting that at

these angles there exists a resonance effect: the incident wave velocity

is equal to the component in the direction of incidence of the

creeping-wave velocity. In Ref. 13 they have been established

rigorously by following the path of a delta-function pulse and correlating its travel times with causality requirements. The picture which

emerges from such an analysis is given in Fig. 4. Figure 4 a shows a

creeping wave (solid line) launched on the cylinder at the critical angle

a (measured from the shadow boundary), then proceeding to the

observer P, leaving the cylinder at the same critical angle a (measured

from N - the normal to the observation direction). The rigorous

theory (141 predicts these angles to be

at*- lim cos-16,k/jx)

-cos-1(c l/j * ) for continuous waves, (27a)

al'- lim cos-1(dV5k/dr)

3-

Scos-'(c 1/cr) for pulses, (27b)

where s is the integration variable in a Laplace transformation used to

obtain a delta-function pulse from the plane wave. These angles in
CREEPING WAVE 45

(b)

Fig. 4 -Geometry of the circumferential

waves: (a) 9 < 2a corresponding to the *unseparated form," (b) 0 > 2a corresponding

to the "separated form." The geometrically

reflected wave is indicated by the dashed

line.

general are complex (except for the rigid or soft cylinder in which case

they become ak - 0, showing that the Franz waves are launched at the

shadow boundary). The approximation of taking their real parts (the

right-hand sides of Eq. (27)) gives meaningful physical angles if lmG'k)

is small, which is the case for all Rayleigh-type zeros. For 9 - 2at the

distance which the upper creeping wave in Fig. 4a has to travel on the

cylinder shrinks to zero. It therefore has to make a full revolution

around the cylinder before proceeding to P, as shown in Fig. 4b. (This

means that m has increased by one in Eq. (21)). The condition 0 - 2

is precisely the same as that in Eq. (18) for the change-over from the

unseparated to the separated forms for the residue contributions. At

this point the saddle point (15) is directly under one of the Rayleightype poles. The separated form of the solution (which is to be used for

9 > 2a) predicts exactly, from our causality arguments 1141, the path

shown in Fig. 4b in agreement with physical intuition.
46 CHAPTER 3

SCHLIEREN OBSERVATION OF FRANZ AND

RAYLEIGH RADIATION

Waves that contribute to the total acoustic field resulting from a

wave incident on the curved circular cylindrical surface and normal to

the cylinder axis can be isolated and identified by hydrophone experiments and also by schlieren visualization of the resultant radiation.

The total field is a combination of many effects resulting from waves

that are diffracted in the outer medium, travel circumferentially inside

the cylinder or on the interface, and travel through the body of the

cylinder. Isolation of specific waves and identification of exact causes

for them has been largely achieved. A schlieren visualization of the

entire radiated field resulting from an incident acoustic pulse is shown

in Fig. 5. In this dark-field schlieren photograph the cylinder is seen in

cross section as a dark disk and the acoustic pulse is seen as a white

area. The schlieren system used to produce the photographs shown

here is described in Chapter 22. In Fig. 5, a pulse of 7 MHz waves

slightly longer than the cylinder diameter causes the complicated field

that cannot be readily sorted out. Using a shorter pulse and insonifving

only one side (quadrant) of the cylinder circumference causes considerable simplification of the radiated field and permits some isolation of

the causative effects. Those are the conditions for Fig. 6 which shows

photographs for a short acoustic incident pulse at two different times

and the reradiated field resulting from that incident pulse.

Fig. 5 - The total field radiated from a

cylinder when the entire cylinder is

insonified with a pulse longer than the

cylinder diameter.
CREEPING WAVE 47

(a) (b)

Fig. 6 - Schlieren photographs of a short (3) pulse of 6.6 MHz insonifying one

quadrant of a solid aluminum cylinder in water (ka - 353). (a) The incident

pulse shown at two different times in a multiple exposure before it strikes the

cylinder. (b) The total radiated field of waves resulting from the incident pulse in

(a).

Selectively insonifying only a small range of incidence angles can

further simplify the radiated field. If the pulse falls incident only near

the tangent to the cylinder the visualization of the Franz wave results in

the shadow region. For cylinder sizes small enough, this diffracted

wave can interfere with the directly reflected backscattering from the

central region of the cylinder. This interference causes the characteristic oscillations in the differential scattering cross section or form function vs x, curves for rigid and elastic cylinders (see Chapter 4, Fig. 2).

This behavior of the creeping or Franz wave will be discussed further in

Chapter 4. In the schlieren photograph in Fig. 7, a pulse was incident

only near the tangent. The creeping or Franz wave is seen to propagate

into the shadow region on the opposite side from the source and

attenuates rapidly from the intersection of the tangent to the point

where it intersects the cylinder surface in the shadow. The striations in

the wavefront to the left of the cylinder tangent have been explained by

Marston and Kingsbury. [241 In the series of schlieren photographs in

Fig. 8 the pulse was incident on the range of angles that included the

Rayleigh angle and the tangent to the cylinder cross section. The

resulting Franz wave radiation and the Rayleigh wave radiation are

labeled in Fig. 8. The first dark gap in the wavefront, indicated by an

arrow, is a result of the known dark strip caused by opposite phase of

t
48 CHAPTER 3

Fig. 7 - Schlieren photograph of the wavefront

resulting from the Franz wave for an aluminum

cylinder (diam - 10.16 cm) in water for ka of

1008.

Fig. 8 - Schlieren photographs of the pulsed creeping

wave propagated into the geometric shadow of an

aluminum cylinder at a frequency of 5 MHz (ka =

324) showing two successive positions of the wave.

The source is at the top radiating downward with its

beam incident at the Rayleigh angle and on the

tangent of the cross section at the far left of the

cylinder.

the specular and Rayleigh radiation which will be discussed in detail for

a flat surface in Chapter 18. One advantage of schlieren visualization is

that it gives one an ability to examine the entire radiated field. However, shortcomings exist for this method of examining acoustic fields.

It is largely qualitative rather than quantitative in amplitude, since

schlieren is not a linear process over all of its dynamic range.
CREEPING WAVE 49

Accurate quantitative information about the radiated waves can be

obtained from hydrophone experiments. A diagram of an experiment

that permitted determination of creeping wave attenuation and circumferential speed is shown in Fig. 9a. The pulse in Fig. 9b is the one that

was used, having a 1 MHz center frequency. The pulse was measured

at six different positions for which the circumferential path on the

cylinder, that the wave travels from source to receiver, was successively

increased. A multiple exposure of the oscilloscope traces received at

positions 2 through 6 is given in Fig. 10. The dashed line connects the

peaks of the signals indicating the decay of the creeping wave.

3

2

SOURCE

(a)

(b)

Fig. 9 - (a) Diagram of an experiment used to measure circumferential wave properties. (b) The pulse received at position (1)

in (a) in the absence of the cylinder.

\2 Fig. 10 -A multiple exposure of successive3

receptions of the "Franz-type" or "creeping" . *.4

waves at receiver positions 2-6 in Fig. 9(a)., .. -5 4

showing the attenuation that occurs in the

path along the cylinder.
50 CHAPTER 3

Accurate measurement of the time differences associated with the

difference in path between the adjacent receiver positions allows a

determination of circumferential wave speed. If peaks of the individual

cycles of the pulse do not change their position within the pulse for the

different path lengths on the cylinder, the speed that is measured

represents both the phase and the group velocity value to within the

accuracy of this experiment. A double exposure in Fig. I I of the pulse

at position I and at position 6 shows that to be the case for this experiment.

Fig. I I - Double exposure of a received

acoustic pulse in the absence of a cylinder

and the same pulse with part of its transmis- V sion path along the circumference of the

cylinder.

The creeping wave speed determined by this means along with

theoretically computed values and other experimental values are plotted

in Fig. 12. A similar plot for creeping wave attenuation is shown in

Fig. 13.

1.00 1

0.95

0.90

0 EXPERIMENTAL (ALUMINUM)

ALUMINUM (UGINIIUS) "' PHASE

0.85- THEORY 0 RIGID (FRANE) VELOCITY

0 RIGID (DOOLITTLt ET.AL.)J VLI

v ALUMINUM (UGIN IUS)- GROUP VELOCITY

0 50 100 150 200 250 300 350

ke

Fig. 12 - The comparison of experimental measurements and various

theoretical calculations of the Franz-wave velocity is a function of size

parameter (ka).
CREEPING WAVE 5

5

2

14-

W R -o A LUM INUM ~ : i ;: 300 350US

REE EE IETL QAUINM(EBUR

1. iA. Sommerheldomparialn ofexeriental eutions nd hics cade

Press, New York-London 1964. See particularly the appendix

4 after Chapter 6.

2. B. van der Pol and H. Bremmer, Phil. Mag. 24, 141, 825 (1937).

3. W. Franz and K. Deppermann, Ann. Phys. Leipzig 10, 361

(1952).

4. W. Franz, Z. Naturforsch. 9a, 705 (1954).

5. W. Franz and P. Beckmann, IRE Trans. Antennas and Propagation AP-4, 203 (1956).

6. P. Beckmann and W. Franz, Z. Naturforsch. 12a, 257 (1957).

7. W. Franz, "Theorie der Beugung Elektromagnetischer Wellen," I Springer-Verlag, Berlin 1957.

8. G. R. Barnard and C. M. McKinney, J. Acoust. Soc. Am. 33, 226

(1961).
52 CHAPTER 3

9. See the bibliographies in Refs. 1I and 13.

10. A. J. Rudgers, J. Acoust. Soc. Am. 45, 900-910 (1969).

11. R. D. Doolittle and H. Uberall, J. Acoust. Soc. Am. 39, 272

(1966).

12. H. Uberall, R. D. Doolittle, and J. V. McNicholas, J. Acoust.

Soc. Am. 39, 564 (1966).

13. R. D. Doolittle, J. V. McNicholas, H. Uberall, and P. Ugincius, J.

Acoust. Soc. Am. 42, 522 (L) (1967).

14. R. D. Doolittle, H. Uberall, and P. Ugincius, J. Acoust. Soc. Am.

43, 1 (1968).

15. P. Ugincius and H. Uberall, J. Acoust. Soc. Am. 43, 1025 (1968).

16. J. V. McNicholas, H. Uberall, and K. Choate, J. Acoust Soc. Am.

44, 752 (1968).

17. P. Uginvius, Ph.D. thesis, Phys. Dept., The Catholic Univ. of

America, 1968; and U. S. Naval Weapons Lab. Tech. Report No.

2128, Feb. 1968, Dahlgren, VA 22448.

18. W. G. Neubauer, J. Acoust. Soc. Am. 44, 298 (L) (1968).

19. W. G. Neubauer, Ph.D thesis, Physics Dept., The Catholic Univ.

of America, 1968; Experimental Examination, By Means of

Pulses, of Circumferential Waves on Aluminum Cylinders, Naval

Research Laboratory Report 6791, November 15, 1968, and I.

Acoust. Soc. Am. 45, 1134-1144 (1969).

20. These differ from the corresponding elements in Refs. 10 and 13

by the overall factor of I, thereby making them dimensionless.

See also Refs. 14 or 16 where the correctly nondimensionalized

elements are given for the more general 6X6 determinants for a

cylindrical shell.

j. 21. G. N. Watson, Proc. Roy. Soc. London A59, 83, 546 (1919).

22. J. J. Faran, J. Acoust. Soc. Am. 23, 405 (1951).
CREEPING WAVE 53

23. B. R. Levy and J. B. Keller, Commun. Pure Appi. Math. 12, 159

(1959).

24. P. L. Marston and D. L. Kingsbury, J. Acoust. Soc. Am. 70,

1488-1495 (1981).
Chapter 4

CIRCUMFERENTIAL WAVE INTERPRETATIONS

IN CYLINDER REFLECTIONS'

I. INTRODUCTION

Theoretical solutions to the problem of the scattering of sound by

rigid, immovable cylinders, nonrigid cylinders in a fluid medium, and

small cylindrical obstacles in a solid medium were formulated by Rayleigh [11. The solutions he presented described goemetries in which

the diameters of the cylinders were small compared to the acoustic

wavelength in the surrounding medium, although he outlined a more

general method for finding the solution for larger liameter cylinders in

terms of cylindrical harmonics. This method, called the harmonic

series or the Rayleigh series method of solving acoustic scattering problems is in theory applicable to targets whose shape conforms to any of

the eleven separable coordinate systems. In practice it has been extensively applied only to scattering from spherical and infinite cylindrical

geometries, since cylindrical and spherical harmonics are readily available. Solutions to the problems of the scattering from rigid cylinders

and rigid spheres which have radii up to the order of a wavelength (ka - 6) were given by Morse, [21; here ka - 2ira/k, a is the radius of

the scatterer, and A is the acoustic wavelength in water. Exact solutions to the scattering of a plane sound wave by homogeneous, isotropic cylinders and spheres capable of supporting both shear and compressional waves (elastic scatterers) were first given by Faran, 131 who

obtained expressions in terms of normal mode series. Faran presented

comparisons of computed bistatic patterns and experimental measurements at icc - 5. Extensions of the normal mode calculations to

higher icc [4,51 and experimental measurements to determine the

degree to which the normal mode theory and experiment agreed, over a

broad ka range, were first made on solid elastic spheres and spherical

shells [6-91. Hickling [41 was the first to make extensive use of a digital computer to evaluate the normal mode series expressions, although

Mlesa resuts fit appeared as NRL Report 3216 by Louis Rt. Dragonette (1973).

55
56 CHAPTrER 4

his computations were hampered by the slow convergence of the harmonic series solution, which led to computation difficulties with the

computers available at that time. Hickling gave computed curves that

describe the steady-state- backscattered pressure vs ka, which results

when the target is a solid-elastic sphere in water. The computations in

his work ranged generally from 0 < ka < 30, and he extended the formulation to include the scattering of incident spherical as well as

incident plane waves. He also included both near-field and far-field formulations. The results presented by Hickling are given in terms of the

form function, f,.. This dimensionless quantity is obtained by normalizing the reflected pressure with respect to the radius (a) of the target

and the range (r) of the field point from the center of the target. Hicklig also computed acoustic reflections from elastic spherical shells 151.

Empirical results on solid metal spheres in water were given by Hampton and McKinney 16), who demonstrated that the reflection from

metal spheres immersed in water could not be described by purely

geometric theory, and by Diercks 171, who demonstrated qualitative

agreement between the computations of Hickling and measurements

made in a lake. Precise quantitative comparisons between normal mode

theory and experiment were first carried out by Neubauer et al, [81 who

performed a series of precise steady-state measurements on solid metal

spheres in a controlled acoustic tank facility. These measurements

demonstrated quantitative agreement, between computations, based on

the normal mode series, and experiment, to within the known accuracy

of the shear wave speeds of the materials used in the sphere fabrications. This work (81 covered the ka range 0 < Ira K, 30. Dragonette et

al. (91 demonstrated empirically that quantitative steady-state results

could be obtained from measurements made with short broadbandincident-acoustic pulses. Comparisons between normal-mode theory and

experiment for elastic cylinders in water are more recent, [101 but again

demonstrated excellent agreement between the theory based on the

infinite elastic cylinder and near-real-time experiments performed with

finite length cylinders in a laboratory tank. The preceding theoretical

and empirical papers (I1-101 established that the normal-mode series

formulation of the acoustic reflection from elastic-metal targets quantitatively describes measured results up to at least ka - 30, without the

necessity of material absorption being included in the theory.

Empirical observations by Barnard and McKinney [111 demonstrated periodic, multiple echo returns when solid and hollow brass

cylinders (Ira -40) were illuminated by short acoustic pulses. Subsequent empirical work and analysis by Diercks et al. 1121, Horton et al.

1131, and others [14,15) proposed the existence of two types of circumferential waves that were compared to flexural and longitudinal modes
CIRCUMFERENTIAL WAVES 57

on infinite plates. A similarity was recognized between the circumferential behavior of the waves they observed and the waves discussed

by Franz [161 in his work on the diffraction of electromagnetic waves

by conducting cylinders and spheres. The original analogy between the

acoustic waves observed on cylinders and the purely geometrically

diffracted circumferential waves considered by Franz broke down,

because the speeds of the observed acoustic circumferential waves were

from 33% to 300% higher than the speed of sound in the medium surrounding the targets; whereas, analogy with the "creeping waves" of

Franz would have predicted a speed slower than that in the surrounding

medium.

Uberall and collaborators at Catholic University also noted the

similarity between the circumferential behavior of empirically observed

acoustic waves and waves studied in electromagnetic theory. A modified

Sommerfeld-Watson [17] transformation had been used in the study of

the propagation of radio waves around the earth [181 and in the studies

of the diffraction of electromagnetic waves by cylinders 1161. The

Sommerfeld-Watson transformation offered certain advantages, namely, the opportunity to isolate the individual mechanisms responsible for the

empirically observed circumferential waves and rapid convergence of

the solution. This latter advantage was particularly significant at the

time since the normal mode series was considered to be practical only

at low ka, because of its slow convergence and the expense of computation. (Advances in computer technology make present high ka Ray- leigh series computations both possible and economical [191.) Uberall

et al. applied the Sommerfeld-Watson transformation to cylinders with

rigid and soft boundary conditions and predicted the existence of true

Franz type, or purely geometrically diffracted, circumferential waves

1201. Application of the Sommerfeld-Watson method to solid elastic

cylinders (211 revealed two groups of poles corresponding to two types

of circumferential waves discussed in Chapter 3. The Franz-type or

geometrically diffracted waves again appeared, and, in addition, poles related to elastic circumferential waves, called R or Rayleigh-type

waves, whose speed and properties depend primarily on the elastic constants of the target, were found. Grace and Goodman [221 also

presented theoretical evidence for the existence of R-type waves.

Experimental detection of the acoustic Franz-type or purely

geometrically diffracted wave was accomplished by Neubauer [231 and

by Harbold and Steinberg [241. The first experiments designed to

demonstrate elastic R-type circumferential waves, that is, those related

to R-type poles, were performed by Bunney et al. 1251 and by Neubauer

1261. Both of these researchers [25,261 used short incident pulses and
58 CHAPTER 4

narrow beam sources to observe the scattering from solid aluminum

cylinders. Their results demonstrated the existence of a train of

periodic echoes with a circumferential speed close to the shear wave

speed in aluminum. Neubauer's [261 work included schlieren visualization of wavefronts resulting from the circumferentially traveling waves.

These experiments gave mainly high ka results (/ea values between 50

and 500). Originally the mechanism responsible for the periodic pulse

trains observed [25,261 was, in fact, considered to be multiple circumnavigations of the cylinder by the Rayleigh wave. Later work by Neubauer and Dragonette [271 showed that multiple internal reflections of

shear waves could produce the observed effect, and this multiple reflection analysis was supported by the theoretical work of Brill and

Uberall [281, who demonstrated the circumferential behavior of the

radiation from multiply internally reflected waves. Theoretical [291 and

measured attenuation [301 of the Rayleigh wave on submerged flat surfaces also gave attenuations too large to support the conclusion that

Rayleigh waves were the source of the multiple returns observed at

high ka in Refs. 25 and 26. Dragonette [311 first predicted and

observed the ka range at which a Rayleigh circumferential wave can be

significant.

The normal mode solutions give a straightforward method of

obtaining the scattered acoustic pressure vs frequency, limited only by

the expense involved in summing a slowly convergent series. Experimental results have been obtained which agree with the computation to

a high degree of accuracy. The major disadvantage of this approach is

that individual physical phenomena, such as surface waves, which make

up the solution are not immediately obvious. The Sommerfeld-Watson

transformation of the normal mode series has the advantage that it isolates individual circumferential waves and the disadvantage that the

poles must be found and interpreted and their significance judged.

GENERAL THEORETICAL FOUNDATION

The Rayleigh series expression for the scattered acoustic pressure,

p,(9), which results when a plane wave, poe k

l, illuminates an infinite

F elastic cylinder, in the geometry described by Fig. 1, is given in many

publications (3,10,21,32,331. The following form is found in Refs. 32

and 33:

pe)-p 3 EM J, (Z) L.- Z J.(Z)IH,(r)on 1)

I H.(Z) L.-ZH(Z)

The time dependence e'"' is suppressed. In Eq. 1, e. is the Neumann

factor (e. - 2, n - 0; e. I. n > 0), J, is a Bessel function, H, is a
CIRCUMFERENTIAL WAVES 59

Z

Y

+P

Fig. I - The geometry used in the description of the scattering

of a plane wave by an infinitely long cylinder.

Hankei function of the first kind, Z /ka and the L, are the quotients

of two 2-by-2 matrices: ala3I

_p D,111Z] la21 9a) a 3 1--- 

a31 a33

where the matrix elements a are given in Ref. 32. In the far field

where r >> a, H.(kr) may be written in its asymptotic form

H. (kr) ,- 1'_-' r 2 e~rin'-wr2-414 (2)

and, defining

J. (Z) L. - Z J.,(Z) G.M(3

H. (Z) L. Z H.,(Z) -G() 3

the far-field preasure scattered by an infinite cylinder illuminated by a

plane incident wav e fs written n 

|21'/2 " e *i rkie4 e.G.(Z) cos(nO). (4)

For backscattering, 0 - er and

e,, 1--L '

/2 .1 ,,-4 a.(z). (5)

H vkr ~ A nf2i/ 2
60 CHAPTER 4

A quantity called the far field form function, f., is defined to give a

nondimensional representation of the scattered pressure. In keeping

with the definition used extensively in the literature [4,5,7-10,32,331

f(0) - r o (6) IaJ P

This definition is chosen since it results in - IA®t 1 for the case of a

purely rigid cylinder in the high frequency limit. From Eqs. 4 and 5

the expressions for f0 *(O) and f*(7r) for an elastic cylinder are given

by

(0) -2 / E. G.(Z) cos (nO) (7a)

(ir Z) 12

and

f() -2 - E 1 G(-l) G.(Z). (7b)

Using Eq. 7b, the individual normal modes or partial waves which make

up the backscattered form function are defined as

-2

f( (iO) Z) 2 E,(-l)" G,(Z) (7c)

where

f,,(n') f0(.r) (7d) n-0

Computed plots of f** vs ka, obtained from Eq. 7b are called reflection

function plots, and such curves give a dimensionless representation of

the scattered steady state pressure vs frequency. This representation

can describe the scattering at any combination of radius and frequency

within the ka limits of the calculation. Equations 7a and 7b give steady

state values of f., so that a continuous wave or very long pulse experiment can be used to obtain a direct comparison between experimental

and calculated results [8]. Such an experimental method is tedious and

excessively time consuming, as each experiment at each single frequency gives one point on the reflection function curve. To overcome

this practical deficiency, methods to obtain the steady state quantity,

f., from short broadband incident pulses were developed [9,101.

If the incident sound wave in the geometry described by Fig. I is

not steady state but a pulse, pi(T), with a Fourier transform gi(ka)

given by

g,(ka) - f p1&) eika r dr (8)
-7 - _

CIRCUMFERENTIAL WAVES 61

then from Eqs. 6 and 8, the backscattered pressure has a Fourier

transform g,(ka, v) given by

g,(ka, a fr) (ka, r) g,(ka) (9)

and If. (ka, r) I can be obtained from

I L 

!/

If. (ka, 70)1 - r1JI/2 1 g(ka, 7)I (10) a Igi(ka)l (0

The quantity r is a dimensionless time parameter

ct-r 011)

a

which is normalized to be zero when the incident pulse is coincident

with the position of the center of the cylinder. The wave speed in the

ambient fluid is c and t is the time. Equation 10 is the basis by which

a steady state quantity If. (ka, 7) can be obtained over a broad frequency range by a single short pulse experiment. The incident and

reflected pulses are digitized, their transforms computed, and the division indicated in Eq. 10 carried out [9]. With present minicomputer

technology this entire procedure can be accomplished in a near real

time framework [101. The ka range over which f., is obtained depends

of course on the bandwidth of the incident pulse. Theoretical computations of the scattered echoes which result when a short incident pulse

with a known spectrum, Ig (ka)I, is used to insonify a target with a

known f..(O, ka) can be obtained by using Eq. 9. This computational

procedure allows the isolation of the individual mechanisms which contribute to the steady state scattered pressure. These pulse calculations

are of significant value for many reasons, the most important of which

is, that the theoretically formulated incident pulses used, can be made

shorter than any which can be reasonably achieved in the laboratory.

This allows isolation of closely spaced echoes which cannot be accomplished at a reasonable cost in the laboratory. In addition theoretical

computations can simulate experimental measurements over a large

number of frequencies, target materials, target sizes, and target shell

thickness that would be impossible to duplicate economically in a

laboratory. In the theoretical procedure an incident pulse pj (r) with a

known spectrum, Ig(ka)l, is used to insonify a target whose form

function can be computed. Computation of the form function and the

procedure indicated in Eq. 9 are accomplished by the computer, and the

scattered echo, p, (r), is described by

P,(7) - 1/2w f g,(ka) e- , "r dka. (12)

I mmmmmN •m
62 CHAPTER 4

Solutions to Eq. 12 are obtained by using fast Fourier transform techniques in the computer.

NORMAL MODES OF VIBRATION OF A CYLINDER

AND CREEPING WAVES

The acoustic reflection from a rigid cylinder has been well understood since the prediction [201 and empirical observation [23,24] of the

Franz wave, and only a few ideas relating to the direct use of the form

function curves to derive Franz wave properties can be added. In the

case of a rigid cylinder, G,(Z) as defined in Eq. 3 reduces to

G R) (z) = J" (Z) (13)

H,' (Z)

and the form function for a rigid cylinder is given by

f(R) (Z) _ -2

(irka)112 e (- G R) (Z). (14)

A plot of f**(ir) vs ka computed from Eq. 14 is given in Fig. 2. Since

by definition the boundary conditions imposed to obtain the curve in

Fig. 2 preclude penetration into the cylinder, the backscattered

reflection function curve can include contributions only from specular

reflection and diffraction.

1.5111

1.2

0.9- .

0.6

0.3

0.0 ,,t

0 2 4 6 a 10

ka

Fig. 2 - The form function for a rigid cylinder.

It
CIRCUMFERENTIAL WAVES 63

In the creeping wave solution Eq. 14 is transformed from an

infinite series of n terms into a series of "creeping surface waves" by the

Sommerfeld-Watson transformation [18,201. The creeping waves arise

as the residue of poles in the complex v plane determined from the

equation

Hi (Z) = 0 (15)

with solutions

v,(w)= Z+(Z/6)1 3

e 1 3 q,- 16 Z e1 /3 1 -L +-12 (16) l + 180 ' 180](6

where the q, are the zeroes of the first derivates of the Airy function as

defined by Franz [181. The index I = 1, 2, 3..., and I increases in the

direction of increasing real and imaginary parts of P1. The attenuation,

a f, of the Ph Franz wave in Np/rad is given by

af(Np/rad) = Im v1 (17a)

and the phase speed, c,, is given by (201

C;/c - Z (17b) Re P/

where c is the speed of sound in water. Calculations based on Eqs. 16

and 17a demonstrate that only the I = 1, or first Franz wave is of

significant magnitude, and Figs. 3 and 4 give computations of the

attenuation and phase velocity of this first Franz wave as a function of

ka as computed from Eqs. 17b and 17a.

1.0

0.8

0.6

0.4

0.2

0.0 1'

0 1 2 3 4 5 6 7 8

ka

Fig. 3 - The Franz wave phase velocity vs ka for a rigid cylinder.
64 CHAPTER 4

1.4

1.2

, 1.0

0.8

0.6

0.41

0 1 2 3 4 5 6 7 8

ka

Fig. 4 - The Franz wave attenuation vs ka for a rigid cylinder.

In the case of the rigid cylinder the connection between the steady

state form function f!, given in Fig. 2 and the Franz wave with properties described by Figs. 3 and 4 is not difficult to determine. Sound does

not penetrate a rigid cylinder, thus the form function must be made up

entirely of specular reflection plus a pure geometrically diffracted contribution. The backscattered return must then be as described by Fig. 5.

A specular reflection begins at point A of Fig. 5, and two Franz waves

begin at points B and C and take the paths shown. This well understood result can, however, be taken further. The reflection function

given in Fig. 2 is a steady state function, ad the knowledge that this

reflection function results from sound waves taking the circumferential

paths shown in Fig. 5 leads to the following analysis. Because the two

diffracted waves BC and CB take the same path and travel at the same

speed, they are always in phase with each other in the backscattered

direction. Computations based on Eqs. 16 and 17a demonstrate that for

ka > I only these first Franz returns need be consijered; that is, the

contribution from succeeding circumnavigations of the cylinder are too

small in amplitude to be significant. The difference in the time of

arrival, at the field point P, between the specular and diffracted waves is

t= 2._a + -a

c + (18)

The difference in path lengths traveled by the backscattered specular

and Franz wave contributions at the field point P is expressed as

ld = 2a + F. . (19)

• . m m -- .mum==m=C
CIRCUMFERENTIAL WAVES 65

Y

Fig. 5 - Source of backscattered echoes from a rigid cylinder.

If it is assumed that the peaks in the reflection function curve (Fig. 2)

occur when the specular and Franz contributions add in phase at P,

then peaks occur when

Ad = a(2 + 7rc/c) -P nX = n 2r (20) k

which leads to

C/Cp (ka) 2n 2 (21)

The notation (ka)peak is used to indicate the (ka) values at which a

peak in If I as seen in Fig. 2 occurs. A similar expression can be

derived by assuming that the nulls in the reflection function curve of (2n - 1)

Fig. 2 occur when Ad- 2 1X. This expression is

c/cF= (2n - 1) 2 (22)

A Franz wave speed can than be calculated directly from the reflection

function curve by determination of (ka),eak and (ka) ,11. The normalized phase velocity c[/c (which is the reciprocal of the left-hand side of

Eqs. 21 and 22) is computed from these equations and compared to the

direct "creeping wave" theory computation, from Eq. 17b, in Fig. 6.

Agreement between the two methods is excellent, demonstrating the

possibility of obtaining Franz wave velocity directly from the normal

mode form function curves. Thus Franz wave velocities can be I,
66 10CHAPTER 4

1..

0.6-

'-A.

0.0

0 1 2 3 4 5 6 7 8

ka

Fig. 6 - A comparison between the computed Franz wave wave speed for a rigid

cylinder (-) and values estimated from the form function curve (x.

obtained for bodies for which no creeping wave analysis exists. Examples are given in Figs. 7 and 8, where computed form functions and

derived Franz wave velocity for a rigid sphere and an aluminum oxide

cylinder are given. The rigid sphere shows a much more rapid rise in

Franz wave velocity with increasing frequency, than is observed for the

rigid cylinder. The velocity for the aluminum oxide cylinder shows

only minor deviations from the rigid cylinder curve.

The attenuation of the Franz wave as a function of ka can also be

investigated directly from the normal mode calculation. Here the

reduction in amplitude of the successive oscillations is assumed due to

the increase in attenuation of the Franz wave as a function of ka. The

reduction in magnitude of the oscillations in Fig. 2 with increasing ka

should then give a measure of the Franz wave attenuation vs ka. A

comparison of attenuation values obtained from creeping wave theory

for a rigid cylinder and values obtained from the form function curve is

given in Fig. 9a. In Fig. 9b the attenuation is given for an aluminum

oxide cylinder for which no direct creeping wave data are available. The

rigid cylinder curve is included in Fig. 9b, for comparison.

Empirical observations of Franz waves on a rigid cylinder were

obtained in an air acoustic range. A solid 3.18-cm-diameter cylinder

was used as the target, and the frequency of the incident pulse was 34.4

kHz. In air the impedance mismatch between the aluminum and surrounding air medium is so great that rigid boundary condition assumptions are successfully achieved.

X
CIRCUMFERENTIAL WAVES 67

1.5 I

1.2

0.9 " "•" ;

0.6

0.3 -

0 2 4 8 8 10

ka

Fig. 7(a) - The form function vs ka for a rigid sphere.

1.0

0.1-

0.6

0.4

0.2-

0 1 2 3 4 5 6 7 8

ka

Fig. 7(b) -Franz wave velocity estimated from P7al for a rigid sphere

()and Franz wave velocity for a rigid cylinder -
68 CHAPTER 4

1.

0.9J

0.80 -

-i

0.4

0 2 4 a 10

ka

Fig. 8(a) - The form function for an aluminum oxide cylinder.

1.0 , , , , ,

0.8 -

I0.4

0.2

0.01 1 L 'L '

ka

Fig. 8(b) - Franz wave velocity estimated from the form function( )

~~~and Franz wave velocity for a rigid cylinder ( )
CIRCUMFERENTIAL WAVES 69

1.2-

1.0.

CL1.0.

0.8-

0.6-

0.4 ________

0 1 2 3 4 5 6 7 a 9

ka

Fig. 9(a) -Franz wave attenuation for a rigid cylinder obtained directly -

and estimates obtained from the form function curve .

14-

12-

0

10.

0

08a

06-

0.41 L '

1 2 3 4 5 6 7 8

ha

Fig. 9(b) - Franz attenuation estimates for an aluminum oxide cylin- i (o) ~der compared to the rigid cylinder ()
70 CHAPTER 4

Experimental observations of Franz waves on a rigid cylinder are

given in Fig. 10. The measurements are made at aspect angles of 450

and 750 so that the changes, both in relative amplitude and time separation, between the specular reflection and the Franz wave can be clearly

observed. The incident pulse is seen in Fig. 10(a) and the scattered

echoes at 45* and 75" are seen in Figs. 10(b) and 10(c). The time

separations between the specular and Franz waves are measured from

the large positive going peak in each echo. The measured time differences are 81 pIs at 450 and 125 uss at 75°. The ratio of the Franz wave

to specular amplitude is PFlPpc - 0.56 at 450 and - 0.13 at 75 °.

Plots of theoretical computations, based on Eqs. 9 and 12, for the

echoes from a rigid cylinder are shown in Fig. 11. The theoretical computations agree closely with experiments in the relative magnitudes and

positions of the Franz waves. The values obtained from Fig. 11 are a

time separation of 78 jus at 45° and 125 jus at 75°, and a ratio PF/P..

of 0.57 at 450 and 0.14 at 75°. The center dimensionless frequency of

the pulse is koa - 10. Echo computations agree with experimental

measurements, and can be used to isolate mechanisms or to supplement measurements when f. is known or can be computed.

---Oil 16 me 10-- 1*- 16 "s- 243 Ve

(a) The incident pulse, (b) The specular and Franz (c) The specular and Franz

echoes at 0 = 45% echoc% at 6 = 75*,

Fig. 10 - The experimental observation of the klanz wave

radiation from a rigid cylinder.

For elastic cylinders, resonances in the normal mode solution can

be identified with specific R-type circumferential waves predicted by

creeping wave theory. The ka range over which the Rayleigh wave

makes a significant backscattering contribution has been the subject of

much conjecture [25-28,341 in the literature. There is a limited, low

ka, region of importance for the Rayleigh circumferential wave in contrast to previous hypotheses [25,26,41. The first experimental observation of backscattered circumferential radiation from the true Rayleigh
CIRCUMFERENTIAL WAVES 71

rr

(a) The inczident puls. (b) The specutar an Franz (c) The specular and Franz

echoes at e a 450. echoes at 0 - 75.

Fig. I I - Computation of the scattering of a two cycle

incident pulse by a rigid cylinder.

wave was described by Dragonette 1311 and was accomplished in the

low ka region where it is predicted. Computation and analysis of the

effects of the normal mode resonances on the backscattered L(.1 demonstrate further that the form function is made up of a rigid background

on which narrow resonances are superimposed.

The backscattered form function for an elastic aluminum cylinder

in water is calculated from Eq. 7b and given in Fig. 12 over the range

from 0.2 4 ka 4 20. The curve is calculated in ka steps of Aka -

0.01. A comparison between this theoretical computation and an experiment using the short pulse experimental technique is given in Fig. 13.

The theoretical curve is computed in intervals of Aka - 0.05 which is

compatible with the Aka resolution that can be achieved experimentally. Agreement between theory and experiment is within 2%. The

form function curve seen in Fig. 12, shows that over the range 0.2 4

ka < 4.7 the aluminum reflection curve is very similar to that of the

rigid cylinder (Fig. 2). This region of similarity is followed by marked

irregular oscillations and these oscillations continue as (see Chapter 13)

ka - o, if no absorption is included in the theory. Damping of the

oscillations as ka increases will occur when absorption becomes

significant [351, but for metals such as aluminum, experimental results

indicate [8-101 that absorption need not be included over the ranges of

ka that will be considered here. I
72 CHAPTER 4

IA.3

1.1

1.2

S0.9

0.6-

0.31.3

[r 2.1) (1.2) 13.1) (4.1) (3.2) (4.2 (3,3)

0.0 2.0 4.0 6.o .0-10.0 12.0 14.0 0 18.40 20.0

ka

Fig. 12 - The form function for an aluminum cylinder in water.

120 40 160 t80 700

10

40 60 80 000 l2 0

ko

Fig. 13 - Comparison of theory (-) and experimental observations (the points)

for an aluminum cylinder in water.

The irregular oscillations in the aluminum form function, which

begin at ka - 4.5, are related to excitation of the individual normal

modes, f., as defined in Eq. 7c. Resonances occur at ka values at

which

D,,21 [kal = D (2) In. kal - 0 (23)

with D. 2" In, ka) the matrix introduced in Eq. ]a. Solutions to Eq. 23

give the ka position of the free modes of vibration (resonances) of the

cylinder. The correspondence between the ka values at which irregularities in the form function occur and the ka values at which resonances

occur is indicated in Fig. 12, where the resonances are identified by the

subscript (n, 0. Here n is the mode number and I is the eigenfrequency, for example (n, 1) means the fundamental resonance of the
CIRCUMFERENTIAL WAVES 73

nth normal mode, (n, 2) the first harmonic, etc. The "creeping wave"

solution to the problem of the scattering of an incident plane wave by a

solid aluminum cylinder was carried out by Doolittle et al. 1211, who

transformed Eq. 7b using the Sommerfeld-Watson technique. Doolittle

found poles in the complex , plane al the positions

D 2 ) Is,, kaI - 0. (24)

He computed a series of R type, or elasticity related, poles and gave a

table of the positions of the first six R-type poles in the complex s,

plane as a function of ka. Each pole gives rise to a circumferentially

traveling wave 121). The relationship between Eqs. 23 and 24 form the

basis for the correspondence between normal mode resonances and the

individual circumferential waves predicted by the creeping wave theory.

To obtain this relationship, it is necessary to consider the properties of

the normal modes individually. Equation 7c describes the nth normal

mode. The n - 0 term is the breathing mode, n - I the dipole term,

n - 2 the quadrupole, etc. The individual motions can be represented

by a pair of standing waves ej '

* - *6 ) traveling in the opposite directions with phase velocities

c.(ka) kac (25) n

and group velocities

cS(ka) d(k ) (26) dn

At a resonance of the nth mode exactly n wavelengths fit over the circumference of the body, and the Ith .eigenfrequency of the nth mode,

(ka),.t is the /th solution to Eq. (23). A comparison of Eqs. (23) and

(24) leads to the connection between the creeping wave solution. The

complex quantity ,, - a',(ka), (27)

which is related to R type circumferential waves with phase velocities

c,(ka) - Re &, (28)

and group velocities - C

"(ka)- dRe C,/dka (29)

If now Re &,ka - n, Eqs. 23 and 24 become identical in form and the

modal velocities (Eqs. 25, 26) are identical to the creeping wave velocities (Eqs. 28, 29). Thus, when Re v, - n, the Ith Rayleigh type circumferential wave coincides with the wave speed, c,(ka), of the Nth

modal vibration. This hypothesis is demonstrated below. Table I gives
74 CHAPTER 4

Table 1 - Modal Eilgenvalues

(ka)., and Mode Speeds c./c for an

Aluminum Cylinder in Water

(kay.) c./c

1 2 3 1 2 3

0 - - 9.43 - - oo

- 5.87 13.44 - 5.87 13.44

2 4.78 9.17 16.31 2.39 4.59 8.16

3 7.38 12.53 19.12 2.46 4.18 6.37

4 9.65 15.84 - 2.41 3.96 -

5 11.78 19.02 - 2.36 3.80 -

the modal eigenfrequencies obtained from Eq. 23, that are identified in

Fig. 12, and the corresponding modal phase velocities are computed

from Eq. 25. The breathing modes (0, 0 are not strongly excited, as

evidenced from Fig. 12, but are included in Table 1. The (1, 1) mode

is generated in the region where the Franz wave or rigid reflection

predominates and is also not observed in the curve of Fig. 12. The ka

values at which Re v (ka) - n, are extrapolated from the work of

Doolittle et al (Table 2 of Ref. 21), and comparisons are made between

these extrapolated values and the normal mode resonances identified in

Fig. 12. This comparison is shown in Table 2, which also gives the

computed values of the creeping wave phase velocity 1211. Tables I

and 2 demonstrate that the ka values at which resonances occur

correspond to ka values at which Re v, (ka) - n. They demonstrate

further the equality of the modal and circumferential wave velocities c.

(ka) and cl(ka). The relationship is thus established between the (in,

6 normal modes and the elastic or R-type poles found by the "creeping

wave" theory. For it - 2 (i.e., the (2,1) mode), the circumference of

the cylinder is 2 wavelengths of the R, type wave, at the (3, i) resonance, the circumference of the cylinder is exactly 3 wavelengths, etc.

The R2 circumferential wave is similarly related to the (n, 2) normal

mode resonances, and so on, with the (n, 0 normal mode resonances
CIRCUMFERENTIAL WAVES 75

Table 2 - The correspondence between the normal mode

resonances and the ka values at which Rep, - n.

The target is and elastic aluminum cylinder in water.

(ka) values at which

Normal Rep I - n from the Phase velocities c1/c of the

mode Sommerfeld-Watson RL circumferential wave when

resonances formulation of Rep, - n

Ref. 21

N I (ka)w L (ka). R, R2 R 3 R, R5

0 3 09.43 3 09.40 c 00 c - -

4 10.46 4 10.40 _ _ -

5 17.14 5 17.03 00

1 2 05.87 2 05.80 5.85 .-. .

3 13.44 3 13.39 - 13.24 - - -

4 16.02 4 15.90 - - 16.04 - -

2 1 04.78 I 04.85 2.37 - -

2 09.17 2 09.10 - 04.58 - - -

3 16.31 3 16.28 - - 08.16 - -

3 1 07.38 1 07.30 2.45 - -

2 12.53 2 12.47 - 04.15 - - -

3 19.12 3 19.05 - - 06.36 - -

4 1 09.65 1 09.70 2.40 - -

2 15-84 2 15.80 - 03.94 - - -

5 1 11.73 1 11.80 2.35 - -

2 19.02 2 18.90 - 03.79 - - -

related to the Ith order Rayleigh or R, Rayleigh type circumferential

wave.

The R, circumferential wave is related to the leaky Rayleigh wave

[361 on a flat surface (see Chapter 18). This is the surface wave which,

as ka increases, approaches the phase velocity of the Rayleigh wave on

a flat infinite half space. The higher order R-type waves are called

"whispering gallery" waves and become lateral waves in the limit as ka

-. 0o [351. Since the R, or Rayleigh wave is related to the (n, 1) resonances in the normal mode solution, the influence of the circumferentially traveling Rayleigh wave on the backscattering from an aluminum

cylinder can be inferred from the relative influence of the (n, 1) modes

on the form function seen in the curve of Fig. 12. The (2, 1), (3, 1),

and (4, 1) resonances are observed to have a marked effect on the form

function in the 1a range from 4 4 1a < 10. For n > 4 the effects of

the (n, 1) resonances on [V.1 are small, and in fact for ka > 20 no (n,

I) modes were observed to influence V.j The result of the calculations

plotted in Fig. 12 and calculations of [V.1 carried out between 20 < ka

< 40 (not shown) strongly indicate that the Rayleigh wave will not
76 CHAPTER 4

contribute significantly to the observed backscattering from an

aluminim cylinder at ka values above ka - 20. This is a significant

point, since the possibility of Rayleigh wave generation at high (ka) has

been a matter of dispute in the literature [25-28,331. This point will be

explored in more detail. Figure 12 does show, however, that the (2,1),

(3,1), and (4,1) resonances are major features of the form function

curve for ka < 10. Nulls in the form function at the ka positions of

these three resonances should be related to the interference between

specular reflection and the circumferentially traveling Rayleigh wave.

Specular reflection and Rayleigh wave radiation are known to be 180 °

out of phase with each other in the flat surface case 130,36,37).

Verification of the above explanation of the nulls at the (2,1), (3,1),

and (4,1) resonance values should be possible both by computing the

low ka echo response of an aluminum cylinder to an incident short

acoustic pulse and by experimentally determining the echo response of

an aluminum cylinder at low ka. Both measurement and computation

were done, and this experimental observation of the backscattered reradiation from a circumferentialy traveling Rayleigh wave was the first

observation of this phenomenon 131 ).

The major difficulty in achieving an experimental observation of

the Rayleigh wave at low ka is in obtaining a practical and possible

combination of cylinder radius, frequency, and pulse length that allows

the Rayleigh wave to be separated from the specular reflection. The

best available combinations were an aluminum cylinder of radius a -

0.635 cm measured with a short pulse centered at frequencies f. - 386

kHz and f. - 500 kHz. Commercially available lead zirconate titinate

transducers, with active elements 1.905 cm square were driven with a

rectangular pulse, and the achieved pulse length was 5 cycles.

Figure 14 shows the experimentally observed backscattering from

the 0.625-cm-radius-aluminum cylinder at f. - 500 kHz, or ka -

13.5. The backscattered echo consists of a specular return followed by

a Rayleigh circumferential wave which is 1800 out of phase with the

specular return. Even at this relatively low ka value the second traversal of the cylinder by the Rayleigh wave is already almost entirely in the

noise 25 dB below specular reflection. The experimental backscattering

result at f, - 386 kHz, (ka) - 10.4 is given in Fig. 15. Here there is

a slight overlap of the final cycle of the specular reflection and the first

cycle of the Rayleigh wave, but the 180* phase shift can still be

observed. At this ka, direct measurement can be obtained of the group

velocity and attenuation, as the second transversal of the Rayleigh wave

is now visible. The path difference between the first and second manifestations of the Rayleigh wave is Ad - 2wa. The measured group
CIRCUMFERENTIAL WAVES 77

(P)

0)i (P)

Fig. 14 - Experimental observation of the Rayleigh circumferential wave

on an aluminum cylinder at koa - 13.5.

A

06

IX

.': ::

0 25.6 51.2

TIME (,,s)

Fig. 15 - Experimental observation of the Rayleigh circmferential

wave on an aluminum cylinder at koa - 10.4.

velocity is cj/c - 1.9, in comparison to the estimated value of cj/c - 2.0 obtained from Ref. 21. It is expected that the group velocity would

be higher [39,401 at these ka values than the infinite half space Ray- leigh wave velocity, which is cl/c - 1.8 [30]. Equality between the

group velocities of the flat surface and circumferential Rayleigh wave

does not occur until the ka value of the cylinder reaches, at least ka - 30, when the cylinder circumference is greater than 10 wavelengths of

*the Rayleigh wave [391.
78 CHAPTER 4

The fact that the circumferential speed of the R, wave is a function of frequency, or ka, means that changes in the pulse shape should

occur between the specular reflection and the R, circumferential wave.

The speed of sound in water is not a dispersive quantity, i.e. not a

function of frequency, and the specular reflection has the same pulse

shape as the incident wave. The Rayleigh velocity is a function of frequency over the range 0 < ka - 30, and thus is not constant over the

range of frequencies represented by the incident pulse. Changes in

pulse shape between the Rayleigh wave and the incident and specularly

reflected waves are expected. Figures 14 and 15 are digital representations of the experimental recovered signal sampled at intervals of 0.02

jss, or at about 100 points per cycle of the received pulse. No two digi- tal representations are exactly alike, but with so many points having

been taken, the only differences that are noticeable occur in the flattening effects at some of the peaks and valleys. In amplitude measurements these effects are negated by averaging many measurements. The

phase shift of 1800 was determined by comparing the pulse cycles

labeled (1) and (P) in Fig. 14. Despite the slight change in pulse shape

as discussed above, the beginning cf the pulse labeled (1) and the

characteristically large amplitude at the center of the pulse labeled (P)

are present in both the specular and Rayleigh echoes and are 1800 out

of phase for these two echoes. The attenuation of the circumferential

Rayleigh wave at ka = 10.4 is measured from Fig. 15 to be aR - 1.69

Np/revoiution where one revolution equals a travel path of one circumference. Computations of the response of an aluminum cylinder to

an incident pulse should allow examination of the Rayleigh circumferential wave properties at even lower ka values than ko a - 10.4 (the

experimental conditions for Fig. 15). A one-cycle pulse can be pro- grammed as the incident pulse for a computer even though it is not

readily attainable in a laboratory with normally available transducers.

The computation of the reflection of a single-cycle pulse centered at

koa -- 8.8 is shown in Fig. 16. Again specular and Rayleigh wave

echoes are 1800 out ,1 phase. The group velcity obtained from Fig. 16

is ch/c - 2.4, which compares to cf/c - 2.3 estimated from Ref. 21.

Estimates of the expected Rayleigh wave attenuation due to radiation into the water can be made from the flat surface formula given by

Dransfeld [291:

p ,R PC (30)

P5s CR XR

The nonsubscripted variables refer to water, and the subscript R refers

to the Rayleigh wave. Just as the limiting velocity is not achieved until

ka > 30 for aluminum, it is not expected that computations made from
CIRCUMFERENTIAL WAVES 79

60

40

it

20-

- 0 L I #

-20-I 'I

SPECULAR RAYLEIGH WAVE REFLECTION

-40 REFLECTION

INCIDENT PULSE

-60 I I I i I I I I I I

0 I 2 3 4 5 6 7 8 9 10 II

T a ct/

Fig. 16 - Computation of the echoes scattered by an aluminum

cylinder at koa - 8.8.

Eq. 30 be exact for low ka. For aluminum the flat surface values for

the variables in Eq. 30 are

p/p, - 0.37; c/CR - 0.55, and XR - 1.8X

yielding

aR 0.113 " 0.113k 2 cm-(3a (31a)

X - 21r C

or

(21ra) aR cR - (0.113) (ka) (31b)

where aR is dimensionless.

Equation 31b gives R - 1.0 at ka - 8.8 in comparison with the

value aR - 1.0 obtained from Fig. 16. At ka - 10.4, Eq. 31b gives

a; - 1.2 compared with aR - 1.7 obtained from the measurement in

Fig. 15. The results indicate that a reasonable estimate of af can be

made using the flat surface attenuation formula, even at low ka. The

empirical observations given in Figs. 14 and 15 are the first observations of the true Rayleigh circumferential wave. Previous observations

of circumferential waves on solid elastic cylinders have been made at

ka values in the range 40 K ka K( 400 [25,261. This high ka range

was chosen because of the ease of pulse separation of any circumferential effects and, also, because Rayleigh-like circumferential wave properties would have more closely approximated the flat.surface Rayleigh

wave at high ka. The circumferential waves that were observed on
80 CHAPTER 4

aluminum were identified as Rayleigh [25] or "Rayleigh type" 1261

waves. Further analysis of high ka circumferential results lead to an

explanation of the effects seen in Refs. 25 and 26 in terms of multiple

internal reflections of shear waves in the cylinder (271. This view was

further supported by theoretical calculations of the circumferentially

radiated wavefronts which result from internal reflections 1281. The

subject has remained however a matter of some conjecture, [25-28,34],

but the results given here show that the Rayleigh wave has large

enough amplitude to contribute significantly to the backscattering by

the cylinder only at ka values below 20.

The excitation of the (2,1) resonance, which corresponds to the

ka value at which the cylinder circumference is two wavelengths of the

Rayleigh circumferential wave, marks the highest ka at which the Franz

wave contribution can be isolated. A comparison of the form function

curves for the rigid cylinder (Fig. 2), the aluminum oxide cylinder (Fig.

8a), and the aluminum cylinder (Fig. 12) shows that, in all these cases,

there exists a region where the behavior of the form function is purely

rigid, i.e., dominated by the interference of specular reflection and the

Franz wave. For an aluminum cylinder this behavior exists up to ka - 4.5, where the (2,1) resonance minima begin. For aluminum oxide the

specular plus Franz wave behavior persists up to ka - 9.9, and the

resonance null at ka = 9.9 in Fig. 8a is the (2,1) mode for aluminum

oxide. Similar curves were computed for copper, brass, and tungsten

carbide, and in all cases the generation of the (2,1) mode marks the

end of the purely rigid behavior. The ka value at which the purely rigid

behavior will end for a cylinder of a given material can be inferred by

using aluminum as a reference. The (2,1) mode will be excited at

Z 2 I (material) - Z 2, I (A/) • cR (material)

cR (aluminum) (3lc)

The Rayleigh wave speeds used in Eq. 31c are flat surface numbers,

and the equation assumes that the effect of curvature is the same for all

materials, i.e., that the flat surface limit is reached when the circumference is greater than 10 Rayleigh wavelengths. Using the simple formula given in Eq. 31c, the ka position of the (2,1) resonance was

predicted to within 1% for the materials discussed above.

SEPARATION INTO RIGID BACKGROUND

AND RESONANCE PARTS

Junger and Feit [401 qualitatively considered the resonance

features of the acoustic scattering by elastic bodies. They showed that
CIRCUMFERENTIAL WAVES 81

resonances should appear where the sum of the mechanical and radiation impedance goes to zero. The acoustic scattering from a submerged

aluminum cylinder can be described in terms of a rigid background

term, with a resonance contribution superimposed on that background.

The observed phenomenon can be formalized by using the methods of

nuclear scattering theory, so that mathematically explicit forms for the

resonances and background, as well as expressions for the resonance

widths, are obtained. It is necessary to establish the nature of the background before the formalism is developed because a parallel formalism

could have been developed, using a soft or an intermediate background, which would have had no physical significance for the problem

of the solid metal cylinder. The conclusions and formalism developed

for the scattering from an aluminum cylinder apply to the scattering

from any submerged solid cylinder whose density is greater then that of

the surrounding fluid and whose shear and compressional wave speeds

are greater than the speed of sound in the fluid.

It has been shown in the previous section that the irregular

characteristics of the form function for solid elastic cylinders are related

to the normal modes of free vibration of the body, and these resonances often occur over a narrow frequency range as was seen in Fig.

12. It will be demonstrated in Figs. 17 through 20 that the resonances

are superimposed on a background of reflection resulting from rigid

boundary conditions, so that the elastic body can be regarded as a rigid

body except in the frequency interval over which the resonances occur.

In Fig. 17, the individual partial waves, If, 1 from n =0 through n =

5 are plotted vs ka for an aluminum cylinder. The f, are described by

Eq. 7c. The curves in Fig. 17 show that the amplitude of the individual

partial waves If,~ I have distinctive behavior in regions where the resonances occur. The eigenvalues 1, are labeled along the curve in Fig. 17.

The individual partial waves for both the infinitely rigid and the

infinitely soft cylinder have no resonance irregularities, as seen in Fig.

18(a) and 18(b) respectively. The demonstration that the individual

partial waves for a metal elastic cylinder consist of resonances superimposed on a rigid background is seen in Fig. 19. Here the 1f21 and I3

individual partial waves for rigid, soft, and elastic boundary conditions

are plotted. It is clearly observed in Fig. 19 that the rigid and elastic

curves are the same except in the region where resonances occur. The

resonances for I>2 are narrow resonances; the 1=I eigenvalue

which corresponds to the R, or Rayleigh surface wave is a broader

resonance. A more dramatic example of the relationship between the

elastic and the rigid solutions for a solid cylinder is seen in Fig. 20.

Here the quantity 1f 2(#r)elastic _f 2(7f)iidj is plotted vs ka, and the

(2,1), (2,2), and (2,3) resonances are clearly isolated. As was noted,
82 CHAPTER 4

n .0

1.0 -

0.5 -

1.0

0.5

n-.2

1.0 -

0.5-

n-~3

1.0-

0.5

0-.4

1.0 -

0.5 -

1.0-

0.5-

0 2.0 4.0 60 60 O 1 12.0 14.0 160 18.0 20.0

ko

Fig. 17 - The individual partial wave amplitudes from

n - 0 to nt - 5 for the aluminum cylinder.
CIRCUMFERENTIAL WAVES 83

1.2.

0.0-N

/" " ,-\ 2 \ ('\ 

o ,,' .'2 /. , ,. .. , . ,

0.1

o"

So.e /

80.0 4

• , ." •N

! .o

0.1.

0 2 4 a 8 10 12 14 16 is 20

ka

Fig. 18(a) - The individual partial wave amplitudes from

n - 0 to n - 5 for a rigid cylinder.
84 CHAPTER 4

1.8•

n-3

-0A" %%. ..

0...

2.4

Fig 18 4 Th idvual parti l amltd s romo

1f

... 6. " f"l " " " "' "

Fig 18b The indvda paria wav pi tue from i ,

n" .ot - on 5 o yidr
CIRCUMFERENTIAL WAVES 85

3.0-

RIGID

2.5 - =2 ------- ELASTIC

2.0 .. SOFT

4 1.0 (2)

1.0 - i. ' 

0.5 1 " --.

(a)

3.0

2.5 n =3

2.0

1.5 (1)

. 0 (2)

0.5

0.0V

(b)

Fig. 19 - A comparison of the rigid (-), soft (- -) and elastic --- -)

partial wave amplitudes for (a) n - 2 and (b) n - 3.

- 1.5

1.2

0.9

I-0.6

. yO.3

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

ka

Fig. 20 - A plot of the difference between the elastic and rigid partial wave amplitudes for n - 2, over the range 0.2 < ka < 20.

the (2,1) resonance and in fact the (n, 1) resonances in general are broader than the narrow resonances which occur for higher order eigen- frequencies, i.e. I > 1. A method of computing resonance widths will

be described later. The results seen in Figs. 17 through 20 are, of

course, not restricted to the backscattered direction.

Fig. 21a shows a bistatic form function If.*(O) curve for a rigid cylinder at ka - 12.53, which is the ka value at which the (3,2)

r resonance occurs in the aluminum case. In Fig. 21b this bistatic form function for a rigid cylinder is compared with the bistatic form function
86 CHAPTER 4

0o

A

2700 - 90

10 20 30

180,

Fig. 21 (a) -A polar plot of ~. vs 0 for a

rigid cylinder at ka -12.53.

3I

2-%

i. 21a '- 'olrplto If*lv o

0 20 40 60 80 100 120 140 160 180

ASPECT ANGLE IN DEGREES

Fig. 21(b) - A comparison of If,(9)I for rigid and

aluminum (-) cylinders at ka - 12.53.
CIRCUMFERENTIAL WAVES 87

3

2-

0 20 40 8 s0 100 120 140 I8 0 S8

ASPECT ANGLE IN DEGREES

Fig. 21(c) - The same comparison as (b) except that the n - 3 term

in the rigid series is replaced by the n - 3 elastic term.

for an aluminum cylinder. Here the results are plotted on a linear

rather than polar plot, and due to the symmetry apparent in Fig. 21a

only the range 00 < 0 < 1800 is plotted. The rigid and elastic solutions plotted in Fig. 21b were obtained with 23 terms (n - 0 through 22 from Eq. 7a). If the single n - 3 term from the elastic solution is

substituted for the n - 3 term in the rigid solution, the result seen in

Fig. 21c is obtained. A similar procedure was carried out by Vogt and

Neubauer [411 for a sphere in a monostatic geometry. In Fig. 21c the

exact bistatic solution for the aluminum cylinder at ka - 12.53 is compared with the hybrid solution formed by taking 22 terms from the

solution for the rigid cylinder (n - 0 through 2 and n - 4 through

22) and adding the n - 3 term from the solution for the elastic

cylinder. The form function resulting from the modified solution for

the rigid cylinder vs aspect curve and the same plot for the elastic form

function are seen in Fig. 21c as very similar. The differences between

Figs. 21b and 21c are most noticeable in the backscattered half space,

90* < 0 < 270 ° (recall that by symmetry the results seen in Fig. 21c

from 900 < 0 < 1800 are exactly the same as the results between 1800

< 0 < 2700). Figure 21 again indicates the probability that the scattering from an aluminum cylinder can be treated as resulting from a rigid

background term with resonances superimposed. It indicates further

that this behavior is not limited to the single monostatic angle (0 - 180) but can be utilized at any bistatic aspect angle.

f These results indicate that the solution to the scattering from solid

elastic bodies as expressed in Eqs. 4, 5, and 7 should be separable into

two terms, the rigid background term and the resonance term.
88 CHAPTER 4

Consider the scattering function

S, - exp(2i 8,). (32)

Now the solution describing the scattering by an elastic cylinder may be

written in the form familiar to nuclear reaction theory [431 as

- 1/2 ; e, i"(S, - 1) H.(kr) cos nO. (33)

The 8,, of Eq. (32) are called scattering phase shifts, and a comparison

of Eqs. I, 3, and 33 shows that

S,- 1 -- 2G,(ka). (34)

For the case of a rigid cylinder, defining ,, as the scattering phase shift

for a rigid cylinder and defining

S(' R) = exp(2i ed) (35a)

as the scattering function for a rigid cylinder using Eqs. 34 and 13,

leads to

S R ) - 1 =- 2GR)(ka) (35b)

and thus

H(Z)(35c)

If the rigid portion of the scattering function as given in Eq. 35c is

factored out of the expression for the elastic scattering function Eq.

(34), we have using Eqs. 3 and 35c

S SR){ ' - - z (36)

The L,,'s were previously defined in Eq. (la), and the is are defined by

z- (Z) H (Z) (37a)

and H, (Z)

zj- - (Z) H, 2 ,(Z) (37b)

the primes in Eqs. 37 and 35c represent derivatives with respect to the

argument, and as defined previously H and H(2) are Hankel function of

the first and second kind respectively. The quantities z'1 of Eq. 37 can

be separated into real and imaginary parts:

z12 - An ± is, (38)
CIRCUMFERENTIAL WAVES 89

with

& 1/(Z) J"(Z) Jn(Z) + Y'(Z) Y(Z) (39) [J(Z)j2 + IY(Z))

and

-2 [J'(Z)]I + [y-(Z)] 2 (40)

Equation 36 may be rewritten using Eq. 38 as

S. -SR) {

L - A A n} is. (41)

The linear approximation method of nuclear resonance theory is

used in which the resonance frequencies Z. are defined by the condition

Ln- I (Z,,) - A,,. (42a)

The quantity (L, -

' - A.) is assumed to be linearly varying with frequency so that it can be expanded in a Taylor series in Z in the vicinity

of any one of the resonance frequencies:

L - 1 _- An +)9,(Z - Zn). (42b)

A resonance width is defined by

r- -2s, (43)

and the scattering-function S,,, may be rewritten in resonance form as

Sn = e 24S._ S"(R) Z - Z n- iF , 2 } (4

Z z z + iF,/2 I.44

From Eq. 44 the S, are seen to have resonance poles at the complex

frequencies Z - Z. , given by

Zpoie - Zn - iFn/2 (45)

and a resonance zero, Z = Zzero, at

Zzero Z, + iFn12. (46)

The resonance width rn defined in Eq. 43 is a positive quantity. Thus

Zpole is located in the lower half of the complex Z-plane a distance

(I/2)F from the real axis, and Zzero is located in the upper half plane

at the same distance above the axis.

vA
90 CHAPTER 4

The quantity S, - 1 which appears in Eq. 33 can be written,

recalling the definition in Eq. 32, as

S, - I = 2 i e'8 " sin 8,. (46a)

Using Eq. 36 and the expressions for zj' given in Eq. 38 (Sn - 1) can

be rewritten in the form

S,, - I - 2i e't( -An + e i " sin (46b)

or, using Eqs. 44 and 46a, S,, - I may be written in the resonance

form

2i 2i Z 1/2 F,, +ee nsin j. (47)

The individual partial waves, fn(0), of Eq. 7c can thus be written, using

Eq. 47, as

f"(0)- 2i e) / 1 _/2 + e'( sin fi cos (no).

(48)

The first term of Eq. 48 represents the resonance contribution, and the

second term represents the rigid boundary contributions. The representation of f,(O) given in Eq. 48 shows that the complex eigenfrequencies

of the scatterers are the locations of the resonance poles in the complex

frequency plane, whose real parts determine the resonance frequencies

in the scattering amplitudes.

A consideration of the field within the elastic cylinder can also be

made in light of the above results. The displacement u within the

cylinder is represented by a scalar potential IV and a vector potential A,

and is written

u -- V P +V x A (49)

with solutions [201

' P ,, il C,, J,,(kL r)cos nO (50a)

n-0

and

Az - e,, in B, J,(kT r)sin nO. (50b)

n-O
CIRCUMFERENTIAL WAVES 91

The subscripts L and T are longitudinal and shear respectively. The

coefficients C,, and B, are given by

. 2i 2 1 a 3 3 (51a)

rp w2 ZH,(Z) D," z- ' - L oi

and B1 2i I a32 (5 1b)

B 7rp W,2 ZH(Z) D,(') z- 1 - L- 1 "

The expansion of L - 1 in Eq. 42 leads in to the resonance expressions

for 'P and A, which are

= 2 . in a33 J,(kLr) Cos no (52a)

Hr p 2 fln ZH,(Z) D, 1

) Z - Zn + 1/2 ir,

and

2 ., e i" a32 J T(krr) sin no

Az - irrpo _ I n/J, (52b) 2 -o" fl Z'(Z) D,11 Z - Z, + 1/2 iF,

Equations 52a and 52b show that the internal solutions are of a

pure resonance form only. This is as expected, since by definition a

rigid body is impenetrable; thus no rigid background term is expected

for the internal solution.

CIRCUMFERENTIAL WAVES ON CYLINDRICAL SHELLS

Numerous empirical observations of circumferential radiation

from cylindrical shells exist in the literature [11-15,25-271, but these

have left many serious voids in the understanding of circumferential

waves as well as erroneous information concerning the properties of the

waves. A connection can be made between circumferential waves and

the exact Rayleigh series solution. It can be demonstrated that the

Lamb wave dispersion curves on plates predict the range of possible

excitation of circumferential waves and that the velocity of circumferential waves may be obtained directly from the form function vs ka

curves. Lamb dispersion curves also give immediate knowledge of the

ka region over which a particular circumferential mode is significant.

Calculations of the backscattered echoes from shells can be used to

obtain curves relating the amplitude of the specular and circumferential

contributions as a function of ka, and, contrary to earlier literatre, the

circumferential waves will be shown to be of most importance in the

low ka region generally avoided in that work. A target classification

scheme has been proposed in the literature which relied on the assumption that a hollow shell acts as a soft body, in that its specular reflection
92 CHAPTER 4

is 1800 out of phase with an incident wave [441. By consideration of

the interference between specular and circumferential radiation, the

actual ka range over which such a hypothesis is valid is determined.

Barnard and McKinney [II] were the first to observe backscattered acoustic echoes with circumferential properties. This observation

was a significant contribution done in the absence of guiding theory.

They attempted to link the observed acoustic phenomenon with the

geometric diffraction phenomenon observed by Franz and Doypermann

[181 in the electromagnetic domain; however, the analogy broke down,

since the acoustic phenomenon was a predominately elastic effect, the

geometry serving in the capacity of a waveguide. Horton, et al. [131

made the initial attempt' to relate circumferential waves on cylindrical

shells to the elastic properties of flat plates. They compared their

observation of a circumferential wave on an aluminum shell to a

theoretically computed flexural plate mode and found a 10% difference

between measured and predicted velocities. A similar comparison was

attempted with a brass shell [131, but the circumferential wave could

not be excited. The ka range considered was 21 < ka < 38, with b/a

= 0.96, where b and a are the inner and outer shell radius respectively.

The circumferential wave observed by Horton et al. will be related in

this work to the first antisymmetric Lamb mode, whose properties and

ka region of possible excitation are discussed later. Diercks, et al. [121

made empirical observations of circumferential waves on both brass and

aluminum shells near ka - 50, with b/a - 0.96. They established the

existence of circumferential waves with two different group velocities.

The faster velocity wave was called a longitudinal mode, and the lower

velocity wave was called the flexural mode. This paper by Diercks et al.

[121 was significant in that it was the first to clearly state that more than

one circumferential mode existed. Their conclusions, concerning which

mode is dominant and whether both modes can be simultaneously

excited, were thought to be general but are limited strictly to the conditions of the observations. It has been demonstrated [31] that the socalled longitudinal mode is actually many different modes, and therein

lies much of the confusion about the frequency range of excitation

and/or dominance of a particular mode. Goldsberry [141 demonstrated

that the circumferential waves observed previously 111-131 would

reflect from slits cut in the shell. He called the wave with lower group

velocity a low frequency wave, and the faster wave a high frequency

wave. Again these generalizations do not survive beyond his experimental conditions. The ranges he considered were 32 < ka < 38

(flexural, slow, low frequency wave) and 70 < ka < 76 (compressional, fast, high frequency wave). The b/a was 0.96. Uberall et al.
CIRCUMFERENTIAL WAVES 93

predicted circumferential waves of different types on rigid [201 and elastic cylinders (211 and shells (451. These predicted wave types were

Franz-type waves [201 with properties similar to the electromagnetic

case of Franz (161, and R-type or Rayleigh-type [21,451 waves, which

depend on the elastic properties of the target. Neubauer [231 empirically isolated the Franz-type wave on a solid elastic cylinder in water,

and Harbold and Steinberg (241 isolated the wave on a rigid cylinder in

air.

Bunney, et al. [251 used narrow beam sources to illuminate

cylindrical shells over narrow ranges of incidence angles and "directional" receivers to observe circumferential-wave radiation. The ka

range they considered was between 50 < ka < 320 with b/a - 0.95.

Many observations of a low velocity circumferential wave were compared to the antisymmetric Lamb mode, and the single observation of a

higher velocity mode was related to the symmetric plate mode.

Neubauer and Dragonette [271 and Dragonette 1461 empirically

demonstrated that the velocity of the observed circumferential waves

on cylindrical shells [271 and the velocity of Lamb waves on plates [461

could be predicted by considering guided wave propagation within

cylindrical shells and plates. Dragonette [461 also established that

Lamb modes were most easily excited in the frequency thickness

regions where the phase velocity reached a constant plateau. This

result is significant in a consideration of the so-called fast circumferential wave on cylindrical shells and was the basis for correcting some

erroneous conclusions in the literature.

Shirley and Diercks [47] compared measured and predicted values

of the steady state response of spherical shells over the range 25 < ka

< 65 with b/a - 0.95. Differences of the order of 10 dB, or 300% in

pressure amplitude, were found between theory and prediction, but

similarities in shape between the theoretical and empirical curves were

observed.

Horton and Mechler [151 attempted to measure phase velocity of

circumferential waves on aluminum cylindrical shells by setting up a

long pulse or steady state interference pattern between the successive

circumferential pulses. The significance of their paper was that it

offered a possible approach to phase velocity determination which, as

will be discussed later, is a difficult parameter to obtain when waves are

excited on a curved surface.

Figure 22 shows the geometry of the cylindrical shell problem. It

is similar to Fig. I except that the target now has a finite thickness h
94 CHAPTER 4

Y

w

LU

0

Fig. 22 - The geometry used in the solution to the

reflection from a cylindrical shell.

given by h -a - b, where b is the inner radius and a is the outer

radius. The shell is air filled. An experimental observation of circumferential waves using a pressure sensor is seen in Fig. 23. Here the target is a stainless steel cylindrical shell with b/a -0.96. The radius of

the shell is 1.27 cm, and the center frequency of the pulses seen in Fig.

23 is 1.5 MHz, leading to k.~ a =80. The specular reflection is not

shown, since it is 40 dB greater than the largest echo seen in Fig. 23,

and was gated out of the return so that the echoes seen in the figure

could be amplified to the maximum extent for display. The

source/receiver is located 15 diameters from the target. The backscattered echo seen in Fig. 23 was digitized at the rate of 13 points per

cycle and stored on magnetic tape. The first, third, and all successive

echoes in Fig. 23 result from a circumferential wave which circumnavigates the cylinder, with little attenuation, continually radiating into the

water as it travels. The second echo in Fig. 23 is the result of a second

type of circumferential wave so highly attenuated at the ka and/or k/i

value of this experiment that only its first traversal around the cylinder

is observed before it attenuates into the noise. (Observation of Fig. 23

alone would not allow identification of the second echo as a circumferential wave. This identification was based on many experimental

measurements, some of which will be given below.) Measurements of

the circumferential velocity of the persistent series of equally spaced

echoes (1-7) in Fig. 23 straightforward. The circumferential group

velocity c; is obtained from Fig. 23 by

where At is the time between echoes and 27ra is the circumference of

the shell. The measured value from Fig. 23 is c; 5 .48 x 105 cm/s or
CIRCUMFERENTIAL WAVES 95

I.-

(2) (3) (4) (5) (6) (7)

05 102

Fig. 23 - Experimental observation of circumferential waves on a

stainless steel shell at k0a - 80.

=c 3.7. This value for c; identifies this wave as that which had

been called the "fast", "high frequency", or "compressional" wave [13-

151. As will be discussed, these names can be misleading or in error.

Attenuation measurements from Fig. 23 are also straightforward. The

successive amplitudes from Fig. 23 are plotted on semilog paper in Fig.

24, yielding an attenuation of

ot" - 0.14 Np/revolution.

The use of the digitizing procedure and display makes possible observation of the individual cycles of the successive echoes in Fig. 23. The

empirical observations in Figs. 25 through 28 are photographs of scope

traces. Time scales needed to show many successive echoes do not

allow observation of the individual cycles within the echo.

Figures 25, 26, 27, and 28 show further results of reflection measurements on steel shells, and the experimental conditions and results

are summarized in Table 3, which also includes the results described

for Fig. 23. The hydrophone measurements seen in Figs. 23 and 25

through 28 show the acoostrreflection in the backscattered direction, 9

- Jr. Observation of the entire scattered field, 0 < 9 < 21r can be

obtained simultaneously by schlieren visualization. Figure 29 shows a

schlieren visualization of the scattered field of a stainless steel cylindrical shell of radius 0.9525 cm with b/a - 0.95, and koa - 202. The

incident pulse is seen in Fig. 29a, and the time sequence of photographs shows the scattered field at later times. The specular reflection
96 CHAPTER 4

10 ,

0

I

.I

'C

SI i I

0 2 4 a B 10

REVOLUTIONS

Fig. 24 - The amplitude of the circumferential

waves seen in Fig. 23 plotted on semilog paper.

Fig. 25 - Experimental observation of a "fast" circumferential wave

at koa-11, on a stainless steel shell. The time scale is

20pus/division; the amplitude scale is 5 mv/division.

Fig. 26 - Experimental observation of a "fast" circumferential wave

at ka-69, on a stainless steel shell. The time scale is

20ps/division; the amplitude scale is 2 mv/division.
CIRCUMFERENTIAL WAVES 97

Fig. 27 - Experimental observation of a "slow" circumferential wave

at k0 a - 17, on a stainless steel shell. The time scale is 20

ps/division, the amplitude scale is 200 mv/division.

Fig. 28 - Experimental observation of a *slow" circumferential wave

at ka - 43, on a stainless steel shell. The time scale is 20

pus/division; the amplitude scale is 2 mv/division.

Table 3 - Summary of Cylindrical

Shell Observations

Radius k c Figure (cm) koa b/a c,/c Np/revolution

23 1.27 80.8 .96 3.7 0.142

25 1.59 113.0 .97 3.5 0.209

26 0.9525 69.0 .95 3.6 0.126

27 0.9525 17.0 .95 1.35 1.000

28 1.905 43.0 .97 1.35 1 0.600

______
98 CHAPTER 4

Fig. 29 - Schlieren visualization of a "fast* circumferential

wave at k~a - 202 on a

stainless steel shell.
CIRCUMFERENTIAL WAVES 99

and the beginning of the radiation from a circumferential wave are seen

in Fig. 29b, and Fig. 29c shows the reradiation from the first complete

traversal of the circumferential wave into the backscattered direction.

At the bottom of Fig. 29c the diffraction around the shell can also be

observed. The group velocity of the wave seen in Fig. 29 is c,*'c - 3.6.

In Fig. 29 the incident pulse insonified the entire cylinder, so that the

same effect was generated on both the upper left and upper right quadrants of the cylinder.

The properties of the circumferential waves seen in Figs. 23 and

25 through 29 are similar to circumferential wave properties previously

reported in the literature [11-15,25-271. Specifically, circumferential

waves with group velocities c/*c = 3.6 (Figs. 23, 25, and 26) and c/*c

-1.3 (Figs. 27 and 28) are observed, with the faster group velocity

observed at the higher frequency (or higher ka) and the slower group

velocity wave at the lower frequency. This simple interpretation of the

results is, however, misleading and demonstrates some of the practical

difficulties in a predominantly empirical approach to this problem,

where broad generalizations are made, based on limited measurements.

The experimental measurements here, and reported previously, are

generally made at high ka values. At very low frequencies, or ka

values below ka - 20, it is a practical impossibility to achieve short

enough pulse lengths to separate circumferential waves with transducers

generally available; hence, high ka measurements are made as a matter

of necessity. This low-ka limitation on the isolation of separate echoes

is especially true of high speed circumferential waves. In addition,

most empirical measurements are made on shells with b/a < 0.96,

since thinner shells are more difficult to fabricate and maintain.

Finally, as a practical matter it is not possible to measure enough combinations of shells and frequencies to do a complete empirical study.

This latter statement is true not only because of the low ka separation

difficulty mentioned above but also because in the case of a curved

shell there are two frequency variables. For a flat plate, frequency

times thickness, fh may be considered a single variable. The radiation

from a given plate may be examined as a function of fh, simply by

varying the frequency of the incident pulse or continuous wave. A

similar experiment on a curved shell is not as unambiguous. As the

frequency of the incoming wave is varied, the fh or kh of the shell

changes accordingly, but in addition the ka value changes. As will be

seen, there are certain effects which are strong functions of ka, and

others which depend almost entirely on k/i. These had not previously

been differentiated successfully in the literature, and this could not

have been reasonably accomplished empirically. Because previous work
100 CHAPTER 4

has been limited to the high ka region, where the flat surface limit is

approached, the kh variable has generally been considered most

significant in all empirical observations of isolated circumferential

waves. In fact the low ka region which has been avoided is the only ka

region where a high velocity circumferential wave plays a significant

role in the acoustic scattering by a cylindrical shell, as first shown by

Dragonette [311.

Analysis of circumferential waves on cylindrical shells can best be

accomplished by determining the relationship between the circumferential waves and exact steady state theory. In addition, Lamb theory for

plates can be used to predict the possible ranges of excitation of circumferential waves on shells.

The geometry of the cylindrical shell problem was given in Fig.

22. The formulation of the exact normal mode solution to the scattering of sound by an elastic cylindrical shell exists in the literature and

can be presented in a form similar to that of Eq. 1, which described the

scattering from solid elastic cylinders [331:

ps(O) _po %(i){ J" (Z Q " Z J'

(Z)}

p, (0) - -p. G) J.(Z) Q"- Z HI(Z) H, (kr)cosnO.(53)

This expression differs from Eq. (1) only in the replacement of L0,

which involved the division of two 2-by-2 matrices, by Q., which

involves the division of two, 4-by-4 matrices. The larger matrix results

from the extra boundary condition on the surface r = b, and the

expression for Q, is

a 21 a22 a 23 a 24

al a32 a33 a34

a 4 l a42 a 43 044

Q. - _9 a61 a 6 2 a6 3 a 64

PS all a 12 a 13 a 14

a31 a 32 a33 a 34 1

a41 a42 a43 a4 (54)

a61 a 6 2 a 6 3 a 64

with the matrix elements aj given in Chapter 5 [33]. Computations are

given in Fig. 30 of the form function vs ka for stainless steel shells

with b/a = 0.99 (Fig. 30a) and b/a - 0.98 (Fig. 30b). The curves

cover the ka range 0.2 < ka < 50. A similar set of curves for aluminum are given in Fig. 31, and the two figures demonstrate the similarity
CIRCUMFERENTIAL WAVES 101

2.0

1.5

1.0

0.5

(a)

2.0

1.x

1.,

'I,

0.0 'I . I p

0 5 10 15 20 25 30 36 40 45 0

ka

(b)

Fig. 30 - The form function for stainless steel shells over the range 0.2 K< ka < 50

for shell thickness of (a) 0.99 and (b) 0.98.

2.0 

1 , ,

1.5-

0.5

(a)

2.0[

2. C - I i I

0 5 10 15 20 25 30 35 40 45 50

ka

(b,)

Fig. 31 - The form function for aluminum shells over the range 0.2 < /ca 50

for shell thicknesses of (a) 0.99 and (b) 0.98.
1i

102 CHAPTER 4

of results obtained on metals quite different in density, but with shear

and longitudinal wave speeds approximately twice and four times the

water wave speed, respectively. The elastic constants used in obtaining

the curves seen in Figs. 30 and 31 are given in Table 4.

Table 4 - Table of Constants

MaterialCL CT P (10 cm/s) (105 cm/s) (g/cc)

air 00.343 0.000 0.00

aluminum 06.370 3.120 2.17

aluminum oxide 10.700 6.300 3.92

iron 05.950 3.240 7.70

stainless steel 05.5940 3.106 7.90

water 01.493 0.000 1.00

The curves in Figs. 30 and 31 are made up of the steady state

interference of specular reflection and a single circumferential mode

which makes many circumnavigations of the cylinder before attenuating

into the noise. The above explanation of Figs. 30 and 31 can be

demonstrated analytically by computing the response of the shell to an

incident pulse. The computation is made using Eqs. 9 and 12 with the

procedures described earlier. Figure 32 shows the response of an

aluminum shell, with b/a - 0.99, to an incident acoustic pulse. The

pulse is centered at a dimensionless frequency koa = 10. The backscattered echoes are seen in Fig. 32(a), and the incident wave is seen in

Fig. 32b. The backscattered return is made up of the specular reflection followed by a series of equally spaced echoes which result from

multiple circumnavigations of a circumferential wave. The ratio of the

circumferential wave group velocity c; to the water wave speed c is

given by

C;/C - 2w (55) AT

where Ar is the dimensionless time between successive circumferential

echoes. The result obtained from Eqs. 32 and 55 is c;/c - 3.7. The

deviations from - I.4 1 in Figs. 30 and 31 occur when the circumference, 2wra, is an integral number of circumferential wavelengths. For

ka values at which 21ra - nWA, where X* is the wavelength of the circumferential wave and n is an integer, the long pulse or steady state

interference of the circumferential waves gives a maximum contribution, since all add in phase with one another. At these ka values peaks
CIRCUMFERENTIAL WAVES 103

50-

40- Al b/a 0.99 k a z10

30-

20

I

o I,-o 10

10 -10

-20

-30

-40

-50

-2 -1 0 1 2 3 4 5 6 7 8 9 0 1 r (ct -r) /a T

(a) (b)

Fig. 32 - Computations of the echoes scattered when a short incident pulse

impinges on an aluminum shell with b/a = 0.98. The pulse is centered at k a

10. The scattered echoes are seen in (a) the incident wave in (b).

will occur in the form function if the specular reflection and circumferential waves are in phase and nulls will occur if they are out of

phase. The ka difference, Aka, between the successive fluctuations in

the Ifo.1 vs ka curve are directly related to the circumferential wave

phase velocity c; by

c / c = A ka (56)

which for Figs. 30 and 31 gives cp/c - 3.7. The nearly constant spacing of the fluctuations Aka - 3.7 indicates a constant or slowly varying

phase velocity, so that the approximation cg - c; is valid over the ka

range 0 < ka < 50 seen in Figs. 30 and 31. The phase velocity of the

circumferential wave on an aluminum cylindrical shell with b/a = 0.99,

is obtained from Fig. 31a and Eq. 56 as cpc = 3.7, which is identical to

the group velocity obtained from Fig. 32 and Eq. 55.

The previous paragraphs demonstrated the significance of the Aka

spacing between the deviations in the form function for cylindrical

shells. The direction of the deviations from (f.oo = I in the form

function curves, as seen in Figs. 30 and 31, is also significant. The
104 CHAPTER 4

hypothesis of Tucker and Barnickle [441 was based on the assumption

that a hollow air-filled shell will act as a soft body in that its specular

reflection will be 1800 out of phase with an incident wave. This would

then distinguish hollow body echoes from echoes scattered by solid

bodies, which would act rigidly, that is, give a specular return in phase

with the incident wave. In fact, the return from solid bodies whose

density is greater than the density of water and whose shear and

compressional wave speeds are greater than the speed of sound in water

can be described in terms of a rigid background term plus a resonance

term over all ka > 0. Thus, for solid bodies with these elastic properties, the rigid background portion of the hypothesis of Tucker and Barnickle would be correct. With regard to the "soft" scattering by a hollow shell, however, the hypothesis breaks down, as can be determined

form the work described in the previous paragraph. Figures 30 and 31

demonstrate that the ka range over which cylindrical shells will act as a

soft body is a function of frequency. As frequency is increased, the

thickness h of a given shell becomes greater with respect to a

wavelength, and whether a shell acts as a "soft" body (specular

reflection 180' out of phase with the incident wave) or a "rigid" body

(specular reflection in phase with the incident wave) depends both on

the frequency and shell thickness. For example, as was discussed the

fluctuations in [fol for the shell described in Fig. 31b occur at intervals

Aka -- 3.7. The deviations from [f.1 = 1 are, however, not uniform in

direction either in Fig. 31b or in any of the other form function curves

shown in Figs. 30 and 31. Three separate background regions exist.

The shell acts as a soft body over the ka range where the fluctuations in If-1, at Aka = 3.7, are in the negative direction. Here the

specular and incident wave are 1800 out of phase. Recall that the

incident wave and the circumferential wave are in phase (as seen in Fig.

32), and further pulse calculations such as that in Fig. 32 show that

they remain in phase over the ka range from 0.2 < ka < 50. For an

aluminum shell with b/a = 0.98 the ka region over which the shell acts

as a "soft" body (specular reflection and the incident wave 180 ° out of

phase) is seen from Fig. 31b to be 0 K< ka < 23.

As ka increases, the shell passes through a transition region during which a single fluctuation has both positive and negative aspects.

This occurs over the range 23 < ka < 37 for the 0.98 Aluminum shell.

Finally, for ka > 40 the deviations from f.1 - I are positive and the

shell is a rigid reflector with respect to its specular reflection. The

extent in ka of the three background regions will vary with thickness

and with material. Later in this section the advent of higher order

modes will be discussed, but it can generally be said that if the product
CIRCUMFERENTIAL WAVES 105

of frequency and thickness is large enough to allow more than one circumferential mode to be excited, the shell has already reached the rigid

background region. Development of the formalism for the resonance

scattering from hollow shells with intermediate background is the subject of work by Murphy et al. [48].

The generation of a single circumferential wave under the conditions present in the computation of the results seen in Figs. 30 through

32 are consistent with the dispersion curves for Lamb waves on plates.

Figure 33 shows dispersion curves for the first four symmetric and

antisymmetric Lamb [491 waves for aluminum plates. The symmetric

Lamb modes satisfy the frequency equation [46,49,501

tanhI (lVfh/ V) I(C¢2- V2

)/c 21/ 2

)

tanh{(7rfh/V) [(cZ- V2)/cZL" 2

4{[(cL2- V2)/cI]I/ 2.

- [C2- V2)/C2-]1 /2}

T= (57) 1(2c 2 - V2)Ic 2l

2

and the antisymmetric mode satisfies the equation

tanhf(irfh/V) [(C2 - V2

)/c2]11 2

)

tanh{(irfh/ V) [(c2 - V2)/C ]1/2)

1(2c'- V2)/c 2]

2

- __ (58) 4(1(c t- V2)/C2J1 L / 2"[(Cr - V2)/cI]I1 2}

In Eqs. 57 and 58, V is the Lamb phase velocity. The group velocity of

the Lamb wave, V., is related to the phase velocity by

Kg = V - 1- (fh) dV/d(fh) I59

It has been demonstrated that Eqs. 57 through 59 describing Lamb

waves on plates in vacuo are not strongly modified when the plate is

immersed in water [46,50,511 and that Lamb waves can be generated by illuminating a plate in water by an incident pulse [46,50. Radiation

of the Lamb wave into the water can be observed either with a hydrophone [501 or by schlieren visualization [461.

Grigsby and Tajchman [521 gave dimensionless curves for the

phase and group velocities of Lamb waves on a plate whose ratio of

longitudinal to shear speeds is 1.8. Their curves are seen in Fig. 34.

Special attention is directed to the group velocity curves in Fig. 34b,

where all the modes show a flat peak in the group velocity at V" -

1.8. In Fig. 34 the ordinate V is Lamb phase velocity divided by shear

wave speed i.e. V - V/c r , and the abscissa (fh), is fh/cT. In Fig.

34b the ordinate Vg,, = g/cT.
106 CHAPTER 4

x

U

LI &. 2.54 - . - . . .

2.54 5.08 T 62 10.16 12.7

FREQUENCY x PLATE THICKNESS x IO5CM CYCLES/SEC

Fig. 33 - The Lamb dispersion curves for the first four symmetric (- ) and

antisymmetric (- - -) Lamb waves on aluminum plates.

5-

(a)

4

3

>2-(1) (4)

(f On3 5 6 7 8

(fh) n

(b)

2-

q I I L I

I 2 3 4 5 6 7 S

(fh) n

Fig. 34 - Dimensionless curves of (a) Lamb phase velocity and (b) Lamb

velocity curves for materials with CL/CT - 1.8.
CIRCUMFERENTIAL WAVES 107

Dragonette (461 demonstrated that strong generation of a Lamb

mode takes place in the region where the phase velocity curves reach a

flat plateau (at approximately V, - 1.8 in Fig. 34a). This plateau

region corresponds to the frequency thickness region where the group

velocity curve for a particular mode reaches a flat maximum (at approximately V ' - 1.8 in Fig. 34b). Dragonette (461 demonstrated further

that this strong generation of a Lamb mode, in the fh region where

phase velocity is approximately equal to the group velocity, persisted as

the plate was curved.

In Fig. 35 the Lamb phase velocity curves describing the first

symmetric and first antisymmetric mode curves for an aluminum plate

are isolated. The ordinate is given in terms of the Lamb phase velocity

V, and also in terms of the angle of incidence 6, at which a Lamb wave,

with that phase velocity, can be generated by a plane wave incident

from water onto the plate surface. This angle 9, satisfies the equation

sin 0, - c/V, (60)

and a Lamb mode cannot be generated by an acoustic wave incident

from water to the plate unless V > c. The frequency thickness variable, fh, which is the abscissa of the Lamb curves seen in Fig. 35 may

be written in terms of ka for a specific cylindrical shell by a simple algebraic manipulation:

kh = 2rfh - kal - (b/a)] (61a)

C

(ka) c(l - b/a) (61b) fh== 261r

Using Eq. 61b the abscissa, fh, for the flat plate case may be

transformed from fh into ka for shells with various b/a values. Figure

35 shows the abscissa written in equivalent ka values for an aluminum

shell with b/a = 0.98 (see Fig. 31b). The results seen in Fig. 35

predict that the first symmetric mode can be generated at any ka value,

and in fact since the flat plateau in group velocity occurs at the low end

of the frequency thickness or ka scale, this wave should be strongly

generated at low ka. The arrow in Fig. 35 points to the place where V

- c for the first antisymmetric mode. For the shell with b/a = 0.98,

this curve predicts that the antisymmetric mode cannot be excited at ka

values below ka t 50. The phase and group velocity of the symmetric

mode is predicted to be V/c - 3.7 by the curves in Fig. 35, in excellent agreement with the circumferential wave observed for the cylindrical shell (Fig. 32). The circumferential wave related to the first symmetric Lamb mode has died out by ka z 50, as seen in Fig. 31b, and,

as seen above, the results in Fig. 35 predict that the onset of a circumferential wave related to the first antisymmetric mode cannot occur at a
108 CHAPTER 4

ALUMINUM

I&e -V85X0C/

64

S22-o-V4XIO5

2-4-

Fig 35 - The angle of incidence at which the first symmetric and

~antisymmetric Lamb modes may be generated.

1 value lower than ka := 50. Such a wave would have a lower group

velocity; i.e., oscillations in If~ vs ka would occur at closer intervals

than those observed in Fig. 3lb. Recall that c,/c - Aka.

! A plot of Ifi( vs ka for an aluminum shell with b/a ="0.98 is

given over the range 50 < ka <( 90 in Fig. 36. This is then a continuai tion of the curve given in Fig. 31b, and it shows the onset of a circumferential wave with the properties related to the first antisymmetric - mode. The oscillation in the form function curve predict a circum-

! ferential wave with c;/c - Aka - 1.3.

I

0.¢2X 5

kG

Figl. 36 -The form function vs ka over the range 50 (; ka 4; 90

for an aluminum shell with b/a - 0.91.

I

!32r( l~~ 

~ 4,,am~m,, 0 -
CIRCUMFERENTIAL WAVES 109

In general, then, a circumferential wave related to the first symmetric mode should always be generated for a shell of any thickness.

Its influence is restricted to the low ka region over which the phase

velocity has a flat plateau. The location of this region is a function of

thickness, as will be described below. A circumferential wave related to

the first antisymmetric mode can only be generated at ka values higher

than the coincidence frequency ka - (ka)c, where V - c. Thus the

thicker the shell, the lower the ka value at which this mode can be generated.

These conclusions should be reflected in the form function curves

for aluminum cylindrical shells of various thicknesses. Differences

from If.I = 1 should occur at intervals Aka = 3.7 for all thin shells,

and these differences should die out more quickly with ka as thickness

increases, since the plateau region in Fig. 35 corresponds to a smaller

ka range for thicker shells. Differences at Aka =- 1.3 should begin to

occur at lower ka values as thickness is increased. Fig. 37 shows the

form function curves for aluminum cylindrical shells with b/a - 0.99,

0.98, 0.96, 0.94, 0.92, 0.90, and 0.85. The above conclusions are

demonstrated by the curves in Fig. 37.

As thickness increases, the antisymmetric mode is seen to occur

at lower ka values. The low velocity circumferential wave observed in

Figs. 27 and 28, and by others [12-15,25-71, is related to the first

antisymmetric Lamb mode for a plate. The ka range over which it is

generated depends on the thickness of the shell. The first symmetric

mode is strongly generated on aluminum plates in the thickness region

where it has a phase and group velocity ratio of V/c - 3.7. This is carried over to the shell case, where a circumferential wave having the

properties of the first symmetric mode are observed at low ka on all

thicknesses of shells from 0.85 < b/a < 0.99. Closer spaced oscillations in [/1 vs ka are also seen to occur at relatively lower ka values as

shell thickness is increased. These oscillations are related to a circumferential wave with cYc -- 1.3. If one returns now to the GrigsbyTajchman 1521 group velocity curves seen in Fig. 34b, the reason for

the association of low frequency, with low velocity and high frequency

with high velocity in the literature becomes apparent. All the higher

order Lamb modes in Fig. 34b are most strongly generated when their

group velocity is V.. - 1.8. This is also the group velocity of the first

symmetric mode at its region of strong excitation. Moreover, the same

is true of all higher order antisymmetric modes (see Fig. 33). Thus the

first symmetric mode and all higher order symmetric and antisymmetric

modes cannot be distinguished from one another by measurement of

group velocity alone, and these waves collectively have been identified

I
110 CHAPTER 4

2- | " b/m.Oi

(a)

0.,

oI . , , b' / , ,

(b)

o. J" ,t' , . . . . . .

(€)

01.

(d)

O.I ~bin.O.U' i

/

0.1

(e)

I.I

I b/bO.S

OA

0'o 20 4. 60 S.0 00 12.0 140 A6.0 16O 20

Fig. 37 - The form function vs ka over the range 0.2 4 Ica

4 20 for aluminum shells with b/a values of (a) 0.85, (b)

0.90 (c) 0.92 (d) 0.94 (c) 0.% (f) 0.98 (5) 0.99.
CIRCUMFERENTIAL WAVES 11

as the fast circumferential wave. The particular mode generated

depends on the shell thickness and frequency, but, as discussed earlier,

practical considerations generally preclude isolation of the first symmetric mode. All of the measurements of a low velocity wave are

related to the first antisymmetric mode, which for thin shells is generated at a higher frequency than the first symmetric mode but at a

lower frequency than any of the higher order modes. In the empirical

observations of circumferential waves, the wave related to the first

antisymmetric mode is obtained at a frequency which depends on the

thickness (see Fig. 37).

Because of the slow speed of this wave, there is a greater time

difference between successive traversals of the circumferential wave (a factor of 3 as compared to the faster waves); hence, this mode when

present can be isolated at lower ka values than a mode traveling with a

velocity 3 times higher. Thus in past pulse hydrophone measurements

112-15,271 low velocity corresponded to low frequency in the experimental observations. As frequency was increased it became possible to

isolate higher velocity modes, all of which were strongly generated with

the same group velocity; hence, high velocity corresponded experimentally to high frequency.

The circumferential wave related to the first symmetric mode is

the only one of the "fast" circumferential waves whose amplitude

approaches the amplitude of the specular reflection, and, while it is not

practical to isolate it experimentally, its contribution to the steady state

pressure or form function at low ka is apparent. As demonstrated in

Fig. 33, it is possible to isolate the first symmetric mode by computation of the response of a shell to a short incident pulse. The results of

computations similar to those carried out to produce Fig. 33 are given in Figs. 38 and 39 for various thicknesses shell at various center frequencies. The purpose of these calculations is to demonstrate that the

relative amplitude of the circumferential and specular contributions is a

function of ka. In Figs. 38 the responses of three aluminum shells with

thickness b/a - 0.99, 0.96, and 0.90 are presented. The center dimensionless frequency of the calculation is koa - 10. A circumferential

wave related to the first symmetric mode is seen in Fig. 38a (b/a - 0.99) and Fig. 38b (b/a - 0.96). The measured attenuation in these

two cases is 0.50 Np/revolution (Fig. 38b) and 0.43 Np/revolution (Fig. 38a). In Fig. 38c the first antisymmetric mode is generated simultaneously with the symmetric mode. This figure demonstrates that the

antisymmetric mode is in fact generated at lower frequency as thickness

is increased, as predicted in the discussion of Fig. 37. It also shows that

at the same koa the attenuation of the symmetric mode increases with
112 CHAPTER 4

.75

.50

.-25

0 0

S-.25

-50

-.75

-2 .1 0 1 2 3 4 5 6

Fig. 38(a) - Computation of the echoes scattered by an aluminum shell

at k~a - 10. the shell thickness is b/a -0.99.

1.0-

75

05

IL .25

w 0

-.25

-0.5

*75

.10

Fil. 38(b) - Computation of the echoes scattered by an aluminum shell

at k,,a - 10; the shell thickness is b/a -0..
CIRCUMFERENTIAL WAVES 113

.F5

.50

-3 .25 a.

o 0 N

-.25

0

z

-,50

(1I) 1A) (4S) (2A)

-.75

-1.0

-2 -1 0 1 2 3 4 5 6 7

Fig. 38(c) - Computation of the echoes scattered by an aluminum shell at

koa - 10; the shell thickness is b/a = 0.90.

increasing thickness, and finally it demonstrates that the large oscillations in If(01)I which appear with increasing thickness in Fig. 37 are

due to the larger magnitude with which the antisymmetric mode is generated. The attenuation of the antisymmetric mode in Fig. 38c is 1.0

Np/revolution, which is much larger then that of the symmetric mode,

but the magnitude of the first-antisymmetric-mode-echo is more than 6

dB larger than that of the first-symmetric-mode-echo.

The responses of an iron shell with b/a = 0.99 at koa values of

It and 20 are seen in Figs. 39(a) and 39(b) respectively. Here the

attenuation is 0.46 Np/revolution at koa = 11 and 0.23 Np/revolution

at koa - 20. Thus the attenuation decreases with higher frequency, an

observation similar to that of Horton and Mechler [15], who observed

this phenomenon at ka - 30 for the wave identified here as the

antisymmetric wave.

As may be observed in Fig. 38c, a determination of the attenuation of a particular circumferential wave is not necessarily a measure of

its relative importance to the total scattered field. If the ratio of the
114 CHAPTER 4

i 0.5 0.5

-2 -10 1 2 3 4 5 -2 -1 0 1 2 345

T T

Fig. 39(a) - Computation of the Fig. 39(b) - Computation of the

echoes scattered by an iron shell with echoes scattered by an iron shell with

b/a - 0.99 at a koa value of 11. b/a - 0.99 a ta koa value of 20.

amplitude of the first circumferential echo, pl, to the specular echo,

p5 c, is taken from Figs. 39(a) and 39(b), the result obtained is

Pl/Ps - 0.2 at koa - 11 and pl/pm - 0.1 at koa - 20. These ratios

show that a particular circumferential wave is more strongly generated

and gives a larger contribution to the steady state scattering at low ka,

as was observed in Figs. 30, 31, and 37. The ratio p/pec is directly

related to the relative contribution or importance of the wave to the

steady state solution. Figure 40 shows a plot of p/psc for various

thicknesses of stainless steel shells as a function of ka. The points were

obtained from pulse computations such as those in Figs. 38-39. In general for shells with b/a < 0.99 the circumferential wave related to the

first symmetric mode has a backscattered amplitude more than 20 dB

down from specular for ka > 20.

.5 .. . . .5 5 - -, • , - .5 .5

b/b0.99 /a0.98 b/ b/a =0.9 4 4 .4 .4

3 .3 .3

.2 2 .2 .2-

1.1 .1.1

O00 0.10 230 0 10 20 00 20 0 10 20 30

(a) (b) (c) (d)

Fig. 40 - The ratio (pj/pz,,) vs ka for stainless steel shells

of thicknesses (a) 0.99 (b) 0.98 (c) 0.95 (d) 0.90.
CIRCUMFERENTIAL WAVES 115

For higher order modes generated above ka - 20, the ratio

pl/ps. is more than 20 dB down from specular. This is consistent with

the observations made here of a "fast" circumferential wave (Figs. 23,

25, 26), where pl is greater than 30 dB below specular, and with all

reported observations in the literature (12-15,261.

The circumferential wave related to the fundamental antisymmetric mode also has its largest influence on V1f' at low ka. It can,

however, be generated only at low ka for thick shells. The general

observation in the literature that the "slow" circumferential wave is

more strongly generated in thicker shells, is simply because of the

results discussed in connection with Fig. 37, namely that as shells

become thicker, it is possible to excite the first antisymmetric mode at

lower ka.

The results considered here were for thin shells. Figures 33 and

34 show that as fh becomes larger, all the Lamb modes tend toward a

final velocity V -- 2.0c. Since the phase velocity curves again level off

for large fh, the Lamb modes are again strongly excited. Each higher

order mode is first strongly excited at V/c - 3.7 and then in the limit

of a thick shell at V/c - 2.0. Therefore it should be possible to find

an intermediate frequency range at which a Lamb type mode is generated with V/c - 3.7 simultaneously with a lower order mode which

has reached its high frequency limit V/c - 2.0. Such a situation is

seen in Fig. 41. Here a schlieren visualization is made at an fh value

of 11.2. The target is a 3.4-cm-diameter-aluminum cylinder with b/a - 0.9. This corresponds to ka - 476 and kh - 47.6. Simultaneous

observation of circumferential waves with c;ic - 3.7 and C90/c - 2.0

are seen. Figures 37 and 41 explain what has been referred to in the

literature [121 as rare occurrences when slow and fast circumferential

waves are observed simultaneously.

CONCLUSION

The steady state acoustic response of infinitely long solid elastic

cylinders and cylindrical shells can be exactly computed in terms of a

normal mode series, For rigid cylinders the backscattered form function can be described in terms of the interference of a specularly

reflected wave and a Franz-type circumferential wave whose speed and

attenuation are related to the peaks and nulls in the form function.

Solid metal cylinders in water exhibit this purely rigid behavior in the

low ka region, after which region the form function is dominated by

minima related to resonances in the individual normal modes. For

cylinders made of metals whose shear and compressional wave speeds
116 CHAPTER 4

Fig. 41 - Schlieren visulization of circumferential

waves generated on an aluminum shell, with b/a -

0.9 at koa - 476.

are greater than the wave speed of sound in water, the first resonance

minimum observed is the (2,1) resonance. This occurs at ka = 4.78

for aluminum, and the ka value at which it occurs for other metals can

be computed using aluminum as a reference.

The normal mode resonances are related to circumferential waves

predicted by creeping wave theory. A mode resonates when its modal

velocity is matched by the velocity of a circumferential wave. A single

circumferential wave generates a given eigenfrequency in successive

modes. The (n, 1) resonances are related to the Rayleigh wave, and

the (2,1), (3,1), and (4,1) modes are generated at ka values at which

the cylinder circumference is 2, 3, or 4 Rayleigh wavelengths. Similarly

the (n, 2) resonances are related to the R 2 type whispering gallery

mode and so on. The predominant circumferential waves in a given ka

region can be predicted from the dominant resonance minima in

lf.(r)IL The "Rayleigh" wave was experimentally observed on aluminum in the predicted region.
CIRCUMFERENTIAL WAVES 117

The region of oscillations in [V.001) which begins at the position

of the (2,1) resonance persists as ka - 00; however, in reality a frequency will be reached after which absorption must be included in the

theory. In the resonance region (ka > 4.78 for aluminum) the scattering from the cylinder is made up of a rigid background part, on which

the numerous resonances are superimposed. The resonance formalism

of nuclear reaction theory is used to separate the exact normal mode

series solution into rigid background and resonance terms, and resonance widths can be calculated.

For a thin cylindrical shell, the Franz wave does not measurably

affect If.(ff)I even at low ka. This is consistent with soft rather than

rigid scattering behavior, and it is demonstrated that as ka increases

thin shells pass through three background regions. In the softbackground region at low ka the specular reflection is 1800 out of phase

with an incident pulse. This is followed by a region of intermediate

background, and then a rigid-background region at which the specular

and incident pulses are in phase and remain in phase as ka is further

increased. Circumferential waves are isolated theoretically by applying

fast Fourier transform techniques to the Fourier integral representing

the echoes scattered, when a short acoustic pulse is incident on a shell.

The relationship between the observed circumferential waves and the

steady state form function shows that, for thin shells, the number of

circumferential waves present, their velocity, and their relative

significance can be obtained directly from the form function.

Lamb theory for plates is utilized to predict the ka range of possible excitation of specific circumferential waves. A circumferential wave

related to the first symmetric mode is generated for ka > 0, for all thin

shells. A circumferential wave related to the first antisymmetric mode

is generated at ka values which vary with thickness in a predictable way.

As shell thickness is increased, circumferential waves related to all high

order Lamb modes are strongly generated with the same group velocity.

This accounts for observations reported previously in the literature and

thought to be the observation of a single circumferential wave.

REFER ENCES

1. Lord Rayleigh, Theory of Sound, Dover (1945).

2. P. M. Morse, Vibration of Sound, McGraw-Hill (1936).

3. J. 1. Faran, J. Acoust. Soc. Am. 23, 405-418 (1951).
118 CHAPTER 4

4. R. Hickling, J. Acoust. Soc. Am. 34, 1582-1592 (1962).

5. R. Hickling, J. Acoust. Soc. Am. 36, 1124-1137 (1964).

6. L. D. Hampton and C.M. McKinney, J. Acoust. Soc. Am. 33,

664-673 (1961).

7. K. J. Diercks and R. [-ickling, J. Acoust. Soc. Am. 41, 380-393

(1967).

8. W. G. Neubauer, R. H. Vogt, and L. R. Dragonette, J. Acoust.

Soc. Am. 55, 1123-1129 (1974).

9. L. R. Dragonette, R. H. Vogt, L. Flax, and W. G. Neubauer, J.

Acoust. Soc. Am. 55, 1130-1137 (1974).

10. H. D. Dardy, J. A. Bucaro, L. S. Scheutz, and L. R. Dragonette,

J. Acoust. Soc. Am. 62, 1373-1376 (1977).

11. G. R. Barnard and C. M. McKinney, J. Acoust. Soc. Am. 33,

226-238 (1961).

12. K. J. Diercks, T. G. Goldsberry, and C. W. Horton, J. Acoust.

Soc. Am. 35, 59-64 (1963).

13. C. W. Horton, W. R. King, and K. J. Diercks, J. Acoust. Soc.

Am. 34, 1929-1932 (1962).

14. T. G. Goldsberry, J. Acoust. Soc. Am., 42, 1298-1305 (1967).

15. C. W. Horton and M. V. Mechier, J. Acoust. Soc. Am. 51, 295-

303 (1972).

16. W. Franz, Z. Naturforsch 9a, 705-716 (1954).

17. G. N. Watson, Proc. Roy. Soc. (London) A95, 83-99 (1919).

18. W. Franz and K. Deppermann, Ann. Phys. 10, 361-373 (1952).

19. L. Flax, J. Acoust. Soc. Am. 62, 1502-1503 (1977).

20. H. Uberall, R. D. Doolittle, and J. V. McNicholas, J. Acoust.

* Soc. Am. 39, 564-578 (1966).
CIRCUMFERENTIAL WAVES 119

21. R. D. Doolittle, H. Uberall, and P. Ugincius, J. Acoust. Soc. Am.

43, 1-14 (1968).

22. 0. D. Grace and R. R. Goodman, J. Acoust. Soc. Am. 39, 173-

174 (1966).

23. W. G. Neubauer, J. Acoust. Soc. Am. 44, 298-299 (1968).

24. M. L. Harbold and B. N. Steinberg, J. Acoust. Soc. Am. 45,

592-603 (1969).

25. R. E. Bunney, R. R. Goodman, and S. W. Marshall, J. Acoust.

Soc. Am. 46, 1223-1233 (1969).

26. W. G. Neubauer, J. Acoust. Soc. Am. 45, 1134-1144 (1969).

27. W. G. Neubauer and L. R. Dragonette, J. Acoust. Soc. Am. 48,

1135-1149 (1970).

28. D. Brill and H. Uberall, J. Acoust. Soc. Am. 50, 921-939, (1971).

29. K. Dransfeld and E. Salzmann, in Physical Acoustics, Vol. VII,

edited by Warren P. Mason and R. N. Thruston, Academic Press

(1970).

30. W. G. Neubauer and L. R. Dragonette, J. Appl. Phys., 45, 618-

622 (1974).

31. L. R. Dragonette, NRL Report 8216 (1978).

32. L. A. Scheutz and W. G. Neubauer, J. Acoust. Soc. Am. 62,

513-517 (1977).

33. L. Flax and W. G. Neubauer, J. Acoust. Soc. Am. 61, 307-312

(1977).

34. R. E. Bunney and R. R. Goodman, J. Acoust. Soc. Am. 53,

1658-1662 (1973).

35. R. H. Vogt, L. Flax, L. R. Dragonette, and W. G. Neubauer, J.

Acoust. Soc. Am. 57, 558-561 (1975).

36. G. V. Frisk and H. Uberall, J. Acoust. Soc. Am. 59, 46-54

(1976).

• II. mm| IIIIIIIIDIINIIIIIN i K ll lm m mmnnllnn lln 
120 CHAPTER 4

37. W. G. Neubauer, J. Appi. Phys. 44, 48-55 (1973).

38. H. L. Bertoni and T. Tamir, Appi. Phys. 2, 157-172 (1973).

39. E. K. Sittig and C. A. Coquin, J. Acoust, Soc. Am. 48, 1150-

1159 (1970).

40. I. A. Viktorov, Rayleigh and Lamb Waves, Plenum Press (1967).

41. M. C. Junger and D. Feit, Sound Structures and Their Interactions,

M.I.T. Press (1972).

42. R. H. Vogt and W. G. Neubauer, J. Acoust. Soc. Am. 60, 15-22

(1976).

43. S. deBenedetti, Nuclear Interactions, Wiley (1964).

44. D. G. Tucker and N. J. Barnickle, J. Sound Vib. 9, 393-397

(1969).

45. P. Ugincius and H. Uberall, J. Acoust. Soc. Am. 43, 1025-1035

(1968).

46. L. R. Dragonette, J. Acoust. Soc. Am. 51, 920-935 (1972).

47. D. J. Shirley and K. J. Diercks, J. Acoust. Soc. Am. 48, 1275-

1282 (1970).

48. J. D. Murphy, E. D. Breitenback, and H. Uberall, J. Acoust. Soc.

Am. 64, 677-683 (1978).

49. H. Lamb, Proc. Roy. Soc. (London) 93, 114-128 (1917).

50. D. C. Wornton, J. Appi. Phys. 32, 967-971 (1961).

51. M. F. Osborne and S. D. Hart, J. Acoust. Soc. Am. 17, 1-18

(1945).

52. T. N. Grigsby and E, J. Tajchman, IRE Trans. UE-8, 26-33

(1961).
CIRCUMFERENTIAL WAVES 121

Appendix A

LIST OF SYMBOLS

a - the radius of the target

aij - matrix elements defined in Ref. 32

A - the vector potential

b the inner radius of a cylindrical shell

B. - a coefficient in Eq. 50b

c - the velocity of sound in water

Fthe phase velocity of the Franz wave CF - the group velocity of the Franz wave

c4() - the moup velocity f the n wale

c, (ka) - the modal phase velocity for the nd, normal mode

cf(ka) - the modal group velocity for the nib normal mode

c,(ka) - the phase velocity of the th R-type circumferential wave

cf(ka) - the group velocity of the aIh R-type circumferential wave

CA - the phase velocity of the Rayleigh or R, circumferential

4A - the group velocity of the Rayleigh or R, circumferential

wave

CL - the longitudinal wave velocity in a material

CT - the shear wave velocity in a material

- the phase velocity of a circumferential wave in a

cylindrical shell

cg - the group velocity of a circumferential wave in a cylindrical shell

C, - a coefficient in Eq. 50a

d - distance

D') (ka) - A 2-by-2 matrix defined in Eq. la

D,(2)(ka) - A 2-by-2 matrix defined in Eq. la

f. - the far field form function

f! - the far field form function for a rigid cylinder

fA - the nth partial wave or nth modal contribution to the form

function

f - frequency

f4 - the center frequency of an incident pulse

fh - the frequency thickness product

(Ah), - the dimensionless frequency thickness parameter,

(fh), - fh/cT

- '~0'~~~ - -- ~ ~ ~ 
122 CHAPTER 4

g,(ka) - the spectrum of an incident pulse

g5(ka) - the spectrum of a scattered echo

G.(Z) - defined in Eq. 3

G (Z) - the expression to which G.(Z) reduces when the target is

a rigid cylinder

h . the thickness of a cylindrical shell

H(Z) the Hankel function of the first kind (order n,

argument Z)

H. 2)(Z) - the Hankel function of the second kind (order n,

argument Z)

H.(Z) - the derivative of the Hankel function of the first kind

with respect to its argument

H.(2)'(Z) - the derivative of the Hankel function of the second kind

with respect to its argument

J, (Z) - the Bessel function (order n, argument Z)

J"(Z) - the derivative of the Bessel function with respect to

its argument

k . the wavenumber in water given by k - 2w/.k

kL - the longitudinal wavenumber in a material

kT the shear wavenumber in a material

ka - the dimensionless frequency variable, ka - 2ral,

(ka)o the center dimensionless frequency of a pulse in water

(ka)mk - the ka value at which a peak in If.I occurs

I - an integer, I - 1, 2, ..., used to number the

eigenfrequencies of a given mode

LN- defined in Eq. (Ia)

n - an integer, n - 1, 2 ..., used to number the normal modes

pi, () - an incident acoustic pulse

A,- the incident plane-wave amplitude

PI the pressure amplitude of the first backscattered

circumferential echo

p, (9) - the steady state scattered acoustic pressure

at the bistatic angle 9

PsWC - the pressure amplitude of the specular reflection

q,- the zeroes of the first derivatives of the Airy

function

Q.- defined in Eq. 54

r - the range or distance between the scatterer and the

field point

R, - the Ilk order Rayleigh-type circumferential wave

sn - defined in Eq. 40

S:. ) - the elastic scattering function defined by S. - exp (2i 8,)

S(R) - the rigid body scattering function defined as $.2R =- exp (2 i €,
CIRCUMFERENTIAL WAVES 123

t -time

u - the particle displacement

V - the phase velocity of a Lamb wave

V9 . the group velocity of a Lamb wave

V. - dimensionless Lamb phase velocity given by V,= V/cT

V9n . dimensionless Lamb group velocity givey by V,,, .VIc

Y, (Z) - the Neumann function (argument Z, order n)

Y.(Z) - the derivative of the Neumann function with respect to

its argument

Z - short-hand form for the dimensionless frequency variable

Z = ka

Z.,- the dimensionless frequency at a resonance, Z- (ka),

Zpoie - the Z value at which a resonance pole in the scattering

function S,, occurs

Zzero - the Z value at which a resonance zero in the scattering

function S, occurs

zI- defined in Eq. 37a

Z2 - defined in Eq. 37b

aF - the attenuation coefficient for the Franz wave

a-R the attenuation coefficient for the Rayleigh or R,

circumferential wave

aR - the dimensionless attenuation coefficient for the R,

circumferential wave

a - the attenuation coefficient for a circumferential wave in

a shell n- coefficient i-i the Taylor series expansion (Eq. 42b)

r. - the width of a resonance, given by r. - -2s ,

8, scattering phase shift for the elastic scattering function

Ad - change in distance

Aka - change in ka

At - change in time

Ar - change in the dimensionless time parameter

An - defined in Eq. 39

en - the Neumann factor e, -2. n - 0, e. - 1 n > 0

9 - the polar angle

9, - the incidence of angle of a plane wave

X - the wavelength of sound in water

AR - the wavelength of the Rayleigh wave on a flat

surface

- the wavelength of a circumferential wave

, - a complex variable

cot - the phase shifts for the rigid scattering function

p - the density of water

Ps - the density of the target material
124 CHAPTER 

4

ct 

-

a

7 -the dimensionless time parameter r 

-

.0 

- the azimuthal angle

vI 

- a scalar potential . the angular frequency, w =2irf
Chapter 5

LAYERED ELASTIC ABSORPTIVE CYLINDERS*

INTRODUCTION

This scattering of sound by elastic hollow cylinders has been discussed by several authors [1-41. In these papers, and in others mentioned in their references, the absorption of acoustic energy in the shell

was assumed negligible. Although this assumption is valid in many

cases, it is not, for certain cylindrical shells composed of nonmetallic

materials. This was demonstrated [51 for spheres by showing the solution to be altered significantly when absorption was present. Excellent

agreement between experiment and theory was obtained for that case

only after absorption was accounted for. Acoustic reflection from twolayered cylindrical shells is formulated in which either layer of the shell

may absorb wave energy, and different media may be assumed to be

inside and outside the shell.

A mathematical model is developed to predict the scattering of a

plane acoustic wave from a two-layered absorptive cylindrical shell with

different fluid media inside and outside. Either layer or the shell may

absorb wave energy. The hollow space within the cylinder is assumed

to be occupied by a fluid or a vacuum. Solutions to the elastic problems considered here are constructed using scalar and vector potentials.

The resulting equations are solved in terms of Bessel functions of complex argument. Matrix methods are used throughout.

DESCRIPTION OF PROBLEM

Figure I shows the cylindrical coordinate orientation and the

L direction of a plane wave incident on an infinitely long cylindrical shell

in a fluid medium. The two-solid-elastic-absorptive layers are denoted

by I and 2. The axis of the cylindrical shell is taken to be the z-axis of

the cylindrical coordinate system (r, 0, z). There is a plane sound

bem results first appeared in: Lawrence Flax and Werner G. Neubauer, J. Acoust.

Soc. Am. 41, 307-312 (1977).12
126 CHAPTER 5

SOL") NCIOENTLAYER I

LAYER 2

FLUID 2

Fig. I - Geometry used for formulating the sound scattering

from an infinite circular two-layered cylindrical shell.

wave of circular frequency wv incident along the negative x-axis. The

fluid (2) inside the shell has a density of pj and wave propagation

speed c1 . In general, the outer fluid (I) will be different and is

described by parameters p and c..

In an effort to facilitate comparison with previous single-layered

computations, it is convenient to define cylinder parameters which

represent thickness ratios, b/a and c/a. The radius of the inner core is

b, a is the radius to the outside of layer 2, and c is the outside radius.

MATHEMATICAL ANALYSIS

The two fluids outside and inside the shell are labeled by w and f,

respectively. The two layers are described by the densities

p,(i - 1, 2), and longitudinal and shear speeds VL, and Vsj, which are

complex.

In the outside fluid medium, the excess acoustic pressure is the

sum of the incident plane wave

P . PO( e((- )n J, (kr) cos (nW), (la)

and a scattered (outgoing) wave

P O Po eCH,.,n (kr) cos (n ). (Ib)

The incident pressure amplitude is po, a. is the Neumann factor, k -

W/c, H42) - J - lV, where J. and Y. are Bewl functions of the first

and second kind, H,1) are Hankel functions, and C,, are the scattering

I- -i |
LAYERED CYLINDERS 127

coefficients. Throughout, the time dependence (e-'") has been

suppressed.

The solution for the scalar and vector potentials in layer 1 are

* = 'O P- e,, F- [AfJ.(kLlr) + B Y, (kLr)] cos (n0), (2a)

00

AZ am Po A e. F, [D.J.(kIr) + E Y. (ks1r)1 sin (nO), (2b)

and in layer 2

0 - Po e E i- [GJ.(kL2r) + K, Y.(kL2r)] cos (n), (3a)

n-0

B, Po e, F" [1,J,, (ks2r) + L, Y,,(ks 2r)] sin (no). (3b) n-O

In the core, one has again a compressional wave,

- PO e,, i

- M,,J,,(kfr) cos (no). (4) n-O

Since the core contains the origin, the solution must be regular at

r - 0. Thus, the coefficients associated with Y(kfr) must be zero.

At the outside boundary of the shell r - c, the displacements and

normal stresses must be continuous and the tangential stresses must be

zero. Four boundary conditions are prescribed at the interface between

the outer layer and inner layer r - a:

(i) Radial displacements are continuous.

(ii) Tangential displacements are continuous.

(iii) Radial stresses of adjoining material are equal.

(iv) Tangential stresses of adjoining material are equal.

Finally, three conditions are established between the inner layer

and the internal fluid r - b. These are similar to the conditions present

at r - C.

The coefficients C, are determined from the boundary conditions

and ,re of the form
128 CHAPTER 5

J.(Z)LO - Z J 2>'(Z) (5)

c,- H"2)(Z)L. - ZH (

2) (Z)'

where the primes denote the derivative with respect to the argument

Z - kc. The functions L, are given by a division of two ninth-order

determinants of the form

L, - P P (6)

where

a21 a22 a23 a24 0 0 0 0 0

a3l a 32 a 33 a34 0 0 0 0 0

a 41 a42 a43 a44 a45 a 46 a47 a48 0

a51 a52 a53 a54 a55 a 56 a 57 asg 0

Pn a 61 a 62 a 63 a64 a 65 a66 a 67 a 68 0

a71 a72 a73 a74 a75 a76 a77 a78 0

0 0 0 0 a 8 5 086 a 8 7 a88 a8 9

0 0 0 0 a95 a96 a97 a98 0

0 0 0 0 a10, 5 a106 a107 a10,8 a10,9

and

all a 12 a13 a14 0 0 0 0 0

a31 a32 a33 a34 a35 0 0 0 0

04l a 42 a43 a44 a45 a46 047 a48 0

a51 a52 a53 a54 a55 a56 a57 a58 0

Q- a61 a62 a63 a64 a65 a66 a67 a68 0

a71 a72 a73 a74 a75 a76 a77 a78 0

0 0 0 0 a 85 a86 a87 a8 8 a8,

0 0 0 0 ags a% a97 ags 0

0 0 0 0 a10,5 a 1 0 ,6 ao.7 aos al.91

The elements of the determinants are given in the Appendix.

The arguments of the function appearing in these expressions are determined from the following relationships:
LAYERED CYLINDERS 129

r C,

ZL - - ifiLmkLIc. Z,- ks C - i0siksIC;

r - a,

tL - kLja - ii3LikLja, t, - ksla - if3slksla,

qL - kt 2a - iL2kL2a, qs - ks2a - i3s2ks2a;

r - b.

UL = kL2b i3L2kL2b, US - kS2b - il3s2ksb, (7)

where f's are absorption factors given in Np.

In addition, ratios of complex wave numbers appear in some of

the matrix elements and are given by

G- Z/(Zs2 - 24Z), G2 - qj/(qs2 - 2q2),

G/- ( - 2uZ), U - (p/p 1) Zs/t. (8)

As a result of using complex wave numbers in the theory, the elements of the determinant defining L. contain Bessel functions with

complex arguments. Techniques [61 for calculating these functions are

available.

The scattered pressure is obtained using Eqs. (Ib), (5), and (6). I IJ(Z)L Q. n - Z Ji(Z)I Hg2)(Z)

A5 - Po ;~ e A.I (Z) L. - 1 2 )' (Z) Jcs(/) 9

For large distances from the reflector, the asymptotic expressions

for the H(2)(Z) can be used. For monostatic reflection 0 - 7r, making

cos(nm) - (- 1)" and the scattered pressure becomes

A'PO (IrZ)/2 1) HEA( 2H((Z)L ZH(2)'(Z) . (10)

This may be rewritten as

p3 " po(2c/r)'12f.(Z). (11)

From these results it is possible to consider several limiting cases.

If LA -" 0, the solution would apply to scattering by a rigid cylinder. If

Lo - 00, the solution for a soft cylinder is obtained. The single layered

fluid filled shell scattering coefficients are obtained if c/a - 1.00. The
130 CHAPTER 5

solid cylinder scattering coefficients are found by first taking c/a = 1.00

and letting b/a approach zero. One must make sure to take the limits

of Y,(ZL) and Y,,(Z s ) each going to 0. The limiting process must be

used since simply setting ZL = Zs = 0 gives 0 in both numerator and

denominator of L.

CALCULATED RESULTS AND CONCLUSIONS

The calculations were performed for two-layered hollow (vacuum

in the core) cylindrical shells in water. The layers consisted of Lucite

(outer), an absorptive material, and iron (inner) considered to be lossless. Physical parameters for the materials assumed in the calculations

are given in Table 1. Absorption is included in the theory by using

complex wave numbers. Absorption factors expressed in Np were

derived from Ref. 7 where they are given in dB.

Table 1 - Material properties

pX 103 VLxI10 VsxlO3

(kg/m 3

) (in/sec) (m/sec) (0L (Np) (3 (Np)

Iron 7.700 5.950 3.240 ...

Lucite 1.180 2.680 1.380 0.00340 0.00531

Water 0.998 1.482a ......

aReferred to as c. in the text.

Figures 2-8 show the computation of lf,,(ka)l, called

MODULUS in the plots, for various shells. Computation for shells of

three thicknesses (b/a = 0.99, 0.97, and 0.95) without an outer covering (c/a - 1.00) is shown in Figs. 2(a)-2(c), respectively. The functions were computed for intervals of ka of 0.05. Cases of shells having

iron inner layers of the same thicknesses as the above-mentioned

single-layer shells but with a thin Lucite covering layer (c/a - 1.01)

are shown in Fig. 3(a)-3(c), and with a thicker Lucite covering layer

(c/a - 1.10') are shown in Fig. 4(a)-4(c). In all plots the abscissa is ka

where k - 21r/K and A is the wavelength of sound in water. In all

cases as the iron thickness is increased, the curves exhibit similar

behavior at low ka (ka < 5). The first dip becomes less pronounced

and moves to a higher ka. Comparing igs. 2(a) and 3(a), Figs. 2(b)

and 3(b), or Figs. 2(c) and 3(c) would lead one to predict a decrease in

variation of the modulus as ka is increased. This is however, fallacious.

It can be noted that with a thicker Lucite outer layer, even for the case

of the thinnest shell whose reflection was computed |c/a - 1.10 and

b/a - 0.99, Fig. 4(a)], the modulus variation increased at higher
LAYERED CYLINDERS 131

O0 3 6 9 1;2 5

ke

ho

06

Fig. 2 - Theoretical calculations of the monostatic reflection expressed as MODULUS

(f- ) from a two-layered hollow cylindrical shell with vacuum inside (a) c/a I- 1.00

b/a - 0.99, (b) c/a 1.00, b/a - 0.97; and (c) c/a - 1.00, b/a - 0.95.

21 (a)

0 6 2 15 ho

I.

~Fig. 3 - Theoretical calculation of the monostatic reflection expressed as MODULUS ;r(11. 2) from a two-layered hollow cylindrical shell with vacuum inside (a) c/a - 1.01.

~b/a 0 - 0.99; _5 (b) c/a - 1.01, b/a - 0.97; and (c) c/a - 1.01, b/a - 0.95.
132 CHAPTER 5

.(a)

o 5 6 9 15

(c)

3 6 9 I Is

Fig. 4 - Theoretical calculation of the monostatic reflection expressed as MODULUS

(If,) from a two-layered hollow cylindrical shell with vacuum inside (a) c/a = 1.10,

b/a = 0.9k (b) c/a 1 .10, b/a = G.97 and (c) c/a 1.10, b/a 0.9S.

3-

Fig, 5 - Delineation of three regions for the calculation of the reflection from a twolayered hollow cylindrical shell with a vacuum inside (c/a = 1.01, b/a = 0.93).

i'9

Fig. 6 - Theoretical calculation of the monostatic reflection from a hollow cylindrical

Lucite shell with a vacuum inside (c/a - 1.10, b/a - 1.00).
LAYERED CYLINDERS 133

ii

Fig. 7 - Theoretical calculation of the monostatic reflection from a hollow cylindrical

iron shell with a vacuum inside (c/a - 1.10, b/a - 0.90).

92

Fig. 8 - Reflection by a solid Lucite cylinder.

ka (ka > 12) in the same range where only a small variation was

observed for a thinner Lucite covering [c/a = 1.01, Fig. 3(a)]. For a

thick outer Lucite layer (layer I in Fig. 1), increased variation of the

modulus resulted [Figs. 3(b) and 3(c)] over the same range of the ka

values (12 < ka < 15) for which little variation occurred for an outer

layer only one tenth as thick [Figs. 3(b) and 3(c)].

For many cases we can delineate three regions of the function

)f.(ka)l. Figure 5 illustrates these regions Region I is characterized

by a fairly regular slow variation with ka. Region II is an irregular or

mixed region. Region II tends to be regular or periodic and varying

faster with ka than region 1. These regions appear to be bounded

within a certain ka range that can be identified in many cases. Varying

the thickness ratios or complex moduli will change the ka delineation

of these regions and in some cases can eliminate one or more of the

regions.

By taking the appropriate limits in the matrix equations, these

solutions reduce to the single-layered shell or solid cylinder case. The

plot in Fig. 6 shows If.(ka)l for a Lucite shell without any other

material layer computed by taking b/a = 1.00 and c/a - 1.10. Similarly, a simple iron shell of the same thickness is considered by taking

b/a = 0.90 and c/a = 1.00. The resulting computation is shown in

Fig. 7. Allowing b/a - 0 and c/a - 1.00, a solid cylinder case is

approached. Figure 8 shows the computations for a solid Lucite

cylinder.
134 CHAPTER 5

REFERENCES

R. D. Doolittle and H. Uberall, "Sound Scattering by Elastic Cylindrical

Shells," J. Acoust Soc Am. 39, 272-275 (1966).

2. P. Ugincius and H. Uberall, "Creeping-Wave Analysis of Acoustic

Scattering by Elastic Cylindrical Shells," J. Acoust. Soc. Am. 43,

1025-1035 (1968).

3. Miquel C. Junger, "Sound Scattering by Thin Elastic Shells," J.

Acoust Soc. Am. 24, 366-373 (1952).

4. C. W. Horton, W. R. King, and K. J. Diercks, "Theoretical Analysis

of the Scattering of Short Acoustic Pulses by a Thin-Walled Metallic Cylinder in Water," J. Acoust. Soc. Am. 34, 1929-1932 (1962).

5. Richard H. Vogt, Lawrence Flax, Louis R. Dragonette, and Werner

G. Neubauer, "Monostatic Reflection of a Plane Wave From an

Absorbing Sphere," J. Acoust. Soc. Am. 57, 558-561 (1975).

6. Lawrence Flax and Janet P. Mason, "Fortran Subroutines to Evaluate, in Single or Double Precision, Bessel Functions of The First

and Second Kinds for Complex Arguments," Naval Research

Laboratory Report No. 7997, Washington, DC, 1976.

7. Bruce Hartman and Jacek Jarzynski, "Ultrasonic Hysteresis Absorption in Polymers," J. Appi. Phys. 43, 4304-4312 (1972).
LAYERED CYLINDERS 135

APPENDIX: MATRIX ELEMENTS

The following are the expressions for the elements of the determinants appearing in Eqs. (S) -0 1) in the text.

all - IJR(ZL) - 2GuJ,,'(ZL)J/(1 + 2G1 ),

a12 - [ Y.(ZL) - 2G, Y."(ZL)J/(l + 2Gl).

a 13 - - 2 n [Z, J,(Z,) - J,,(Z,) I/ Zs2

a 14 - - 2 n [Z, Y,,(Z 5) - Y (Z,) I/ Zs2,

a 21 - ZL J,,(ZL),

a22 - ZL Y. (ZL),

a23 = nJ,,(ZS),

a 24 - n Y.(ZS),

a31 - 2nIZLJl(ZL) - Jh,(ZL)I,

a32 - 2n fZL YM"(ZL) - Y Z j

a33 = (Zs)2J,;'(Zs) + n2J,(Zs) - ZJZ)

a-(ZS)2 3 y,$(ZS) + n1 y (ZZS)-

a41 - [J,,(tL) - 2G11,(IL)]/(1 + 2Gl),

a42 - ( Y,, L) - 2G I Y,,(IL) 1/(0 + 2 G1),

a 43 = 2 n [isJ,,(0 - .J,(ts) I/ ts,

a 4- 2 n f ty"( t, - y I)t2

a45 - (P2/PI) [J,(qL) - 2G 2J,,(qd)i/(1 + 2G 2),

a46 - (P2/P1) (J,,(qL) - 2GL Y,"(qL)I/(I + 2G 2 ),

a47 - - 2 n(p 2/p 1) tqsJ,(qs) - J,(qs) 1/qs

a48 - - 2n (plp 1) Iqs Y(qs) -Y, q 1S2

a52 - IL Y,(,L),

5 53 - nJ,(is),
136 CHAPTER 5

a5s - qLJ.(qL),

a56 - qL Y (qL).

a57 - nJ.(WS) -

a58 - nY q

a61 - 2n UL J. (IL - JR(L),

a62 - 2n1 :L Y.(tL) - Y(LI

a63 -IUP2J(tQ) + n 2

j. (t) IS,(s

- I:~~(t + 2 . 2 ~(t) IS

a65 - 2 Un fqL J.(qL) -,(

a6- 2 Un [qL Y.(qL) - Y. Wd~1

a67 - 2U[qs 2J."(qs) + n2j.(qs) -qsJ,,(qs)],

am- 2 U[ q

2 Y."(qs) + n' Y.(qs) - qs Y,(qs),

a71 - nJN(IL),

a72 - n YNOtL),

a73 - ti t)

a7 5 - nJ (qL),

a7 6 - n Y.(qL),

a77 - qJ'q)

a78 - qS Y. (qs),

aS3 - [J.(uL) - 2G 2J.,(uL)1( + 2G 2),

a&6 - (Y.(uL) - 2G2Y.'(uL)I1( + 2G2 ),

a87 - - 2n1(usJ,"(US) -j.(US)I/uS2,

as- - 2nIRus y(US) - 2~u)Iu,

as9 - - J 0 f

ag5 - 2n[(ULJ,(UL) -J.(UL)I,

a%- 2 n [(uL Y.,(UL) Yf(uL)J1, .

a97 - - USlj.( US) + n2j"(us) + US4J.(US),

a"- - us Y.(us) + n' Y.(us) + us2 Y'(us),
LAYERED CYLINDERS 137

a10,.5 - ULJ 8(UL),

a10,.6 - UL Y(UL),

a10 7 - 0.(US).

alo.- nY,(us),

alo.9 - uJ.(u),

The argument of the functions are given in Eq. (7) and, as usual, the

prime denotes derivative with respect to the argument.

=4

ii,
Chapter 6

NONABSORBING AND ABSORBING CYLINDERS*

INTRODUCTION

The scattering of a plane sound wave by an infinite elastic cylinder

has been examined by several investigators [1-31. In those papers, the

absorption of acoustic energy in the cylinder is assumed negligible.

Although this assumption is undoubtedly valid in many real situations,

there are, however, cases where absorption plays an important role.

Here an infinitely long cylinder is considered analytically that is composed of a material in which wave absorption may be ignored and also a

material in which both longitudinal and shear waves are significantly

attenuated by absorption.

The material of the cylinder is assumed to be solid and isotropic,

supporting compressional and shear waves with speeds CL and Cs,

respectively, and immersed in a fluid of sound speed cf.

The pressure Ps reflected by a cylinder is given by Eq. (9) of

Chapter 5 141.

' J,(ka)L, - kaJ;(ka)H'(2)(kr) ) Ps - Po H- Hn2 (ka )L, - kaHn2)' (ka cos (nO), (1

where Po is the incident acoustic pressure, J and H,,(') are the Bessel

function of the first kind and a Hankel function of the second kind,

respectively, and e. is the Neumann factor. Primes indicate a derivative with respect to the argument. The polar coordinates of the field

point are r and 0, a is the cylinder radius and k is the incident wave

number. Equation (1) can be specialized to the farfield by substituting

an asymptotic form for H,,2 (kr) for large kr. The farfield form function magnitude is then

IF.(ka, 0)1

2 P/ 2 2 a)L~-kaJ,(k) . kH'(ka) J(ka)L. cos (nO),

whreth (k:)/ H2M(ka)L, -kaH 2 . where the L. are defined by Eq. (6) of Chapter 5.'(ka)

*These results first appeared in: Luise S. Schuetz and Werner G. Neubauer, J. Acoust.

Soc. Am., 62, 513-517 (1977)
140 CHAPTER 6

Absorption has been included, by introducing complex shear and

compressional wave numbers in the solid as

ks M k(C/Cs)(l - ifs), and kL E-- k(cf/cL)(l - iPL).

The P's are absorption factors, with the dimensions of Np, derived

from the measurements of Hartmann and Jarzynski, 151 who reported

their measurements as absorption coefficients expressed as aLkL and

asks in dB. Conversion between these factors is accomplished according to the relations

PL - aLkL/ 40r log e and P, - asXs/40r log e.

The nonaborptive case is given by Ps - PL - 0.

The chief difficulty in this solution lies in the evaluation. Unexpected difficulties are encountered in calculating the Bessel functions

over the required wide range of complex arguments 161. Comparisons

were made with experimental monostatic and bistatic form functions

measured using lucite (absorbing) and aluminum (essentially nonabsorbing) cylinders.

EXPERIMENTS

The theory is two-dimensional, i.e., an infinite cylinder is

assumed which, of course, can not be realized in a real so-called free

field. However, reasonable agreement between two-dimensional theory

and experiments was shown in Chapter 4 using finite metallic cylinders

if reasonable care is exercised in performing the experiments. Planewave incidence demanded by the theoretical solution was approximated

using directional sources. Schlieren photographs were taken to allow a

preliminary examination of the entire acoustic field. Figure I shows

the dark-field schlieren observation for an aluminum cylinder, with the

magnitude of the theoretical bistatic form function, If(ka, 0)1, superimposed. The cylinder is seen end-on as a dark disk in the center and

the source is visible as a black rectangular area at the top of the picture.

A long incident pulse was used, and the photograph is taken at the

instant when the pulse is just leaving the cylinder. The theoretical

curve in the forward-scatter direction is not shown, since the theoretical

results do not include the incident field.

This photograph indicates good agreement between theory and

experiment at a distance of two or three diameters from the scatterer, r implying that the cylinder is placed in the plane-wave field of the

source.
CYLINDERS 141

Fig. I - Schlieren photograph of the field scattered by a aluminum cylinder (ka - 24 58) with the calculated bstatic form

function superimposed

A water-tank facility similar to the one described by Neubauer [71

was used for the quantitative experiments. Cylinders with a large ratio

of length to radius were insonified using directional sources. Conventional piezoelectric transducers with square or circular active areas were

used. In fact, some of the results at 200 kHz were obtained using a

"fish-finder" source available at boating equipment stores for little cost.

In Fig. 1, the cylinder was placed in the nearfield, i.e., in the

plane-wave field, of the source. This was impossible in the quantitative

experiments. Instead the cylinder was placed at a large enough distance

so that the center of the main lobe was a sufficient approximation to a

plane wave. Figure 2 shows typical beam patterns of sources that were

used. Their use resulted in an incident field which was approximately

plane over a section of the cylinder between 2.5 and 5 cm in length.

Transducers were suspended in the water from two movable

tracks. One of these held the source, on the other the receiver was

mounted from a rotating arm, with the cylinder suspended at its axis of

rotation. Source-to-cylinder distances ranged from 15 to 30 cylinder

diameters. A fine nichrome wire mounted in the upper end cap was

used to hang the cylinder.
142 CHAPTER 6

2 1.2

I-

.4

0 - I

40 so 120 160

ANGLE

Fig 2 - Typical measured beam patterns of

sources used in quantiutive experiments

The incident field was pulsed to reduce interfering reflections.

The steady-state results were in some cases derived from the pulsed

data by Fourier transforming short-pulse returns [81; in others they

were measured directly by approximating the steady state by a very long

pulse. A repeated harmonic pulse was generated by gating the output

of a synthesizer. A counter operating from the signal frequency was

used to trigger the gate, and the same trigger, with a delay, initiated

display of the received pulse. The resulting return was stable enough to

permit signal averaging over long periods of time.

None of the following data are absolute. All are normalized in

such a way that the averages of the theoretical and empirical curves are

equal.

NONABSORBING CYLINDERS

The facility permitted the taking of bistatic data in the backscatter

half-space. Figures 3-6 are plots of the scattering form function versus

scattering angle at various ka, for cylinders composed of 6061 aluminum. Data are taken using cylinders ranging in diameter from 0.90 to

2.54 cm with acoustic frequencies of f rom 0.2 to 1.2 MHz. Previous

work using aluminum spheres 18) at similar frequencies gave good

agreement between theory and experiment when the theory neglected

absorption. Therefore, it is reasonable to assume that at these frequencies aluminum is virtually nonabsorbing. For the theoretical curves

shown here that assumption is made. The theoretical curves are calculated taking the longitudinal and sheftr wave speeds of 6370 and 3120
CYLINDERS 

143

1,2

1.0

,8

Fig. 3 - Bistatic form func- _ lion of a 6061 aluminum " cylinder. ka - 10.83, c1473.39 mlsec. Theory

experiment. e.

.4

.2

100 120 140 160 180 0

1.2

1.0

.8

6 5 Fig. 4 - Bistatic form func- tion of a 6061 aluminum

cylinder, ka- 22,72, c,- 1470.00 m/sec. Theory,- 4 , experiment, 0.

2

100 120 140 180 ISO e
144 CHAPTER 6

1.2

1.0

.8

6 Fig. 5 Bistatic form func-

* tion of a 6061 aluminum

* cylinder, ka - 23.41. cf

1477.65 rn/sec. Theory.,-

.4 experiment, 0.

.2

100 120 140 160 ISO

1.2

1.0

Fig. 6 Bistatic form function of a 6061 aluminum

cylinder. ka - 31.31, cf - C?.6

1468.96 rn/sec. Theory,-; a:

experiment 1, x, experimentv

2. o.

.2

100 120 140 160 18
CYLINDERS 145

m/sec, respectively. These wave speeds are believed to be accurate to

within one percent. The sound speed in water was derived from Ref. 9.

As indicated above, the section of the cylinder which is approximately plane-wave illuminated is only about three diameters in length.

In view of this fact, the agreement between theory and experiment is

surprisingly good. The ratio of plane-wave illuminated length to

cylinder diameter decreases as ka increases in these experiments. It is,

therefore, not surprising to see a decrease in agreement with increasing

ka.

Figure 7 is a plot of the backscatter form function versus ka, also

using a cylinder of 6061 aluminum. The experimental curve is the

result of analysis of transient data processed as described in Chapter 8.

Essentially, the form function versus frequency is calculated by fast

Fourier transforming both a short incident pulse and the reflected

return. The form-function magnitude is then given by the normalized

ratio of the reflected to the incident spectrum.

1.2

1.0

.8

.6

.4

.2

I I I 1 I

16 18 20 22 24 : ka

Fig. 7 - Backscatter form function of a 6061 aluminum

cylinder. Theory, experiment, 0.

I4
146 CHAPTER 6

ABSORBING CYLINDERS

In view of the agreement obtained in the above cases, it was

assumed that the two-dimensional geometry of the theory had been

sufficiently approximated. Under the same experimental conditions

similar observations were made using lucite cylinders at ka of 4.30 and

10.90. In this case, longitudinal and shear wave absorption was

included for calculating the theoretical curves. The absorption factors

PL and As were taken equal to -0.00348 and -0.00531, respectively.

By far the greatest uncertainty in this case lies in the values of

wave speeds, which are difficult to measure, and which, furthermore,

have been found to vary greatly from one sample of plastic to the next.

For example, Refs. 5 and 6 arrive at shear wave speeds of 1380 m/sec

and 1340 m/sec, respectively, in two different samples of lucite. As

shown in Fig. 8, the theoretical form function is highly sensitive to

these velocities, so that an error of 1% or 2% completely changes the

character of the curve. With only a limited knowledge of the wave

speeds (within 3%) it was nevertheless possible to test the model in the

least-squares sense. Highly overdetermined systems were used, with

the wave speeds as parameters. In every case, the result was an

extremely good fit, with narrow confidence limits, and a small standard

deviation. Calculated wave speeds fell well within the expected range.

.7

.6

.5 I

./

.3 *

.2 \ / \'I

.1 ' \!

90 100 110 120 130 140 150 160 170 180

a

Fig. 8 - Variation of theoretical form function with small

differences in wave speeds for a lucite cylinder, ka - 10.90, Cf

- 1469.64 m/sec; cL - 2680.00 m/sec, cs - 1380.00 m/sec

-; c, - 2680.00 rn/sec, cs - 1350.00 m/sec -. - -

and CL - 2710.00 rn/sec, cs 1380.00 m/sec.
CYLINDERS 147

Figures 9 and 10 are the bistatic form functions of two different

lucite cylinders. Wave speeds obtained from the least-squares fit are

indicated in the captions. Both cases resulted in an excellent fit, with a

standard deviation of 0.02 in the first case, and 0.01 in the second case.

10

.8

_ .6

.4

2 J

80 100 120 140 160 180 e

Fig. 9 - Bistatic form function for a lucite cylinder,

ka - 4.30, cL - 2717.30 m/sec, cS - 1312.90 m/sec,

and cf- 1468.27 m/sec. Experiment -; theory 0.

.6

- .4

1 .2

I I I I

80 100 120 140 160 180

I,,

Fig. 10 - Bistatic form function for a lucite cylinder, ka -

10.90, cL - 2688.50 m/sec, cS - 1340.00 m/sec, and cf1468.72 m/sec. Experiment-; theory 0.
148 CHAPTER 6

The importance of the absorption in the theory is indicated by

Fig. 11. The two curves are identically calculated backscatter form

functions, except that one allows the absorption coefficient to go to

zero, and the other does not. In the latter case (solid curve) aL -

-0.00348 and as - -0.00531.

1.8-

16-

1.4- A

II

12 -1 -i~ , 1 0I r

0

S 1.0-I

I t

.4

2 I

4 5 6 7 8 9 10 11 12 13 14

ka

Fig. 11 - Effects of absorption on backscatter form function of a lucite cylinder;

- - - neglecting absorption (as - QL - 0) and - including absorption (as -

-0.00531 and aL - -0.00348).

CONCLUSIONS

In summary, we find that very good predictions can be made

theoretically of scattering from absorbing cylinders, provided the acoustic wave speeds and absorption factors are known to the required accuracy. In the case of lucite, in particular, these wave speeds are difficult

to measure, and widely variable from one sample to the next.

Furthermore, these results indicate that experiments using conventional sources can give good approximations of two-dimensional calculations.

I

12

I
CYLINDERS 149

REFERENCES

1. J. J. Faran, Jr., J. Acoust. Soc. Am. 23, 405-418 (1951).

2. C. W. Horton, W. R. King, and K. J. Diercks, J. Acoust. Soc.

Am. 34, 1929-1932 (1962).

3. William W. Ryan, Jr., Applied Research Laboratory, University of

Texas Report ARL TR-73-20, Austin, TX (August 1973) (unpublished).

4. Lawrence Flax and Werner G. Neubauer, J. Acoust. Soc. Am.

61, 307-312 (1977).

5. Bruce Hartmann and Jacek Jarzynski, J. Appl. Phys. 43, 4304

(1972).

6. Lawrence Flax and Janet P. Mason, NRL Report 7997, Washington, DC (April 1976) (unpublished).

7. W. G. Neubauer, NRL Report 5688, Washington, DC (October

1961) (unpublished).

8. Louis R. Dragonette, Richard H. Vogt, Lawrence Flax, and

Werner G. Neubauer, J. Acoust. Soc. Am. 55, 1130-1137 (1974).

9. Werner G. Neubauer and Louis R. Dragonette, J. Acoust. Soc.

Am. 36, 1685-1690 (1964).
Chapter 7

A CYLINDRICAL CAVITY IN AN

ABSORPTIVE MEDIUM*

INTRODUCTION

Formulation and computation of elastic wave reflections by a

cylindrical cavity in a solid homogeneous isotropic medium with no

losses has been given by Lewis et al., [1) among others 12-41. Several

materials have been considered as the wave-supporting medium.

Among these is polyethylene, a material known [5] to have the

property of significant attenuation of both compressional and shear

waves. In the range of low megahertz frequencies most applications

would indeed appropriately ignore absorption in the ambient medium

for the case of most metals. However, grainy metals such as brass can

have extraordinary losses and thus can cause problems in some cases.

If polymers and other nonmetals are considered, wave attenuations

become significant and should be included in the theory. Although a

very large number of materials were not considered, a sufficient

number were considered by Lewis et al. that it seems that materials fall

into two rather arbitrary but convenient categories depending on the

behavior of the total scattering cross section (SCS) versus ka curves.

In this paper k will designate the wave number of the longitudinal

wave, K that of the shear wave, and a is the cavity radius. In their

work some materials approach a limit of SCS with increasing wave

parameter (ka, Ka) from above and some from below when no losses

are considered. The plots of Fig. 1 distinguish those two categories.

The ordinate labeled SCS is the total scattering cross section following,

but deviating slightly from, the definition of White 121. He defines the

total scattering cross section as the total scattered power per unit length

of cylinder divided by the intensity. We imposed the additional modification used by Lewis et al. [11 of normalizing to the cylinder radius by

simply dividing by 2a. The upper curve in Fig. I is computed for

polyethylene, here considered lossiess, approaching a limit from above

with increasing ka of SCS slightly above 2. This case also has a peak at

'These computations first appear in: Lawrence Flax and Werner G. Neubauer, J. Acoust.

Soc. Am. 63, 675-680 (1978).

151
152 CHAPTER 7

4

0

/ 0 " 1 I 

0 2 4 6 6 10

km

Fig. I - A comparison of the scattering cross section (SCS) of a cylindrical cavity

for an incident compressional wave as a function of ka for PMM and PE without attenuation.

low la. Lucite is the material for which the lower curve was computed

and approaches the limiting value of SCS from below with increasing

ka. The parameter that characterizes the scattering-cross-sectionfunction behavior is the ratio of compressional-wave speed cc to shearwave speed c5 . The "critical value" for that ratio (cc/c) which determines to which of the two classes a particular material belongs is very

close to 2. Examples of metals [11 that belong to the same class as

lucite are beryllium (cc/c, - 1.45), silver (1.64), and tungsten (1.82)

and ones that fall in the same category as polyethylene are aluminum

(2.1) and gold (2.7). A thorough parametric study of all possible case

was not attempted; rather in this study samples of Lucite (a brand of

polymethyl methacrylate PMM) were chosen as an example to calculate

as a member of one category and having significant, but small, wave attenuations. Polyethylene (PE) was chosen as the member of the

other category but is known to have significantly greater wave attenuations than PMM.

NOTATION

The elastic material containing the cylindrical cavity will support

shear and compressional waves and both will be considered as incident

waves and mode conversion on reflection will produce shear from

compressional as well as shear, and vice versa. Also, computations

ignoring and including absorption will be presented. Hence, a word

* about notation is in order. Scattering cross sections of separate waves

will be designated with a q. A first subscript will designate the kind of

incident wave, c for compressional and s for shear. A second subscript,

again c or s, will designate the reflected wave type. Whereas an

L
CYLINDRICAL CAVITY 153

unprimed q designates a computation ignoring wave attenuations,

primes designate the same computations including attenuations. The

physical constants used in the computations are given in Table i.

Table 1 - Material Properties

PMMn PE

cC (m/s) 2680 19500

c. (m/s) 1350 540 b

cc/C' 1.94 3.61b

P, (Np) 0.0035 0.007V

P, (Np) 0.0053 0.0212 c

asee Ref. 7

bSee Ref. I

CSee Ref. 5

THEORY

Consider a cylindrical cavity of radius a embedded in an isotropic

absorptive medium characterized by density p and speeds cc and c, as

determined earlier. The axis of the cylindrical cavity is taken to be the

z axis of the cylindrical coordinate system (r, 0, z). Both types of

incident waves of circular frequency ow are considered, i.e., compressional waves and shear waves polarized in a direction normal to the

cylindrical axis.

Compressional-Wave Incidence

In the solid medium the incident plane acoustic wave pi =

Po exp(-ikcx) can be expanded in cylindrical coordinates as

P - P0 I e.(-i)'J,(kcr) cos (n46). (la)

R-0

The scattered pressure is

A - P0 e. C, H2(kcr) cos (n46) (Ib)

00

+ Po SE. B,, H,'21(kr) cos (n4)).

The incident pressure amplitude is Po, e, is the Neumann factor, k, -

w/qc, k, - wv/c, H (

1

) - J. - iY, where J and Y,, are Bessel functions

of the first and second kind. The H.2) are Hankel functions, and C.

II
154 CHAPTER 7

and B. are the scattering coefficients. Throughout the time dependence

exp(iOt) has been suppressed.

At the boundary of the cavity r - a, the tangential stresses must

be zero. The coefficients C. and Bn are determined from the boundary

conditions and are given by a division of two second-order determinates

of the form

all a13

C. a31 a33

a1 2 -a 1 3

a32 -a 33

(2)

a12 -all

B a32 -a 31

a12 -a 13

a32 -a 31

The elements of the determinants are

all - [J.(Zc) - 2GJ,'(Z,)I/(1 + 2G),

a12 - [H12)(ZC) - 2GH,,"2

)(Zc)1/(1 + 2G),

a 13 - - (2n/Z,2)[ZH12)(Zs) - H(2)(Z,)],

a 14 - - (2n/Z,2)[ZJ,(Zs) - J,(Zs)], (3)

a31 - 2n[ZcJ,(Zc) - Jn(Zc)],

a32 - 2n[ ZH.(2)(ZC) - H.2)(ZC)1.

a33 - '2nH'(2)(Z) - ZSH(2)(Zs) + n2H,(2)(Z,).

a34 - Z2J..(Z) - Z3J1(Z,) + n2j.(Z,).

The arguments of the function appearing in these expressions are determined from the following relationships:

Zc kca - ickc a,

(4)

Z,- ka ip,k,a.

where p's are absorption factors given in nepers.

In addition, ratios of complex wave numbers appear in some of

the matrix elements and are given by

G - z,/(z,- 2Z 2 ). (5)
CYLINDRICAL CAVITY 155

As a result of using complex wave numbers in the theory, the elements

of the determinants defining C, and B, contain Bessel functions with

complex arguments. Techniques developed by Flax and Mason [61 for

calculating these functions were used.

Dimensionless scattering cross sections, as defined in the Introduction are

q,= E 1JCI 2IH,, 2

)(Zcr/a)12 , ft-O

qE. = 0O (6) == J B. J2jH'(2)(Z,,r/a) 12.

n-O

The nondimensional parameter r/a is the distance from the cavity axis.

For large ratios of r/a, the asymptotic expression for H. 2) is used:

H (2)(Z) = (2/rZ) /2 exp[- i(Z - n rl2 - 7r/4)], (7)

where Z - x + iy.

Shear-Wave Incidence

For the case of an incident shear wave the same procedure was

used as for the compressional case. The coefficients D. and E. are

again determined from the boundary conditions and are given by the

quotient of two second-order determinants of the form:

a13 a14

a33 a34

a12 -a13

a32 -a 33

(8)

a12 a14

= 3a2 a34

a 12 -a 13

a 3 2 -a 3 3

Similarly, the normalized cross sections for monostatic reflectors

are found to be

q. _ j .EI D" 121 H"21 (Zr/o) 12,

(9)

- tE.I E IH21 (Zr/a) I2.

- --"-
-7--

156 CHAPTER 7

COMPUTATIONAL RESULTS

In all calculations an assumed value of the range at which the field

is computed is taken to be 10 radii of the cylindrical hole, i.e., r/a -

10. This was taken to be the farfield as defined by a comparison of a

computation using an asymptotic form of the Hankel function for that r

and an equivalent explicit evaluation.

The scattering cross sections for compressional-wave incident on a

cavity in PMM and a resultant scattered compressional wave is shown

versus ka in Fig. 2 for calculations ignoring and including attenuation.

In all plots, the wave number plotted on the abscissa is that of the

incident wave. It can be seen in Fig. 2 that for a hole diameter slightly

larger than one incident wavelength (ka - 10), the SCS attributed to a

compressional wave with attenuation is approximately one-half of the

SCS in PMM when attenuation is ignored. The comparable diminution

in SCS because of shear reflection, generated by mode conversion from

the incident compressional wave is shown in Fig. 3. The SCS for a

compressional-wave incident and a combined result of compressional

and shear reflection from the cylindrical hole is shown in Fig. 4 again

for PMM. An interesting behavior of the computation near a ka of

unity is apparent when the curves labeled q and qcs are plotted

together as in Fig. 5. A decreased value or "dip" in the curve for q is

compensated for by an increased value or "bump" in the curve for q,, to

result in a combined smooth curve in that region in the plot of qcc +

qc,.

PMM

1/I I __________

O2 4 6 O1 ke

Fi, 2 - A comparison of the compressional component of the SCS for PMM as a

function of ka with and without wave attenuation for compressional-wave incidence
CYLINDRICAL CAVITY 157

PMM

0 qa I I

0 2 4 6e 6 10

Fig. 3 - A comparison of the shear component of the SCS for PMM as a function

of ka with and without wave attenuation for compressional-wave incidence.

PMM

3-

0 2 4 ka 16 10

Fig. 4 - A comparison of the net SCS for PMM as a function of ka with

and without attenuation for compressional-wave incidence.

PMM

4

2 k

Fi - The net ond components of the SCS for PMM as a function of ka

with wave attenuation for compressional-wave incidence.
158 CHAPTER 7

The results of similar computations are shown in Figs. 6 and 7 for

PE which has, as can be seen in Table 1, somewhat lower wave speeds,

double the compressional-wave attenuation and approximately four

times the shear-wave attenuation. The contribution to the total SCS by

the shear wave generated from mode conversion of a compressional

wave incident on the cavity is very small for all values of ka. This is

apparent from the similarities of the differences between each pair of

curves in Figs. 6 and 7. The importance of including absorption in the

calculations is apparent from the difference in magnitude of SCS

between the computations with and without attenuation. Curves similar to those shown in Fig. I which seem to approach a limiting value of

SCS when no attenuation was included are shown in Fig. 8 for computations including attenuation. The waves in the two categories referred

to previously do not approach a single value. They cross over at a

value of ka between 3 and 4 for these materials.

PE

4'

qCC

2

01 1 _ _ _L __ 1

0 2 4 ka $ I 10

Fig. 6 - A comparison of the compressional component of the SCS and PE as a

function of ka with and without wave attenuation for compressional-wave incidence.

I

PE

2-

II a ___ _ -- ---- q - I- - -

ok2 4 heI I0

Fig. 7 - A comparison of the net SCS for PE as a function of ka with

and without wave attenuations for compresstonal-wave incidence
CYLINDRICAL CAVITY 159

6

4

2 PMM

IPE qcc+ qc

0 2 4 IS 8 10

Fig. 8 - A comparison of the SCS of a cylindrical cavity for an incident compressional

wave as a function of ka for PMM and PE with attenuation.

Computations of SCS similar to those for an incident compressional wave on the cavity were carried out for an incident shear wave.

The results of those computations are plotted in Figs. 9 and 10 for

PMM and PE, respectively. The plots of Fig. 9 show a curious result.

The computations of SCS including wave attenuation results in a magnitude which is, over a small ka range near unity, larger than that not

including attenuation. No explanation is known for the behavior at

present. It should be mentioned that for PE not only is the mode

conversion a negligible contribution to the SCS, as suggested by others,

[1] but the total shear-wave contribution is negligible for K a above 8.

PMM

4

2

: 1

0 2 4 Ka a 1 10

Fig. 9 - A comparison of the net SCS for PMM as a function of ea with

and without wave attenuations for shear-wave incidence.
160 CHAPTER 7

5 PE

4

2 -

2-- qss + qsc

q'ss+ q'sc

0 2 4 6 10

Fig. 10 - A comparison of the net SCS for PE as a function of Ka with

and without wave attenuations for shear-wave incidence.

A caution in conceptualization of the problem treated here is in

order. The scattering of waves by a cavity in a solid does not really

change locally at the cavity. The apparent scattering behavior appears different as expressed by the function of SCS versus ka or Ka because

the waves, compressional and shear, are attenuated in the course of

propagation to a point in the field. Therefore, it is probably more useful to consider the hole and a surrounding elastic medium out to a

given radius as a total system whose characteristics are sought.

REFERENCES

I. T.S. Lewis, D.W. Kraft, and N. Horn, J. AppI. Phys. 47, 1795-1798

(1976).

2. R.M. White, J. Acoust. Soc. Am. 30, 771-785 (1958).

3. R. Truell, C. Elbaum, and B. Chick, Ultrasonic Methods in Solid

State Physics (Academic, New York, 1969).

4. C.F. Ying and R. Truell, J. Appl. Phys. 27, 1086-1097 (1956).

5 B. Hartman and J. Jarzynski, . Appl. Phys. 43, 4304-4312 (1972).

6. L. Flax and J.P. Mason, "Fortran Subroutines to Evaluate, in Single

or Double Precision, Bessel Functions of the First and Second Kind

for Complex Arguments," Naval Research Laboratory Report No.

7997, Washington, DC (1976).

7. L. Flax and W.G. Neubauer, J. Acoust. Soc. Am. 61, 307-312

(1977).
Chapter 8

ELASTIC SPHERES -

STEADY-STATE SIGNALS

INTRODUCTION

The closest approach of experiment to theory in a reflection situation would be expected to be for a problem of finite elastic geometry

for which the wave equation is separable and for which all conditions

are known. In geometries that are not finite, such as cylindrical

geometry, the approximation of the experiment would be in question,

i.e., the lack of experimental satisfaction of the infinite dimension

inherent in the solution of cylindrical geometry, or the approximations

of the theory would be in question, i.e., the lack of theoretical validity

of approximating the finite length of the cylinder. The problem of

reflection from a sphere is the simplest problem of finite geometry suitable for defining the methods and accuracy of the system and methods

used for measuring reflection. Meaningful definitive measurements of

reflected pressure cannot be expected to be any better for other

reflectors than the correspondence that can be achieved for such a

well-defined analytically calculable problem.

The free-field backscattered acoustic pressure amplitude was

measured in the farfield for aluminum and tungsten -carbide spheres

immersed in water. These were compared with the wave harmonic

steady-state computer solutions incorporating the conditions of the

experiments. A close correspondence between theory and experiment

was sought for the relatively simple case of the sphere as well as a

determination of the effect caused by specific experimental parameters,

such as temperature and material wave speeds, on achieving agreement.

This was necessary so that the experimental facilities and methods

could be used to obtain measurements having known limits of error or

reasonably estimated ones for cases in which little or no guiding theory

is available.

161
162 CHAPTER 8

The scattering of acoustic waves by a solid elastic sphere in water

was described theoretically and experimentally by others 11-31. Acoustic scattering by a solid, homogeneous, isotropic, elastic sphere

immersed in a uniform, inviscid, elastic fluid was described quantitatively by Faran [11 before the advent of digital computers for the case

of a plane-harmonic-incident wave. Hampton and McKinney [2] made

an experimental study in the absence of theoretical computation of the

scattering from solid metal spheres in water, for pulses of sine waves

which were both of long and short duration compared to the size of the

sphere. Hickling [31 obtained expressions describing the backscattered

field by the use of a digital computer for incident sinusoidal pulses,

either spherical or plane, and obtained computer solutions for several

sphere materials. Numerical evaluations of the formulas describing

monostatic scattering of harmonic waves used to compare the measurements discussed below with the theory were made by Rudgers [4].

The elements of the theory for the reflection by a sphere are

reproduced here for the purposes of definition. Let the pressure of a

plane wave traveling in the direction of the positive z-axis and incident

on the sphere centered at the origin be

Pi = Po exp i(Wot - kz)],

where k is the wavenumber and &w the angular frequency. The pressure

at a point outside the sphere is the sum of the incident pressure field pi

and a scattered pressure field Ps,

P = P + Ps,

where the scattered pressure amplitude may be written

p, = (poa/2r) exp [i(tot - kr)lfr,

where a is the radius of the sphere, r the distance from the origin to

the field point and f, is called the reflection "form-function." That

function is a complex quantity whose modulus is related to the experimental observations by the expression

IfrI = (2r/a)(1ps l/p0),

where Po is the measured incident pressure amplitude at the origin in

the absence of the sphere and IPj the observed scattered pressure

amplitude.

A sound pulse was used for the measurements, an approximation

to the steady-state amplitude being achieved by making the length of

the pulse train many sphere diameters and appropriately interpreting a

steady-state amplitude at a specific position within the scattered pulse.
SPHERES-STEADY STATE 163

In the monostatic case (scattering at 1800, or backscattering) the

form function is given by [31

fr " - 2(kr/ka)eikr (-i)+1hn(2)(kr)(-1)(2n 0 + 1) sin 7),e ' ,

n-O

where h, 2)(kr) is a spherical Hankel function and ?,, the phase angle of

the n th partial wave. Expressions for qi, are given in Refs. I and 4.

For large distances from the sphere, use of the asymptotic expressions for h, 2

)(kr) gives the farfield form function in the monostatic

case,

- (2/ka) , (-1)n(2n + 1) sin r/,e'n"

n-O

In order to compare the measurements with the theoretical calculations, the reflected pressure amplitude was observed at a distance of

10 or more sphere diameters from the sphere. At that distance the

measured results correspond to the farfield form function in the range

of frequency parameter ka from 0 to 30.

EXPERIMENTAL METHOD

The experimentally determined modulus of the farfield form

function I.f1J was obtained from the relative pressure amplitudes Po

and 1pL1 of the incident and scattered waves, respectively, measured

with the same equipment. The incident pressure was determined at the

frequencies of interest before and after each run with the receiver at

the position normally occupied by the center of the sphere, and the

sphere removed. Usually the size of the sphere was held constant and

the frequency was varied during a run over a range of ka of approxi- mately unity. Then the sphere was changed to be a different size and

the same range of frequency was covered to achieve a different range of

ka. The steady-state condition was approximated by a burst of sine

waves 300 to 800 usec in duration.

It was found necessary to enhance the signal-to-noise ratio of the reflected signal. For that purpose, the box-car integrator was used to

scan and record on a strip chart recorder, five cycles of the sine wave

located at a position in the pulse judged to be a true representation of

the steady state. The average of the 10 peak values so obtained gave

the peak pressure amplitude. The sphere was then removed and the

box-car integrator continued scanning to record background. If a

coherent background was present, a correction for its effect was
164 CHAPTERS8

estimated from the relationship between its amplitude and phase and

that of the total pressure wave measured with the sphere in place.

The pressure wavefront incident on the sphere was slightly distorted by the presence of the receiver situated on the straight line

between the transmitter and the sphere. The result was to reduce the

incident pressure at the sphere by a few percent. When the receiver

replaced the sphere to measure the incident pressure Po a dummy

receiver was placed in the position normally occupied by the receiver

when the scattered pressure Los! was measured and the result was compared with that obtained with no dummy probe. In practice, the pressure pulse incident on the sphere was usually observed with no dummy

probe and the experimentally determined appropriate correction (of the

order of 2%-4%) was applied.

EXPERIMENTAL EQUIPMENT

The water-tank facility is similar to the one described by Neubauer [51. A cypress-wood tank containing a volume of water 12 ft x 6

ft x 6 ft is surrounded by a platform which supports the electronic

equipment and serves as a working area. The temperature of the water

in the tank was measured at six points at different depths and was held

constant to within 0.1 0C for the duration of each measurement. A residual chlorine level between 1 and 4 ppm was maintained by the addition of sodium hypochiorite solution to the water to keep its optical

clarity at a level required to measure distances optically in the water.

The source, receiver, and scattering sphere were placed along a

horizontal line parallel to the long dimension of the tank and held in

position by a system of tracks, rails,, and carriages mounted over the

tank. The placement system made longitudinal, lateral, rotational, and

elevation movements possible. All positioning of the elements was

done by hand. Final determinations of placement and all lateral and

longitudinal positions of objects in the tank were made by optical

squares which moved on precision flat surfaces located on support

beams above two adjacent sides of the tank. The vertical location was

fixed by placing all objects of interest in the same horizontal plane,

determined by a level telescope sighting through a porthole at one end

of the tank.

The aluminum spheres used for the scattering measurements were

precision machined in a range of sizes from 0.25 to 2.5 in. in diameter,

known to within 0.0005 in., and were suspended in the water from one

of the carriages by nichrome wire 0.005 in. in diameter attached to the
SPHERES-STEADY STATE 165

sphere with epoxy resin in a radial hole at the surface about -in.

deep and 0.009 in. in diameter. The tungsten-carbide spheres were

commercially available sizing balls suspended in a similar manner.

The hydrophone was similar to the piezoceramic device described

by Neubauer [5]. The disk, 0.125 in. in diameter by 0.010 in. thick,

was contained in a waterproof housing, covered by a unicellular rubber

jacket and the whole assembly held in place by a titanium tube 0.063 in.

in diameter extending from the receiver assembly to a connector above

the surface of the water. The tube also acted as an electrical shield for

the signal lead. The projectors used were circular pistons from 1 to 3

in. in diameter.

Sine waves were generated by a synthesizer and monitored by a

counter. The frequency was divided by a preset counter determining

the repetition rate of the acoustical pulse by synchronizing the pulse

generator which provides the rectangular modulating pulse for the gate

and also the timing pulse for the receiving oscilloscope. The acoustical

signal received by the pressure sensing probe was amplified, filtered,

and displayed on an oscilloscope. A box-car integrator was operated in

the scan mode to extract the repetitive signal from the background

noise and its output was recorded with a strip chart recorder.

RESULTS AND DISCUSSION

Aluminum

The curve in Fig. 1 shows the modulus of the farfield reflection

form function as a function of ka for an aluminum sphere in water.

The curve starts at the upper part of the figure and is continued below.

The experimentally determined points, shown by open circles, are backscattered relative pressure amplitudes normalized in amplitude. It was

of greatest interest to make the measurements in regions where the

form function is rapidly varying, especially near the sharp minima in

the curve. Calculations showed close correspondence between the frequencies of free vibration of the elastic sphere and the frequencies at

which the minima occur in the form function.

Faran 16] noted and verified experimentally for cylinders that in

some materials nulls in the backscattered pressure amplitude are close

to the frequencies of free vibration of the elastic cylinders and spheres.

In Faran's work, the distribution in angle of the scattering was observed

and computed for relatively few selected values of the frequency. In
166 CHAPTER 8

z 5.0 6.0 7.0 8.0 9.0 I0.0 1.0 120

S1.5

0

1.01

Z

0

0.5-

w

W 13.0 14.0 15.0 16.0 170 8.0 90 260

o 1.5

0

SPHERE SIZE-PARAMETER (ko)

Fig. I - Comparison between the exact theoretical calculation, of the form function

vs ka (solid curve) for an aluminum (110O0) sphere. and experimental measurement

(pointsD

this work, only backscattering was observed and the monostatic form

function was computed for a range of frequency parameters from 0 to

27.5 for aluminum and to 22.5 for tungsten-carbide, thereby showing in

more detail the proximity of the resonant frequencies of the free elastic

sphere to changes in the shape of the monostatic form function.

Faran noted that the ni th term in the partial wave sries expansion for the scattered pressure has a term with inftnites at the frequencies of the free vibrations of the scattering body, which satisfy the conditions of symmetry of the problem. In the case of the cylinder, the

infinities coincide with the zeros of the secular equation giving the

vibrational frequencies. The same conclusion holds in the spherical

case for a class of vibrations called the spheroidal modes. The resonant

frequencies of the spheriodai oscillations were computed by solving

S numerically the equation given by Sato and Usami 171. The spheroidal

modes are herein designated S,1 (there are various conventions), where

':n corresponds to the n th term in the partial wave series expansion of

the pressure and ! is the overtone with ! corresponding to the fundamental.

iD

Ia 0 . iV 0
SPHERES-STEADY STATE 167

Tables I and 2 list, respectively, the computed frequency parameters of the spheroidal modes, S,,, of' interest for the measurements for

aluminum spheres in water at 19.3°C and tungsten-carbide spheres in

water at 20'C. The frequency parameter is given by 2frva/c, where P,

is the frequency of the spheroidal vibration, c is the speed of sound in

water at the temperature of the measurements, and a is the radius of

the sphere. The resonant mode of lowest frequency is seen to be the

oblate-prolate mode S21, as is well known. Mode charts showing the

nondimensional frequency of vibration of the modes as a function of

Poisson's ratio are given by Faran, and more extensively by Fraser and

Table I - Frequencies Expressed in ka of Spherical Modes,

S,,, of an Aluminum Sphere at 19.3°C

n\I I 2 3 4 5

0 11.994 26.355 .........

I ... 7.616 15.320 18.380 22.629

2 5.588 10.792 18.229 23.552 ...

3 8.334 14.187 21.090 27.536 ..

4 10.696 17.564 23.949

5 12.907 20.831 26.832 ..

6 15.043 23.954 ..

7 17.137 26.930 . . . ...

8 19.204 .....

9 21.254 ........

10 23 290 ........

i1 25.316 ......

12 27.335 ...

Table 2 - Frequencies Expressed in ka of

Spherical Modes, S,i of a Tungsten Carbide

Sphere at 20.0°C

n\ I 1 2 3 4

0 11.213 27.604

I ... 9.312 18.019 21.800

2 7.431 13.266 22.752 ...

3 10.983 17.708 ......

4 14.015 22.235 ......

5 16.857 ... ......

6 19.611 ... ......

7 22.317 ...
168 CHAPTER 8

LeCraw [81. In the references, the frequencies are normalized to either

the longitudinal or shear sound speed in the material of the scattering

body. Tables of vibrational frequencies for the incompressible case and

for Poisson's ratio of 0.25 are given by Sato and Usami [71. The vibrations of the free elastic sphere are discussed in Refs. 7-12.

The free vibration of the sphere and the form-function minima

are sensitive to changes in the shear wave speed of the sphere material

and not very sensitive to changes in the longitudinal wave speed. Shear

and longitudinal wave speeds were measured on cylindrical samples of

the same piece of aluminum of which the spheres were fabricated. The

measured values were 3100 ±30 and 6370 ±40 in/sec for shear and

longitudinal waves, respectively.

Both shear and longitudinal wave speeds were varied within the

region of experimental error, until the computed frequencies of free

oscillation of the sphere best corresponded to the experimental minima

in Fig. 1, and these velocities were used in computing the form function. As noted by Faran [61, the position of a minimum or other prominent feature in the form function would be expected to show a shift

from the frequency of free vibration of the sphere, but in the case of

solids with low losses and with densities significantly greater than that

of the surrounding fluid, the frequency shift is small.

The open circles in Fig. I are the results of amplitude

measurements made with an oscilloscope, and the solid circles were

obtained from recordings with the box-car integrator. Both are absolute

mea- surenients and not normalized in amplitude. The points near the

minima at ka equal to 14.1 and 20.8 are based on the most complete

set of measurements and correspond to resonant frequencies of oscillation of the free sphere to within about 0.5%. The measured shear

speed in aluminum was increased by 20 rn/sec, within the known possible error of its independent measurement, to bring the calculated curve

into the agreement with the observed results shown at the two minima

near ka equal to 14.1 and 20.8. There is still some disagreement at ka

of 24. It was found that a 1% change in longitudinal sound speed had

less than one-tenth of the effect of a 1% change in shear wave speed in

shifting the calculated position of the vibration frequencies at ka values

14.1 and 20.8. The wave speeds used for the calculated curve of Fig. I

were 3120 and 6370 rn/sec, respectively, for shear and longitudinal

waves in aluminum and 1480.08 rn/sec for the wave speed in water at

I 9.3*C.
SPHERES-STEADY STATE 169

Figure 2 shows an incident pulse, of 500-Msec duration, and

pulses reflected by an aluminum sphere recorded by the scanning boxcar integrator. The time scale is increasing from left to right. At a ka

of value of 20.78 (Fig.2(b)), the form function has a minimum from

which it rises steeply to a maximum at ka of 21.21 (Fig. 2(c)). In this

case, both reflected pulses show the effect of the elastic vibrations and

the pulses in Figs. 2(a) and 2(b) have a duration considerably greater

than that of the incident pulse.

(a)

(b)

Fig. 2 - (a) A 500-,sec-long incident pulse. The pulse reflected by an aluminum

sphere when the pulse in Fig. 2(a) is incident for (b) ka - 20 78, and (c) ka -

21.21.

In recording the data, only three to five cycles were recorded for

the measurement of the pressure amplitude instead of the complete

pulse as shown. In the case shown, the five cycles would have been

located about 300 jusec from the beginning of the pulse.

Tungsten-Carbide

The curve in Fig. 3 shows the computed farfield form function of

a tungsten-carbide sphere in water at 20*C. The values of the shear

and longitudinal wave speeds in tungsten-carbide used for the computation were 4150 and 6860 m/sec, respectively, as estimated from the
170 CHAPTER 8

1 '5- 05-

I--

0

0 to 2.0 0 40 50 0 .0

', o.

- 05

_J

0 15

0

O J60 17o 180 10 20o 210 22 0

SPHERE SIZE-PARAMETER (w(I)

Vi-t 3 - (omparton hbeween esC% t iheoretical ca|cultion. f Ihc iorm-funtion %,

ka (%oid .ure) for a lungscen carbode sphere. 4nd experimental measuremcnt

(poilntO)

results of Balashov and Voronov (131. No separate sample of the identical material of which the sizing balls were made was available for

independent measurement of wave speeds in tungsten-carbide. The

measured points are Liven as open and solid circles and were obtained

as described for the case of aluminum. Except for the presence of

sharp minima and narrow regions where it rises abruptly, the curve

resembles the form function of the rigid sphere. The minima and

sharp rises in the curve correspond closely to the calculated modes of

oscillation of the free tungsten-carbide sphere, usually to within a few

tenths of a percent in ko. Figure 3 shows the experimentally determined minima slightly to the right of the computed ones. Since they

are close to the frequencies of free oscillation, which are most strongly

dependent on the shear speed of waves in the tungsten-carbide, the

shear speed was adjusted to bring the computed curve into better agreement with the measurements. The effect on the calculation of increasing the shear speed by 35 m/sec in the sphere material is shown in Fig.

4. This measurement is a reasonable value that is within the limits of

probable error for shear wave speed in tungsten-carbide.

I
SPHERES-STEADY STATE 171

10 - G

z 0 0 20 30 40 50 60 70

I

• 15,

U1 05. J

so 90 o00 10 1 o i 0 14, u_0

G 05-

1C 16C, 17 IF i 20C 21 22'

SPHERE 'F PARAME ER ka

1lig 4 - 1he same to.-mparion gjen in Via 3. here Ihe 'heoreiall LurC has been

shilied. due to d 3S m/,ck upard ddllusment in the iear 'e1hlmi, in tungsten

carhide

In Fig. 5 the scattered pulse with carrier frequency, corresponding

to a resonance ai kO equal to 7 3 (Fig, 5(b)), is compared with the scattered pulse corresponding to ka of 8 4 (Fig 5(c)) and the incident

pulse (Fig. (a)) The pulses were recorded by %canning the periodically recurring signal with the bo)x-car integrator The amplitude of the

incident pulse, shown in Fig. 5(a), is not related to that of the reflected

pulses. the length of the incident pulse shown in the figure is about

300 sec and the diameters of the spheres used for scattering at ka of

7 36 and 8 46 were 22/32 and 25/32 in . respectivelv The shape of the

pulse scattered at a ka of 7 .16 (Fig 5(b)) is observed where the amplitude of the form function has a iharp minimum, corresponding in this

case to a resonant frequenc of vibration of the free elastic sphere The

form function is changing rapidls in amplitude and phase and it can he

seen thai the reflected pulse is considerabhl longer than ihe incident

pulse as the sphere vontinues to oscillate and return energ) to the

detector
172 CHAPrTER I

(C)

1 ,- 0A) A )-ms"-oIis incidenI pulse The pulse reflect d h, A tungienLrhmdk iphere when the pulse in -ili '(d) IS incidlen lor (hi - - -3.. and I.) ka

The pulse scattered at a ka of 846 (Fig 5(c)) corresponds to a

region where the form function is not changing rapidi) and is far from

the nearest resonance The shape of the scattered pulse is similar to

that of the incident pulse with little energy going into the elastic vibrations of the sphere

Elfef e1 Temperawre

Figure 6 shows the form function of aluminum measured near the

minimum at ka of 20.8 The measurements, shown with open circles

and error bars, were made at 17 90 and 20.9°, and the solid curves

show the form functions calculated at those two temperatures The

measurements show a shift of approximately -0.2% per *C in the

abscism of the minimum, in agreement with the shift in the computed

curves This is accounted for, as was expected, by the change in the

speed of sound with the temperature of the water. the only parameter

changed in the calculation

From the experimental results in this case. it appears that the

abissa of the minimum near ka of 20 8 can he estimated to the

nearest 0 05 As previously mentioned, it was found that the salues of
SPHERES-STEADY STATE 173 0* I

II

Fig b - The shift of a minimum in the

curve of the form function vs ka for an

aluminum MOO0) sphere to a 3°C

chanse in temperature (a) 179'( and l (Ib) 20 9C The solid curve is theoretical. the poims are experimental for

temperature of la) 17 9( and 1h) ."

20 9(.

I

a
174 (1 RAPlt R

the minima in the Calculated form function were sensitic to changes in

the shear wave speed used in making the calculation For esample. the

abscissa at the minimum in the cakulated form function near 2(0 8 *js

round to change b) 0 12 for a change in %hear w ae speed of 20 m/sei

Assuming hnearit%. it follows that for the hcdtion of the minimum in

the computed cur,e It) fall within the ka range oft 0 05 round experimertid~. the shear wAje speed must he known wo within X rn/sec:. i e

±03* . provided the other parameters used in the caClulation are

known

(Cacludlng Remarks

These results of the ack c attered reflection from spheres used a

pulsed -sine wae signal long enough ihal the experimnental result L ould

he reaCsoahk compared with c:akulation for reflection at a %pecitk Ialue

of Aa The transient portion of the pulse Ads ignored See ( hapter K

in which (onlk transient signals 1i c . relacisekl short pulses are used in

the e'periments

REFERENC'ES

I J J [-ardn. Jr J to ul Sot .\ni 23 4111' I144 1

2 1 1) Hampton and ( M M k inek J V\oui ,, \ni 31

ks94 11961 1

1 R Ilickling. .1 A.i'usi SoK A~ni 34 1 9(2 I11962,

4 A J Rudgers %R1 Rep 6 l June 11t>-

\k 1L, Nuhaucr %RI Rep ks688 (ki 14I

kS J J I ArAn Jr \t oustics ReSeaIrth IaNi~ raim% I ar .11 i I r

Jesth Mem %o 22 1i4dr 101,

Sd IaO and I I sarti (,.Pophs -, Mag -11 I 146q2 r, I

8 1) IA I rasr Ajnti R ( I e( ta* K %e S,.i nrs ri t~ fy

144

S,,aw and I I .imi ( icoph -%ag 1 1' ' r,'

Ii I IfI lose 1h@ Wdur#Mau11ii IAEhi ' I! icj', 'I I ) % -'St'

N )rk 1441 4th ed
SPHERES-STEADY STATE 175

11, P. M. Morse and H.I Feshback, Methods of Theoretical Physics

(McGraw-Hill. New York, 1953), p. 1872.

12 B. A. Auld, Acoustics Fields and Waves in Solids (Wiley, New

York. 1973). Vol. 11.

13. 1). B. Bdlashov' and F. F Voronov, Phys. Metals Metalog. 9, No.

4. 127 (1960).
- ~ ~~ ------- --

Chapter 9

ELASTIC SPHERES AND RIGID SPHERES AND

SPHEROIDS - TRANSIENT SIGNALS*

INTRODUCTION

In Chapter 8 previous comparison of theoretical calculation and

experimental measurements [1], of the reflection by a sphere versus

frequency, required the use of an acoustic pulse long enough so that a

steady-state value of the reflected pressure amplitude was achieved at a

well-defined single frequency. Such an experiment requires a number

of reflection measurements: one at each frequency of interest. Reference to earlier work can be found in the references for Chapter 8.

The reflected pressure amplitude is expressed here as

p = (a/2r)polfr(ka)I, (1)

where a is the sphere radius, r is the distance from the center of the

reflecting sphere to the point of reception of p,, f,(ka) is the so-called

form function, and ka is the size parameter in the fluid surrounding the

sphere. If reflection measurements are made at a far field point of the

sphere, usually defined as a distance of at least 10 sphere diameters,

f,(ka) becomes independent of r and is called the farfield form function f,.(ka). For a long pulse experiment approximating steady state,

f® (ka) is derived from the substitution of measured amplitudes Pr and

p(, in the expression

if .(ka)j = (2r/a) (Pr/Po). (2)

Measurements of p, and Po, made at several frequencies for a given

sphere or for several spheres at a given frequency, are used in Eq. (2)

t,, determine specific values of If.(ka)1.

rhe incident pulse may be regarded as a function of time, so that

t. i* jc)ustic pressure measured at the position of the center of the

,lh-k sng sphere. in its absence, is

ik oct

p, = poe ,

muhed uh ctin Louis R Dragonette, Richard H. Vogt, Lawrence Flax,

-\..', I ',toust Soc. Am 55, 1130-1137 (1974).

177
178 CHAPTER 9

where c is the acoustic wave speed in the liquid in which the sphere is

immersed. Introducing the transformation r - (ct - r)/a and rewriting Eq. (2) in terms of r,

p eikoaor = ikoar I. (3)

2r

The quantity r represents the nondimensionalized time by normalizing the time parameter to a/c and reducing it by the ratio r/a,

i.e., the distance from sphere center to the reception point of pr measured in units of the sphere radius. Using r amounts to beginning time

reckoning at the center of the reflecting sphere and measuring in units

of wave travel time in water over the distance of a sphere radius a.

A pressure pulse containing frequencies expressed in ka space has

a spectrum g(ka) and its instantaneous time history in r is expressed

according to the Fourier theorem by the integral

P,(T) = f- g(ka)e kaTd(ka),

and the inverse transform defining the pulse spectrum is

g(ka) = f f_ p(T)e-ika'dr.

The quantities pi(r) and Pr(T) are now instantaneous pressures

and the relationship between them in terms of If.(ka), similar to the

amplitude equation for a single frequency given by Eq. (3), is

expressed by

f2 p,(r)e-k'dT = r f (ka) f__ p(T)ekad.

or further,

If. f" Pr)eika dr/fA _ p(rleikaTdl. (

Rather than analytically examining the effects of computational

procedures for performing the integrals in Eq. (4), a well-defined case

of reflection was carried through in the mathematical operation indicated by that equation. The response of a rigid sphere to a two-cycle

incident pulse at a ka of 4.328 was computed by exact theory as

described in Ref. 2. The resulting calculated reflected pulse was sampled at intervals Ar = 0.16. These values were used as a sampled

representative of p,(-). A theoretical two-cycle unit-amplitude incident

pulse, of dimensionless frequency k0a = 4.238, was also sampled at the

same AT. This sampling rate corresponds to about 10 samples per

cycle. These values were used as a sampled representative of p,(r).

Figure l(a) shows the spectrum gi(ka) computed from p,(r) and Fig.

l(b) shows the spectrum of g,(ka) of the reflected pulse. Fast Fourier
SPHERES-TRANSIENTS 179

I CCO

075

W

c) 0,501

L,4

N

0

n

G25-

02

4 6 I0

ko

(a)

075-

050 -

z

025-

0

0 2 4 6 8 0

ka

(a)

;. Fig. I - (a) Spectrum of an ideal two-cycle pulse. (b) Spectrum

,. of the computed reflected pulse from a rigid sphere.

- / I I I00
(I - 7 - --

180 CHAPTER 9

transform techniques were used and the results of the evaluation of Eq.

(4) are shown in Fig. 2 by the points and these are compared to the

solid curve, which is the exact calculation of the form function for a

rigid sphere.

10-0~0 I I

05

0 2 4 6 8 10

ka

Fig. 2 - Calculated If_ I vs ka (solid curve) and computed

points derived from sampled ideal two-cycle pulse.

RIGID SPHERE AND SPHEROID

Monostatic reflection from metal shapes was measured in air at a

pulse center frequency of 26 kHz. Rigid boundary conditions are very

nearly satisfied by any solid metal object for airborne acoustic waves.

The acoustic source that generated the pulse was a capacitance speaker

6 in. in diameter. The pulse received at the position later occupied by

the center of the reflector is shown in Fig. 3(a). Throughout the air

experiments a BrUel and Kjaer capacitor microphone was used that was

1/4 in. in diameter. All pulseforms that are shown are the actual sampled outputs used in computation. The reflection of the pulse in Fig.

3(a) by a 0.750-in.-diam metal sphere is shown in Fig. 3(b). The

apparent tail on the reflected pulse is the result of a creeping wave contribution that is partially isolated in time. This monostatically reflected

pulse was received at a distance greater than 10 sphere diameters from

the sphere. No precise distance measurement was available in the airacoustic range, so the pulses are not shown with amplitudes correct

relative to each other. The normalized spectra plotted on the abscissa,

expressed in units of ka for the pulses in Figs. 3(a) and 3(b), are given

in Figs. 4(a) and 4(b), respectively.
SPHERES-TRANSIENTS 181

(a)

(b)

Fig 3 -(a) Incident acoustic pulse used to reflect from sphere and

spheroid in air (b) Reflected pulse from a metal sphere in air
182 CHAPTER 9

075,

w

050

-j

0

o 0506

z

0

10

z

025-

k

a(a

i

0___ L. 1i

0 2 4

•ko

(bi

Fig. 4 - Spectrum of (a) the incident pulse shown in Fig 3

(a), and (b) the reflected pulse shown in Fig. 3(b)
SPHERES.TRANSIENTS 183

Calculation of I by using the 20-samples-per-cycle output plotted in Fig. 3(a) in the denominator and the similarly sampled output in

Fig. 3(b) in the numerator of the right side of Eq. (41 results in the

plot of Fig. 5. The experimental values shown by points are normalized

to the exact calculation shown as a solid curve to the center of the spec- tral band. The values of ka at which the spectral values are 10 dB

below the peak amplitude on both sides of the peak are shown in Fig 5

by vertical lines. A description of the experimental apparatus is aailable in Ref. 3.

100

0 P 0 0 0

oooo 10 . 00 0 0 o o

ooI

Io

05

0 , I i

0 2 4 6 8 ka

1-ig 5 - Theoretical reflection form-iunction It_' vs ka (solid cur,e) and

expertmental poin% derived from ihe single refle ed pulie shown in Fig 3(hi

The same incident pulse (Fig. 3(a)) was reflected from a metal

prolate spheroid with a 5/3 fineness ratio for which half the interfocal

distance was 1.041 cm. The pulse reflected from the end of the

spheroid is given in Fig. 6(a). This is the aspect of the spheroid for

which the major axis is pointing in the monostatic direction. It is

interesting to observe the creeping wave pulse that is distinctly

separated from and smaller than the initial specularly reflected pulse. AThe spheroid was rotated to a position 900 from the position at which

the pulse in Fig. 6(a) was measured, so that the pulse was reflected

It from its side. Figure 6(b) shows a plot of that pulse. Received pulse

amplitudes are set to optimize the electronic measurement of the pulseforms so the plotted relative amplitudes are not meaningful.
184 CHAPTER 9

(a(

lb)

Fig 6 - Pulse reflected from (a) the end of the spheroid in air (long major

axis), and (b) the side of the spheroid in air (along minor axis)

In Fig. 6(b), a small creeping wave pulse can be observed at the

end of the specularly reflected pulse. The path around the small

dimension of the spheroid is, however, insufficient to cause separation

of the specular and creeping wave pulses, which is evident in Fig. 6(a).

Calculation of If,.I by means of Eq. (4), using sampled incident and

reflected pulse data, is plotted in Fig. 7. The x's are data for end

reflection (incidence and reflection along the major axis) and the open

circles are for side reflection (minor axis incidence) from a rigid prolate

spheroid. Theoretical results [4) for the end reflection exist up to a /ca
SPHERES-TRANSIENTS 8-

VA 

A

I. .

OL

2 4

MO

hg . I heorc .iAI redtcIC.1un i0rml111110101n h u 4 (%)hd .u'vel Ihi end

rcflcti n (rom the %pheriid and e~peTtmenta pomn I1)1 end refieLIton (troises) and

',e rfe%-ion (u. t3elCI

of 4.6 and are plotted as the solid curve. The end reflection data are

normalized to the peak in If.,I at ka - 4.47. The side reflection is

arbitrarily normalized to I. I at the ka at the center of the incident pulse

spectrum, since no theoretical values were available. This normalization was convenient so that the two curves would be separated and

could be compared. The large-amplitude oscillations for the cae of end

reflection are consistent with the larger-amplitude creeping wave

interfering with the specular reflection. For the end reflection case, all

creeping wave paths are the same length and are symmetrical around

the body. When the reflection is from the side, the creeping wave

paths are not all the same length. and therefore the signals that take

these paths do not act in concert to produce the successive large oscillations over the frequency range of the pulse, as in the end reflection

case. Between the limits of ka of 3.6 and 6.2, the spectrum of the

incident pulse is no more than 10 dB below its peak.

ELASTIC SPHERES IN WATER

Portions of the reflection form function were obtained by the use

of a single reflected pulse for spheres made of three different materials.

The materials were aluminum (1100), brass (70-30), and tungstencarbide sizing balls and were the same spheres for which steady-state

reflections were measured and described in Chapter 8.

1

The incident pulseform used for the aluminum and brass spheres

over the same ka range is shown in Fig. 8. This pulse was received by

a hydrophone at the position otherwise occupied by the center of the

I , • , -
iII

I 86 CHArriaR 9

Ai Incident Acoustic pulse used lot Aluminlum And

hra%% sphere in, onfic aton. mcAured the pfition ,it ol thc

c renter ol the sphere% in theit ahsne

tVI

sphere when the monostatically reflected pulse was received. The normalized computed spectrum of the pulse in Fig. 8 is plotted in Fig. 9.

The peak of the spectrum occurs at a frequency of 212.8 kHz, which

corresponds to a ka value of 20.05 for the 4.445-cm-diam sphere used

in the experiment. The actual pulse reflected from the aluminum

sphere is given in Fig. 10. Its computed spectrum is given in Fig. 11.

Both incident and reflected pulseforms were digitized at an interval of I

jusec, or about five samples per cycle with 8-bit accuracy.

The calculation resulting from the use of Eq. (4) for the incident

and reflected pulses in Figs. 8 and 10 is plotted in Fig. 12 at specific

values of ka on the same plot with the exact computer solution 141

shown by a solid line. Values of ka where the incident pulse spectrum

was 10 dB below the peak are indicated by vertical broken lines. The

experimental points are normalized to the value of If.. I at the peak frequency of the transmitted pulse, since the reflection measurements
"'111"MRI> TRANSIt' 187

kO

I ' - I" rcqucnri ,pek irum of in.dcnt ,t.u ,k pUIlc

in I ig K

II

Fig 10 Acustic pulse monoi idcaIy reflected from an

aluminum II1I0NO sphere (diam 4 445S cm)
188 (HAPTtR 9

l oot

U

00f

0z

15 17 19 21 23

ko

F-i I I - Frequents'. %p ctrum (d the d..ou 1sis pulC cflek.ctd

Itom an aluminum ,pherc %hown in Fi It)

2

°° D

17 Is 19 20 21 22

t ka

Fig 12 - Theoretical reflection form-function I f. vs ka (solid curve) for an aluminum

sphere and experimental points derived from the single reflected pulse shown in Fig 10

.r......
owe masde in a 1 50,000 gallim pool which was not et instrumented it,

messure wcurtaly, disuance between the acoustical elements, in the

waWf

The acoustic wave speed in water at 20*( in the caicukatKn of

j.was 1482 25 rn/sec Longitudinal and shear wave spoeds used in

comtputation of if- for aluminum and tungsten cwbide were those

given in Chapter 9 The longitudinal wave speed used in the cakula

tion for brs was mneasured. in a separate cy hndraca sample I16 4-%. m

diem x 7 6 cm) by direct propagation between parallel fates. it) he 4 00)

± 47 rn/sec The cylindrical brass was a piece (it the same bar from

which the sphere was made The measured shear wave %peed in the

same sample was 2110 !t 21 rn/sec The pulse reflected in the monostatic direction from a brass sphere the same size as the aluminum

sphere is given in Fig 1.3 This reflection results from the incident

p.ise shown in Fig 8 The computed spectrum of the pulse in Fig 13

is given in Fig 14 with an abscissa in units of ka A calculation similar

1

Fi. 13 - Acoustic pulse monostatically reflected from a brass sphere.
190 

w 

0 

:::l 

f-

_J 

Q 

~ 

<! 

0 

Lu 

r:::! 

_J 

<[ 

~ 

0: 

0 

z 

I.OOr------------r---------, 

0.75 

05t 

025L v v 

ka 

Fig. 14 - Frequency spectrum uf the acoustic pulse 

reflected from a brass sphere shown in Fig. 13. 

CHAPTER 9 

to that for aluminum by the use of Eq. (4) is plotted in Fig. 15 for 

brass. Vertical broken lines on the plot again indicate ka values at 

which the incident pulse spectrum was 10 dB below the peak value. 

The incident pulse and the reflected pulse from the tungstencarbide sphere for a different range of ka than that used for aluminum 

and brass arc shown in Figs. 16(a) and 16(b), respectively. The 

incident pulse has the same center frequency as the pulse incident on 

aluminum and brass but has a slightly difTercnt shape. The diameter of 

the tungsten-carbide sphere used was 2.54 em. The spectra of the 

incident and reflected pulses are gi\len in Figs. 17(a) and 17(b), 

respectively. The theoretical calculation by straightforward wave harmonic mC;<Ins is plotted in Fig. 18 by the soliu line and the result of use 

of pulse data for tungsten carbide in Eq. (4) is shown by points with 

l 0-dB spectral limits indicated. 

··-:-;--............ -----....... - ...... -·---.. -~··~~>- ''"""'' ... ,. •' ·,,·~ ,. ~-/~< 

. ,.·, 

~ 

... 
SPHERES-TRANSIENTS 191

4-

0 L

• I .

018 

1 9 ka 20 21 22

I 

(a)

Fig. 15 - Theoretical reflection form-function 1.f_1 vs ka (solid curve) for a brass

sphere and experimental points derived from the single reflected pulse shown in Fig.

Fig. 16 - (a) Incident acoustic pulse used for tungatencarbide sphere insonification, measured at the position of the

center of the sphere in its absence. (b) Acoustic pulse

monostatically reflected from a tungsten-carbide sphere (diam

- 2.54 cm).
192 CHAPTER 9

I.00 1.00

075- 075

Ui w

_j _ 0 0

_z N

J

02 0 I = 

9 1I 13 9 H 1:3 15 k0 ko

(a) (b)

Figl. 17 - Frequency spectrum of (a) the acoustic pulse incident on a tunglsten

carbide-sphere and (b) the reflected acoustic pulse in Fig., 16(b).

1.5 -II

_ N

4 4

f

0.5-

o 0

0 0

9 0 3 I 12 13

k1

Fig. 17 - Theoetical slection form-function If. vs k Wld curve) for a tungstencarbide sphere and exper(nb)al poits derived from he sinle ewn refected pulse in

Fig. 16(b).

, a m u mm• ____ilnni aiu a nmn uno n m 
SPHERES-TRANSIENTS 193

It is interesting that the deviation of the experiment from theory

in the case of tungsten carbide around a ac of 11.4 was also observed in

other measurements carried out at a different time in a different

method of using a long pulse approximating steady state. This indicates

that some constants of the material of which the sphere is made are

probably not sufficiently well known to define the theory accurately.

All previous comparisons of theory and experimental results using

short pulses have been normalized to the value of If. at a ka value in

the center of the frequency band of the incident pulse. One absolute

measurement, for an aluminum sphere, was made for which all pins in

the electronics were recorded. Also, the sphere-to-receiver distance r

was measured to within 5 mm, so that the coefficient in Eq. (4) al2r

could be evaluated. When all of these factors were taken into account,

the absolute comparisons in a ka range from 9.7 to 12.5, shown in Fig.

19, resulted.

'.5

0.5 6 080

0 , i , j * I A 1

95 10.0 1OS 110 115 ItO 125

Fig. 19 - An absolute comparison between the theoretical

reflection rorm-function If.I vs ka (solid curvel and experimental points derived rrom a single pulse reflected from an

aluminum sphere,

CONCLUSION

Accurate monostatic reflection characteristics, If.I, of rigid

bodies in air and elastic spheres in water were obtained by using short

tone bursts to insonify the targets. Agreement between exact elastic

calculations and underwater experiments was excellent for spheres of

aluminum, tungsten carbide, and bras. Comparisons between rigid

theory and measurements in air on metal spheres and prolate spheroids

also demonstrated excellent agreement. The use of a short pulse,
194 CHAPrER 9

rather than a steady-state acoustic wave or a pulse long enough to

approximate steady-state, has major advantages because of the frequency dependence of If.. I. The tone burst covers a broad frequency

range, allowing a significant portion of the If,1 vs ka curve to be

obtained from one experiment. Steady-state experiments give only the

value of If. I at a specific ka, and numerous such experiments must be

performed to obtain information over a significant ka range. An added

advantage of short pulse experiments, especially under laboratory conditions, is the ease with which the echoes of interest can be separated

from echoes reflected at the test facility boundaries. The pulses used in

air were shorter than those which could be obtained in water, because

of the higher Q of the ceramic transducers used in the underwater

measurements, as compared to the capacitive devices available for

acoustic generation in air.

The nulls in the form function vs ka curves for elastic spheres are

highly dependent on the material constants of the sphere material,

especially the shear speed. Ordinary methods of measurement of the

elastic constants would seldom yield sufficiently accurate theory to allow

exact agreement with experiment. The wave speeds used here were

measured to within 1% by direct propagation measurements, and the

shear speed was varied within this tolerance until theoretical and experimental nulls agreed.

REFERENCES

I W. G. Neubauer, R. H. Vogt, and L. R. Dragonette, J. Acoust. Soc.

Am. 5. 1123-1129 (1974).

2. A. J. Rudgers, J. Acoust. Soc. Am. 45, 900-910 (1%9).

3 W. E. Moore, "Recovery of Acoustic Pulse Waveforms Using Calculator Controlled Signal Processing," NRL Rep. No. 7658 (1974).

4. R. Hickling. J. Acoust. Soc. Am. 30, 137-139 (1958).

5. A. J. Rudgers, "Techniques for Numerically Evaluating the Formulas Describing Monostatic Refection of Acoustic Waves by Elastic

Spheres." NRL Rep. No. 6551 (1%7).
Chapter 10

ABSORBING SPHERES*

INTRODUCTION

The monostatic reflection from elastic spheres has been the subject of numerous theoretical (1-41 and experimental [5-71 treatments.

Two methods demonstrated excellent agreement between exact elastic

theory and experiment for the monostatic reflection from solid metal

spheres in water. The steady-state technique described in Chapter 8,

utilized long acoustic pulses to insonify the spheres, the transient technique described in Chapter 9 utilized short incident acoustic pulses.

The latter method is generally more useful since a single experiment

gives reflected pressure-vs-ka information over a range of ka values,

whereas each steady-state measurement gives a single point on a

reflected-pressure-vs-ka curve. The factors in the size parameter ka are

the wave number k and the radius of the sphere a.

Figure I demonstrates quantitative agreement between the

theoretical results obtained by means of a harmonic series analysis (2,31

(the solid curve) and both steady-state and transient experiments. Each

of the points was obtained from a separate steady-state measurement in

which frequency and/or sphere size was varied, while the dashed curve

resulted from a single short pulse experiment with a single sphere. The

theoretical analysis did not include effects due to absorption of shear

and compressional waves in the material used to fabricate the spheres.

The results, given in Fig. I and other similar comparisons, demonstrate

that ignoring absorption effects is a very good assumption for spheres

of tungsten carbide, aluminum, and brass, which were the three metals

observed, 16,71 and probably for most other metals. In order to determine the possible effects of absorption on the reflection form function,

a lucite sphere was investigated, and no agreement between the harmonic series analysis without absorption in the sphere and experiment

was found. A modification of the theory to include the effects of the

absorption of shear and compressional waves in lucite was demonstrated to be the major solution to the disparity.

'This work was reported earlier in: R. H. Vogt, L. Flax, and W. G. Neubauer, J. Acoust.

Soc. Am. 57, 558 (1975).

195
1% CHAPTER 10

THEORY,

* STEADY STATE EXPERIMENTS

-- TRANSIENT EXPERIMENT

,,.,, "\ ,-

7 S 9 ko 10 11 12

Oho

Fis. I - Theoretical calculation and two different types of experimental

meaurement of the reflection from an aluminum sphere.

EXPERIMENTAL RESULTS AND COMPARISON WITH

HARMONIC SERIES ANALYSIS

Measurements were made on a Lucite sphere, employing the

technique described in Chapter 9. The form function If.. (ka) is

defined in terms of experimentally measurable quantities by

If. (ka)l 2r Ip,

where r is the distance between the sphere center and the receiving

hydrophone, p I amplitude of the reflected pressure, and IpI the

amplitude of the incident pressure measured, at the point occupied by

the center of the sphere, in the absence of the sphere. The pressure

amplitudes in Eq. (1) are steady-state values. The modified expression

for If. (ka) 1, suitable for the transient measurements, is [7)

If.(a)l- 2r Ig,(ka)l

a T&(ka)I'(2

where g, and g, are, respectively, the Fourier transforms of the

reflected and incident pulses. The geometry of the measurement is the

same as described above for the steady-state case. The method of pulse

digitization and transformation is described in Ref. 8.

Figure 2(a) shows a pulse used to insonify a 3.175-cm-dilam

Lucite sphere. The center frequency of the pulse is 210 kHz, and its

Fourier transform Is given in Fig. 2(b). The abscis of the curve in
ABSORBING SPHERES 197

I.0

a 9 t 1 I 12 13 ko

(a) (b)

Fig. 2 - (a) The pulse incident on a Lucite sphere. (b) The normalized frequency

spectrum of the pulse in (a).

Fig. 2(b) has been converted from frequency space to ka, since the

final result of the analysis will be a form function vs Ia plot. The pulse

reflected by the lucite sphere and its Fourier transform are given in

Figs. 3(a) and 3(b). The operations indicated by Eq. (2) were carried

out with a CDC 3800 computer and the resulting, experimentally

obtained, form function is given in Fig. 4, which also shows the

theoretical results obtained from a harmonic series analysis 12,31 which

neglects absorption. The limits on the abscima in Fig. 4,

9 < Ira < 12.5, were determined from the experiment as the ka values

at which the spectrum of the incident pulse is down 10 dB from its

maximum amplitude. This 10-di-down value was empirically determined to be a reaonable choice of a cutoff value from extensive previous measurements (71. Agreement between theory and experiment was

obtained (71 beyond the 10-d1 points, so this is a conservative cutoff

value.

A lack of quantitative agreement between theory and experiment

is evident from Fig. 4. The accuracy of the experimental measurements made on metal spheres, 171 by the same technique, inferred that

the experimental curve in Fig. 4 was correct and that the theory was in

need of modification. The most obvious modification considered was
198 CHAPTER 10

01-

w

OS 06o,

PM03.

02- 0'

0 9 10 It 12 13

ko

(b)

Fig. 3 - (a) The pulse monomsttilcaUy reflected by a Lucite sphere when the pulse

Ahown in Fig. 2 is ncident (b) The normalized frequency spectrum of te pulse

seen in (a).
ABSORBING SPHERES 199

8

-THEORY (NO ABSORPTION)

~-EXPERIMENT

7

6

5

I ~ I

4

3

2

It J I

9 10 II 12 13 14 15

ta

Fig. 4 - A comparison between theory which does not include

absorption (-) and experiment (-) for a Lucite sphere.

the inclusion of the effects of the absorption of shear and compressional waves in lucite.

MODIFICATION OF THEORY TO INCLUDE ABSORPTION

For an elastic sphere in the absence of absorption, the exact series

solution for Ifm(ka)I is given by [2,31

f,(ka)- A (-I)R(2n + l)sin7?,e '

", (3)

with

tan~l,-- [j,(ka)L, - (ka)jA(ka)]/[y.(ka)L. - (ka)y.(ka). (4)
200 CHAPTER 10

The j. and y. are, respectively, spherical Bessel functions of the first

and second kind, and the primes denote derivatives with respect to the

arguments. The function L, is represented by a division of two

second-order determinants of the form

a 21 a23

a 31 a 33 1

L a P/Ps. (5)

a31 a33[

The elements of the determinants are given by Goodman and Stern, [91

and p and ps are, respectively, the densities of water and the sphere

material. Two elements are repeated here for later comparisons in

order to indicate the complication introduced by absorption:

a 21 - kLaj,(kLa) and a23 -= n(n + 1)Ij(ksa). (6)

The arguments of the spherical Bessel function in Eq. (6) include the

compressional and shear wave numbers (kL and ks ) in lucite. These

arguments are real in the case where no absorption is included, and

computation of the spherical Bessel functions is standard procedure.

Absorption is included by the standard method of introducing

complex wave numbers into the theory. The complex compressional

and shear wavenumbers kL and /s are given by

kLa - (ka) - - ika -- -L, (7) CL CL

ksa- (ka) --E-__ ika _._s. (8)

Cs CS

The subscripts L ans S represent compressional and shear in the Lucite;

the nonsubscripted quantities refer to water. The c's are wave speeds

and the P's are absorption factors derived from the measurements of

Hartmann and Jarzynslk [101. In the frequency region of interest here,

the absorption coefficients are given in Ref. 10 as aL 'AL (compressional) - 0.19 dB and as A As (shear) - 0.29 dB. The wave speeds Cs

- 1380 m/sec and CL - 2680 m/sec were obtained from the Ocean

Materials Criteria Branch of NRL. The velocities are measured to an

accuracy of ±1%, but were unfortunately not measured on a sample

from the same stock used to fabricate the sphere, since no such sample

existed. The speed of sound in water was 1482 m/sec [I]. As a result

of introducing complex wave numbers into the theory, the elements of

the determinant defining L. contain spherical Bessel functions with
ABSORBING SPHERES 201

complex arguments. No tabulations of Bessel functions with noninteger order and complex arguments were available, hence these functions were redefined in terms of the hypergeometric function using the

relations [121:

j.(Z) - AoFj[- + 1; - (Z)2/4](Z),-1/ 2

, (9)

where

A-(y + 1)2-+( / 2

)' V -n +- (10)

r is the gamma function, 0F, is the hypergeometric function, and Z is a

complex,quantity, in this case either (kLa) or (ksa). The derivatives

J. and y. are expressed in terms of hypergeometric functions using the

relation

dx"d- OF (b; x) (b) OF F(b + n; x). (1

The matrix elements describing L. (Eq. 5) are given below in terms of

the hypergeometric function. The following definitions are used in the

expressions for the matrix elements:

ZL kLa Z ks a, U - -ZL21/4; 1 --- Zs2/ 4;

FL oF,(, + 1; U); Fs 0F(,(y + 1; 1),

FL I OF, (y + 2; U); F 2 OF(-/ + 2; 1),

FL2 OF,(y + 3; U); F 2 OF,(-y + 3; 1).

In the following expressions, / and A are the Lam constants, and X

and A are defined by Eq. (9-10). With the above definitions, the

matrix elements are

a - a Y+ L i12FL -zzt 3 2 FL1 (2

a21 - aZ /21 2(y + 1) FLJ' (12)

a23 - An(n + l)ZS-t1 2

Fs, (13)

a31 - 21a 2, - AZZ- 11 2

FLJ. (14)

a33 - A [2(n2- l)Z- 2

Fs (15)

.Y Z? + (312

) FsI+(Z5j+(71

2) F (Y + 1) Fs+4(y + M)(y +2)Fs.
202 CHAPTER 10

all - A [ZZ-L(2I'FL - 2/x I(y - 1

/ 2)(v - 3/2)ZZ- 512 FL

y(y+ 1) FLI+ 4(/+ 1)( _2_ ) FL 2 (16)

al2 - 2(n + InA (y - 3/2)Z _-(12)F s 2 (y/+ ) F sJ .(17)

The form function If. (ka)l is obtained from Eqs, (3-5) using

Eqs. (12-17). Calculations were made with a CDC 3800 computer.

A comparison of curves calculated with and without the inclusion

of absorption effects is given in Fig. 5 while Fig. 6 shows a comparison between the predictions of absorption theory and two independent

experiments. The significance of modifying the theory to include

absorption is demonstrated by Figs. 4-6. Quantitative agreement

between theory and experiment is found in Fig. 6.

NONABSORB ING THEORY

.... A8$ RSiNG THEORY

I,, II

4

2I

I - t

,\ }

7 8 t o 1 12 13

Fig - Comparison between calculations of the

reflection from a ilucite sphere with and without the inclusion of absorption.
ABSORBING SPHERES 203

-THEORY

5 --- EXP I

..... EXP 2

4

3

Fig. 6 - The reflection function calculated

from the theory for an absorbing Lucite

sphere is compared to reflection functions 2

obtained from two independent experiments.

9 10 II 12

ka

CONCLUSION

The monostatic reflection from a lucite sphere in water cannot be

described by a plane-wave expansion technique unless the effects due to

absorption are included in the analysis. The need for modification of

the existing theory was demonstrated by a transient measurement technique and theoretical verification carried out. The results indicate that

the measurement technique is capable of producing quantitative results

with reasonable confidence on other elastic shapes not amenable to

exact calculation.

REFERENCES

1 J. J. Faran, Jr., J. Acoust. Soc. Am. 23, 405 (1951).

2. R. Hickling, J. Acoust. Soc. Am. 34, 1582 (1962).

3. A. J. Rudgers, NRL Rep. 6551 (June 1967).

4. 4. T. Hasegawa, Y. Kitagawa, and Y. Watanabe, J. Acoust. Soc.

Am. 62, 1298 (1977).

5. L. D. Hampton and C. M. McKinney, J. Acoust. Soc. Am. 33,

694 (1961). I
204 CHAPTER t0

6. W. G. Neubauer, R. H. Vogt, and L. R. Dragonette, J. Acoust.

Soc. Am. 55, 1123 (1974).

7. L. R. Dragonette, R. H. Vogt, L. Flax, and W. G. Neubauer, .

Acoust. Soc. Am. 55, 1130 (1974).

8. W. E. Moore, NRL Rep. 7658 (1974).

9. R. R. Goodman and R. Stern, J. Acoust. Soc. Am. 34, 338

(1962).

10. B. Hartmann and 1. Jarzynski, J. Appl. Phys. 43, 4304 (1972).

11. W. G. Neubauer and L. R. Dragonette, J. Acoust. Soc. Am. 36,

1685 (1964).

12. M. Abramowitz and 1. A. Stegun, Eds., Handbook of Mathematical

Functions (Natf. Bur. Stand. Appl. Math. Ser. 55, Washington,

DC, 1970), p. 362, Eq. 9.1.69 and p. 437, Eq. 10.1.1.
Chapter 11

LONGITUDINAL WAVES INCIDENT

ON AN ELASTIC SPHERICAL OBSTACLE

IN ANOTHER ELASTIC MATERIAL

INTRODUCTION

Interaction effects of elastic waves and bodies are the basic source

of information used to attempt identification of flaws and inclusions in

materials nondestructively. Even though analytical descriptions of

some systems are possible, useful interpretation has often been severely

limited by the lack of special function evaluations over sufficiently large

ranges of parameters. Even when such calculations are available,

interpretation is limited because basic wave-interface interactions are

not completely understood. The spherical inclusion of one material in

another is a system that is encumbered with all of these limitations.

The problem of the spherical shape is interesting because it is a closed

(finite) surface with a separable boundary value solution and, in some

cases, is a reasonable approximation to an isotropic hole in a solid, or

an inclusion or scatterer of one material in a matrix of a different

material (often called the host material). Advances in computational

capability (11 have permitted computation of a variety of combinations

of hosts and inclusions.

Other solutions of spherical problems 121 indicate that erratic or

unexpected behavior can occur in the scattering by a solid object in an I elastic medium. A number of results, described by their scattering

cross sections (SCS) as a function of frequency for spherically shaped

inclusions in a different host material, are collected in Ref. 3. Some of

the same combinations of materials assumed in the references, are

assumed to be host and inclusion here. Earlier conclusions were

derived from calculations that did not exceed a ka of 10 and in some

cases only went as high as ka - 3. Certain conclusions about higher

frequency behavior were borne out in some cases by higher ka computations. In other cases, the limited computations indicated incorrect

205
206 CHArER I I

conclusions. Direct experimental observations of total scattering cross

sections are difficult, or perhaps impossible, to achieve. In spite of the

fact that total scattering cross section is an experimentally unobtainable

quantity, it has proven to be useful in allowing generalizations about

classes of scattering problems. Unfortunately, the quantity called SCS

does not have a universal definition. Here previously published curves

are compared with alternate descriptive quantities for the cases treated

elsewhere.

SCATTERING DESCRIPTIONS

Three different quantities are commonly used to quantitatively

describe scattering from bodies as a function of frequency: total scatterins cros section (TSCS), differential scattering cross section (DSCS)

and form function If. I. Perhaps the most common of these is TSCS,

which is usually defined 13) as the ratio of total energy scattered per

unit time to the incident wave energy per unit area normal to the

propagation direction per unit time. This is the quantity used to

describe solutions in the area of nondestructive investigation or evaluation. However, even when TSCS is discussed in theoretical treatments

it is not measured in the associated experiments (41. The popularity of

TSCS is probably a result of the special averaging inherent in the definition: i.e., for a body, it represents the total energy around the entire

body. This is difficult to measure. In most cases measurements in a

single direction, or a few directions, are all that can be made, and even

those, often with great difficulty, especially if the medium containing

the scattering body is a solid.

The definition of TSCS for spherical geometries leads to the

expression

TSCS -4ua 2 2 R +~ I1AM12 + + 1)n 52

1i kLj a' kLj ks a 2

where a is the scattering sphere radius, k is the wave number for longitudinal waves when subscripted by L and for shear waves when subscripted with S The additional subscripts I and 2 refer to the outer

(matrix or host) and inner (sphere or inclusion) medium respectively.

The coefficients A, and 8, are derived from the boundary value problem in the spherical geometry in which the wave equation is separable.

The formulation of the problem is treated elsewhere 13,5,61, and in

Chapter 14. so only the solutions and definitions of pertinent quantities

necessary for computation will be given here. The matrices defining A.

and B, the elements in those matrices and the definitions of the quantities contained in them are given in the appendix.
r~~ ---

SPHERE IN SOLID 207

In addition to the above definition of TSCS another quantity

called the normalized (total) scattering cross section (TSCSN) is defined

as the ratio of the TSCS to the geometrical cross section (wa 2) of the

sphere 131. It is therefore the ratio of total power in the scattered wave

to the power of the incident wave through the cross-sectional area of

the scatterer (sphere).

The use of TSCSN seems to have been helpful in the evaluation

of scatterers distributed in a volume since it takes into account all types

of scattering, including mode conversions. It is a measure of the total

scattered field, without taking into account any directional properties.

At the same time, and for the same reasons, there exist limitations in

the physical interpretation of scattering by evaluating TSCSN. To compensate in part for this shortcoming, definitions have been made of

TSCSN for an incident longitudinal wave or shear wave alone and the

resultant longitudinal or shear wave reflected from the scatterer (41.

The DSCS is similar to the TSCS but includes the angular dependence of the scattering. It can be defined as the scattered power

(energy per unit time) per steradian divided by the incident (longitudinal wave) intensity (energy flux). Solving the boundary value problem

results in the expression

DSCS-a 1 2+-- an A-.I pa(cos) + BP. (cos@)k,. (2)

which again combines the eff-cts of longitudinal and shear waves. Such

a combined quantity cannot be evaluated experimentally with present

day sensors.

For the case of backscattering (0 - 7r), since

P. (cos ir) - 0 and P. (cos ir) - W,

DSCS -2,- (-1)22n + I A (r

in which case no shear waves are involved in the scattering.

Another quantity used to describe the scattered field is the form

function If.I which results from the analyses of reflection from

spheres and cylinders after the manner of Hickling 121 or Neubauer et

al. 171 and others 18,91. For a longitudinal wave incident on a sphere,

(a. #)I - 2 (-)(2n + I)A. P. (cos 9) (4) kLI a
208 CHAPTER I I

For b ckscattering (0 - w) this becomes

If.2(ka, v) 2 (-1)- (2n + 1) A. (5) kLI a

so the relationship between DSCS and form function is a

2

DSCS (Wr) - If,.(ka, r)[2. (6)

Elasticity problems have most often been described by TSCSN and only

occasionally by DSCS. In acoustical problems If.. I is probably the most

common description and the one that has been compared quantitatively

with experiments [4].

COMPUTATIONS

In this work backscattering calculations were carried out for combinations of seven different materials. Therefore, 42 cases of a spherical inclusion of one material in another were computed, as well as the

seven materials with a spherical void or hole in each. The materials,

their densities and longitudinal and shear wave speeds are listed in

Table 1. The capability for computation of spherical Bessel functions

achievable for large argument. That capability has since been achieved

by Flax and Mason, [1] whose results were used for these computations. Here only an incident longitudinal wave will be considered in the

host medium but both longitudinal and shear waves are generated on

reflection. Absorption of bulk waves are included in the computations

by using a complex wave number.

Table I

Density Longitudinal Shear

Material (gm/cc) Wave Speed Wave Speed vL/vs

(m/sec) (m/sec)

Polyethylene 0.90 1950 540 3.61

Polymethyl

methacrylate 1.18 2680 1380 1.94

Magnesium 1.74 5770 3050 1.89

Beryllium 1.87 12890 8800 1.46

Aluminum 2.70 6568 3149 2.09

Germanium 5.36 5285 3376 1.57

Iron 7.90 5790 3100 1.87
SPHERE IN SOLID 209

The cases of a spherical cavity in each material computed by

Truell et al. (31 were taken only to a value of k1a - 7. Calculations,

which were carried out up to a k1a of 40, support their conclusion that

as kla increases, values of TSCSN of very nearly 2 are approached by

all curves. For k1a of 40, TSCSN was in all cases between 1% and 2%

of the value 2. Since our curves show only a gradual smooth approach

to those values with increasing kta they will not be reproduced here.

The case of a germanium (Ge) sphere in an aluminum (Al)

matrix was computed (see Fig. 1) to permit a direct comparison with

the curves of Truell et al. 13). Agreement was obtained up to a k1a of

10, which is the largest value calculated by Truell. Truell et al. speculated correctly that the curve was perhaps a "very gradual oscillation,

with a maximum in the vicinity of kIa - 8, upon which a fine structure

of rapid oscillations is superimposed." That is exactly the behavior of

this curve even up to a k1a of 40. The only other features of any

interest are the bursts of sharp, but small, spikes around values of k1a

of 19, 25, and 33. The case for these small oscillations is undoubtedly

interference between the incident reflected wave and waves reemerging

in the direction of incidence after refraction and reflection inside the

sphere.

4

3

0 I I I

0 Is 24 32 40

Fit. I - TSCSN vs kIa ror aGe sphere in Al. (v,/v 1- 1.243.

On the basis of their limited calculations Truell et al. concluded

that TSCSN curves are smoothest for cases in which there are large

differences in longitudinal wave speeds of matrix and sphere material,

while similar wave speeds result in highly oscillatory TSCSN curves.
210 CHAPTER I I

This conclusion is not borne out by the calculations shown here. It was

shown to be true in some cases and not in others, as evidenced by the

results in Figs. 2-7. For a beryllium (Be) sphere in a polyethylene (PE)

matrix, for which the longitudinal wave speed ratio vI/v 2 is 0.152, the

TSCSN curve is shown in Fig. 2 to be relatively smooth up to a k~a of

30. Lest that be considered an isolated case, the example of a Be

sphere in Ge, for which v1/v 2 - 0.410, is given in Fig. 3. Whereas

both of these cases fit the hypothesis of Truell et al., for the inverse

case of PE in Be, for which vI/v 2 - 6.569, the computations result in

the oscillatory TSCSN curve shown in Fig. 4. This is in contradiction to

the hypothesis of Truell et al. Again to eliminate the possibility of an

isolated instance, the TSCSN of a Ge sphere in Be, (vI/v 2 - 2.348), is

shown in Fig. 5 to be oscillatory also. The case of a magnesium (Mg)

sphere in an iron (Fe) matrix, shown in-Fig. 6, is one for which v, is

very close to v2, i.e ., vI/v 2 - 1.004, and does have a TSCSN curve

oscillatory in nature, which is consistent with the hypothesis. On the

other hand the case of a Ge sphere in Fe shown in Fig. 7 which has a

ratio of v,/v 2 of 1.096 results in a smooth curve.

2

2

II I I

612 1824 30

ka

Fig. 2 - TSCSN vs kIa for a Be sphere in PE. (v1/v2) - 0.152.

On the basis of this set of TSCSN curves one might conclude that

for vI/v 2 > 0.9 there seem to be rather regular oscillations with a

period of k1a 4 1. These are superimposed on another larger basic

period which itself decreases as vI/v 2 increases and becomes difficult to

distinguish as the small period superimposed oscillations increase. By

viewing all of the 42 computed curves at one time, an attempt was

made to distinguish between smooth- and oscillatory - (or perhaps,

more correctly, jagged) - curve behavior and we concluded that for

v1/v 2 < 1.3 curves tend to be smooth and for v1/v 2 > 1.3 curves tend

to oscillate rapidly.
SPHERE IN SOLID 211

3-

1 -

o 8 16 24 32 40

ka

Fig. 3 - TSCSN vs kla for a Be sphere in Ge. (Vl/V 2) - 0.410.

z

(n) u 2

I016 24 32 40

ka

Fig. 4 - TSCSN vs kla for a PE sphere in Be. (vI/v 2) - 6.595.
212 CHAPTER II

4-

3- z

2

0

oI I I I

8 16 24 32 40

ka

Fig. 5 - TSCSN vs kla for a Ge sphere in Be. (vl/v 2) - 2.438.

0 a 16 24 32 40

ka

Fig. 6 - TSCSN vs kla for a Mg sphere in Fe. (v1/v 2) - 1.004.

4

3- z 2-

02

I0 8 24 32 40

ka

Fig. 7 - TSCSN vs kIa for a Ge sphere in Fe. (v,/v 21) - 1.096.
SPHERE IN SOLID 213

Similarities were noted among the curves for which the scattering

sphere was assumed to be beryllium. An example of a typical one of

these is given in Fig. 8. Similarly, for the cases for which beryllium

was assumed to be the matrix material curves similar to Fig. 9 were

obtained. These similarities are possibly related to the fact that beryl- lium has unusually large longitudinal and shear wave speeds (see Table

1).

z

(n

-

16 24 32 40

ka

Fig. 8 - TSCSN vs k1a for a Be sphere in Mg. (vl/v 2) - 0.448.

4I

3-

02

I

0

0 8 16 24 32 40

ke

Fig. 9 - TSCSN vs kIa for a Al sphere in Be. (v,/v 2) - 1.958.

...............................-.........
214 CHAPTER II

Many TSCSN curves were particularly devoid of behavior that one

might attempt to interpret physically in terms of resonant or circumferential wave behavior. (See Figs. 7 and 10.) However, curves of

DSCS or If. I for the same cases have more features that may ultimately be subject to such interpretation. The two such cases given in

Figs. I I and 12 correspond respectively to Figs. 7 and 10. Oscillations

of similar, but not the same, periods are found in the plots for TSCSN

and If,. I for a Ge sphere in Al (Figs. I and 13 respectively). The

opposite observation can be made on examination of Fig. 9, for an Al

sphere in Be for which the TSCSN is regular and relatively well

behaved, while the If. curve for the same combination (Fig. 14) is

irregular. These cases indicate that it may be helpful for applications

where characteristic behavior is sought, to calculate both types of functions until distinct physical causes can be attributed to specific features

of curve behavior.

4I I II

3-

U2

0 8 16 24 32 40

ka

Fig. 10 - TSCSN vs kIa for an Mg sphere in Al. (v1/v 2) - 1.138.

2

0 L 0 8 16 24 32 40

ka

Fig. II -- Form function IfI vs k 1a for a Ge sphere in Fe. (v1/v 2) - 1.09

- ~ ~ . . ._
SPHERE IN SOLID 215

3-

-

I

C'I Ii | I n || i

0l

0 8 16 24 32 40

ka

Fig. 12 - Form function f.Ivs kja for a Mg sphere in Al. (V1A'2) -1.138.

4-

3-

-B

0

o 8 1624 32 40

km

Fig. 13 - Form function I.Ivs k1a for at Ge sphere in Al. (V1/V2) -1.243.

- ----.. -.--- 
216 CHAPTER I I

5-

4-

-

2-

0 8 16 24 32 40

ka

Fig. 14 - Form function If-I vs k1a for an Al sphere in Be. (vI/v 2) = 1.958.

Some results claimed to indicate resonant behavior in scattering

by an inclusion have been calculated by Flax and Uberall [101. However, on careful comparison of the curves describing TSCSN and If1 I

for an Fe sphere in Al shown in Figs. 15 and 16 respectively and the

partial wave curves given in Ref. 7, none of the maxima and minima

corresponded in a consistent way to those of the partial waves in the

spherical region.

3-

U)

U

;2

0 8 16 ka 24 32

Fig. 15 - TSCSN vs k1a for an Fe sphere in Al. (v,/V2 ) - 1.134.
SPHERE IN SOLID 217

z

0 86 2432

ka

Fig. 16 - Form function If-~ vs k a for an Fe sphere in Al. (V I/V2 ) =1.134.

The TSCSN curve for a spherical cavity in a solid has been noted

as having a single maximum roughly corresponding to radial resonant

behavior [31. It is also anticipated that radial resonant behavior should

be observable in elastic spherical inclusions 131. However, experience

with the reflection from solid spheres in water [71 leads one to conclude

that the radial mode resonant peak may not be expected to be

significant in the curves describing reflection from elastic spheres

imbedded in another solid. Resonant behavior is an excitation of a

natural mode of vibration of a body, a sphere in this case, and whether

a peak is detected depends on whether that vibrational mode has

significant radiation in the observation direction. For solid spheres and

shells in fluids at low frequencies the radial mode is difficult to excite

by an incident wave [6-12). It is, therefore, also difficult to detect.

There have been indications that resonant behavior cause minima in

lf I for backscattering because of circumferential modes in spheres and

cylinders occurring at values of ka below those required for their radial

or breathing mode [11). The predominant modal indications are likely

to result from the reinforcement of interface modes, as was the case for

cylindrical shells in water R11]. It would, therefore, seem that a reasonable degree of understanding of these scattering curves for spherical

bodies in a different material matrix must await the more detailed

examination and understanding of the properties of waves that exist on

the interface of two different materials.

By means of complex wave numbers, absorption was included in

the solutions for spherical scattering and reflection. (See the appendix.)

Results of computations assuming no absorption are shown in Fig. 17

for TSCSN for a PE sphere in Al. The same case computed assuming a

longitudinal and shear wave absorption P

9

L and Ps of 0.0070 Np and
218 CHAPTER II

3-

u 2

0 8 16 24 32 40

ka

Fig. 17 - TSCSN vs k 1a for a nonabsorbing PE sphere in Al. (vI/v 2) - 3.368.

0.0212 Np respectively for the PE sphere is shown in Fig. 18. A curve

of If.,1 for the same case considered in Fig. 17, of a PE sphere in Al,

again without absorption, is shown in Fig. 19. The form function If**.

for the same case of a PE sphere in Al but with absorption in the

sphere of PE is given in Fig. 20 showing a more dramatic effect in the

comparisons of the absorbing and nonabsorbing cases. Comparison of

Figs. 17 and 18 shows a moderate difference in the TSCSN curves with

and without absorption. However, for the same case the differences

shown in the curves of IfJI in Figs. 19 and 20 indicate large magnitude

differences. The two descriptors, TSCSN and If, l (or DSCS) have

different sensitivities to the damping effects of absorption.

4 I

3

U)

0 o1 I I

0 8 16 24 32 40

ka

Fig. 18 - TSCSN, vs k 1a for an absorbing PE sphere in Al. (v,/v 2) - 3.368.

ii
SPHERE IN SOLID 219

5-

4

S

2-

1

i 9 16 24 32 40

ka

Fig. 19 - Form function I f vs kIa for a nonabsorbing PE sphere in Al.

(vI/V 2) -3.368.

0816 24 32 40

ka

Fig. 20 - Form function lfJ vs k a for an absorbing PE sphere in Al. (vi/v 2) - 3.368.
220 CHAPTER II

APPENDIX

Four boundary conditions are prescribed at the interface between

the host medium and the sphere at r - a:

(i) radial displacements are continuous, (ii) tangential

displacements are continuous, (iii) radial stresses are

equal, and (iv) tangential stresses are equal.

Since only waves in the outer medium will be considered here, the

required coefficients A. and B. for Eqs. (1-6) are

alI a 12 a14 a1 5

a 21 a22 a 24 a25

a 3 l a32 a34 a 35

a 41 a42 a44 a 45

A,, - A

lil a12 a 13 a 15

a 21 a 22 a23 a2 5

a 31 a 32 a33 a35

041 a 42 043 a 4 1

B. -

A

all a12 a1 3 a 14

a2 a22 023 024

where A - Ojil 032 0313 034

a 41 a42 043 a44

L
SPHERE IN SOLID 221

The elements are given below.

a11 - p)/p1 Lj.(ZL2) - 2G~j.v(ZL2PI/(I + 2G2)

a12 - -2#t(,, + 1) (p)/pi) IZ5 j.(Z52) - j.(ZSz)/Z&4

a13 - Ih."2 (ZL,) - 2Gjh. 2)(ZLj)i/(1 + 2Gj)

a14 - -2n(on + 1) IZS1h.(2ViZS1) - h.(2 1 (Z5 1)I/Zj 1

aIs- .(ZL1) - 2GIJ"(ZL1)I1/ (1 + 2Gj)

a2l - AL2j.n(ZL 2)

22 - #I(M + )J.(ZS2)

a23 - ZL Ih." " (ZL d

a24 - nf(i +- 1)h.", %4l)

a2 5 - ZL IJ.(ZLI)

a 31 - 2 (ZL dM( ZL 2) - MZL 2)1 LU2ZS2 1 VZI 

a32 - IZS12j-(ZS2) + (ov + 2) (it - 1)j- (ZS2)I (LU2ZS'/1AZS'2I

a33 - 2IZL~h.2' (ZLI) _ h."' (ZL d I

a4- V 2 11 (4w) + (M + 2) (n - I )A. 21(Z5,)

a - 21ZLj.4(ZL,) -j.(ZL 1)1

a41 - J-(ZL2)

a42 - j.(ZS2) + Z52j.(Z52)

a4 l - Al1'

2

' (ZL 1)

a"- h, 2 '(ZS,) + Z~1h,'2'(Z51)

a4 5 - j.(ZS I)

Where pi and P2 are the densities of medium I and 2 respectively, ju

and ju are the Lami rigidity moduli and j,., h. are the spherical Bessel

and Hankel functions, respectively.

The following definitions are used to simplify the notation somewhat. ZL I kL Ia

ZL2 &Lla

Z2 k52a

G Z/Z I - 4I

G EZU 1 - 2 Z L2
222 CHAPTER I

In the expression for the elements, the primes indicate

differentiation with respect to the argument of the spherical Bessel and

Hankel functions. The effects of material properties that cause attenuation of longitudinal and shear waves are accounted for in the theory by

introducing complex longitudinal and shear wave numbers defined as

KLI - kL1(l - ifi1)

Ks, m ksl(l -isi)

KL2 = kL,(I - 01.2)

Ks2 ks2I - tos)

where P's are absorption factors with dimensions of Np. The relation to

absorption coefficients ,LAL and asks (A - wavelength) in dB is

L -s aL.S/( 40W e)

Formulations apply to lossless cases by letting 18L.S - 0. As a result of

the use of complex wave numbers, the elements of the determinants

defining A, and B. contain spherical Bessel functions of complex arguments. Techniques developed by Flax and Mason (II for calculating

these functions were used. Far field approximations are assumed for

the scattered field.

RIEFERENCES

I L. Flax and J.P. Mason. *FORTRAN Subroutines to Evaluate, in

Single or Double Precision. Bessel Functions of the First and

Second Kinds for Complex Arguments," NRL Report No. 7997.

Washington, DC (1976)

2. R. Hickling. J Acoust. Soc Am 34, 1582-1592 (1%2)

.u. R. Truell, C. Elbeum. and B.B. Chick, (Academic Press. Inc.,

New York, 1969).

4. R.M. White, J. Acoust. Soc, Am. 30. 771-785 (1958)

5. C.F Ying and R, Truell, J Appl. Phys. 27. 1086-1097 (1956)

6. L Flax and H. Uberall, J Acoust. Soc. Am. 67, 1432-1442

(1980).

7 W.G. Neubauer. R.H Vogt. and L.R Dragonette. J Acoust Soc

Am. 55, 1123-1129 (1974)
SPHFRE IN SOLID 223

8 LR. Dragonette, RAI. Vogt. L. Flax, and W.G. Neubauer, J.

Acoust. Soc. Am. 55, 1130-1137 (1974).

9 R.H. Vogt and W.G. Neubauer, J, Acoust Soc Am. ". 15-22

(1976)

10. L. Flax and II. Uberall. J Acoust. Soc. Am. 64, 5130 (1978).

II L.R. Dragoneue, *Evaluation of The Relative Importance of Circumferenil or Creeping Waves in The Acoustic Scattering From

Rigid and Elastic Solid Cylinders and From Cylindrical Shells,"

NRL Report No. 8216, Washington. DC (1978)

12- L Flax, L.R. Dragonette, and H. Uberall, J. Acoust. Soc. Am.

63, 723-731 (1978)
Chapter 12

RUBBER CYLINDERS AND SPHERES'

INTRODUCTION

The scattering of an infinite plane acoustic wave by isotropic

cylinders and spheres fabricated of lossless materials has been considered by numerous authors both theoretically [1-31 and experimentally [4-61. The harmonic series analysis modified to include the effect

of absorption in the elastic solid [7-91 were found to agree with experiment (7,101 to within the uncertainty in the elastic constants of the

material used.

Here a parametric study will be described of the reflection and

scattering of sound by silicone rubber cylinders and spheres in water.

As will be shown, the use of silicone rubber allows a significant

simplification of the theory. These materials are of special interest due

to the fact that their longitudinal sound speed I111 cL is significantly

lower than that of water. Furthermore, the product of the density p,

and CL can be made to approach that of water. Finally, at ultrasonic

frequencies the longitudinal transmission loss is low. From a practical

standpoint materials such as these have proven useful in the construction of acoustic-slow waveguides 1121 and lenses 1131.

FORM FUNCTION CALCULATIONS

The ratio of the absolute values of the scattered, IpI, to the

incident, po, pressures can be expressed as

fo ee p WIP /IhlI- (a/2.)lf.I, (1)

for spheres and

1po, I/ Ipol - jIf. 1, (2)

Trh work rm appeared i: C. M. Davis, L. I. Drasasais and L. Fin, J. Acoust. Soc.

AM. 63, 104-16 (IM7).

225
226 CHAPTER 12

for infinite cylinders, where a is the radius of the sphere or cylinder, r

is the distance from the center of the scatterer to the receiver and If-. I

is defined as the amplitude of the form function. Derivation of the farfield expressions for If.I are given in Chapter 10 and Chapter 4 for

spheres and cylinders, respectively.

The variation of If. I with ka (k - 2w/k,, where A is the

wavelength of sound in water) for the case of monostatic scattering of

sound from silicone rubber spheres and cylinders in water is calculated.

Values of p, elastic moduli, and sound-absorption coefficients characteristic of commercially available material 1111 are considered. Following Mason, 114) the value of shear speed, c5, is taken to be maximum

of cLI 10 and the shear absorption coefficient a minimum of 25 times

the corresponding longitudinal absorption coefficient a. As is generally

the case for rubber, the value of a for silicone rubber has been shown

(I1I1 to be proportional to frequency.

Curves of If.I, obtained with and without shear rigidity and

absorption were indistinguishable from each other. This is a result of

the abnormally low shear speed which in turn leads to negligible shear

conversion at the water-rubber interface. Thus, all subsequent calculations have omitted the effect of shear and so correspond to a liquid

cylinder. In Fig. I a comparison is made with and without absorption

for values of cL and p, typical of silicone rubber. The curves with a ;9

0 were obtained using a value approximately equal to the average of

those reported by Folds [1I11: specifically a/ k - 5 x 10-3 where a is

expressed in Np/cm. As can be seen, the effect of absorption in these

materials is simply to attenuate the amplitude of the oscillations. As

will be shown, these oscillations are due to the interaction between

various lens waves which suffer attenuation in the material and, in

some cases, specular reflections which do not In order to simplify

both the calculations and the interpretation, the investigation will be

restricted for the most part to the lossless case.

In Fig. 2, If -1, is represented for CL -1.0 km/s and variable p,

such that pICL - p, Silicone rubber with p, > 1.5 g/cml can be

achieved in practice by loading with metal oxide. Curves from Fig. I

corresponding to p, - 1.0 and 1.5 U/cm3 are repeated. For p, (and

therefore P, CL) 1. 5 the value of If .1, increases linearly with ka up

to a ka value of approximately 5, at which point the value of I.I

begins to oscillate about a value of approximately 0.5. For values of

PSCL different than 1.5, an oscillation appears in the low ka region

whose amplitude increases with increasang deviation of p, L from 1. 5.
RUBBER CYLINDERS AND SPHERES 227

-Mm Mmwna

2A - - TMWJ ASUWTOM

Ia.. 1 A

t3 1)45oc. 05=0 1-Y

Fig. I - A comparison of computed curves of If_. I vs ka for silicone

rubber cylinders without and with longitudinal absorption included.

IN.2.0

0k 07 0 105 S 1i 504 

asa.

IFa -Fomunto skcuescmuefocyidr

w0h6 10k/ nd10,p (2 /m

FiA.aiu 0In omiain fp ad ALwr hseuhta

t of 0e vau ismi1ie2tavleo 15 necaetefr0-C 

f u n ctio in r a e l n Al w t kA , t h th e l n A r- e inAx e di g t

both~ig an 2L Forequaltn corsponin curves inteo walnersh ylne

becoe ieseiy la waler cy wtelindriwaead asio suchtprdces no

scattering.
228 CHAPTER 12

05- VL *J.

0 ~~~ I I 81,15 

05~~ VL - 0 1.I 136

0.S_ VL .1. 0

P l. 1.4

0 " .5 .045 0 5.0.5 0.0 3 1.5

ho

Fill. 3 - Form-function vs/ka curves computed for cylinders with combinations of p, and c, livinpc o -P (water).

Similar behavior should occur for other geometries. To check this

assumption, calculations were made for spherical geometries. The

results obtained for CL " 1.0 km/s and P5 values 1.3, 1.5, and 1.7 g/cr 3 are shown in Fig. 4. For P5 PsCL 1.5, the monotonic

increase in ifun with ska is observed to be parabolic rather than linear

as in the cylindrical case. For values of ca > 8, the value of Ifr).

exhibits an oscillation about a value of approximately 4.0, compared to

the 0.5 value observed for cylinders. The effect of sound absorption on

the curve for If.. It corresponding to this latter case is shown in Fig. 5.

The results are essentially the same as in the cylindrical case (Fig. 1).

PULSE CALCULATIONS

The form function curves seen in Figs. 1-5 describe the steady

state, backscattered response of the cylinders to an incident, infinite

plane wave, and give no obvious clues to the individual mechanisms

which account for the return. Isolation of the individual backscattered

echoes, which contribute to the steady-state response, is achieved by

calculating the response of the cylinders to an incident acoustic pulse

short enough in duration that the beckscattered return is a series of

separate echoes whose magnitude and time of arrival can be observed

and analyzed. If g(ka) is the wave-number spectrum of the incident

pulse, the beckscattered response at a distance r much greater than the

radius a is given by (2,3,61
RUBBER CYLINDERS AND SPHERES 229

o SPHERE

5.

i h VL - 1.0 km /s d

- W ABRI

40

3.0'

2.10 -

0.5

O 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 150 ko

Fig. 4 - Form-function vsa curves computed for spherical tarets

with si - 1.0 km s and - 3, L.w and 1.7 g/cm3 .

75WI 

H A S R TO 6.- WITHOU ABSORPTION

4.5 --

(3.0c I %(

1.5 L

.01.5 3.0 4.5 6.0 7.15 9.0 10.5 12.0 M35 15.0

Ito

Fig. 5 - A comparison of computed form-function vs ka curves

for silicone rubber spheres without and with absorption included.

I 1/2

i where If. 1, has been described previously in Eq. (2), and 7- is a

!dimensionless time parameter

7 - (cwt- r)la (4)

normalized to be zero at the time the incident wave would reach the

position occupied by the center of the cylinder in the absence of the

cylinder, and c. is the speed of sound in water.
230 CHAPTER 12

Calculations were made of the response of rubber cylinders to an

incident gated sine-wave pulse. The densities and sound speed of the

cylinders were chosen to be within the values considered in the form

function versus ka curves shown in Figs. 1-3. Figures 6(b)-6(d) give the response of three targets to the incident pulse shown in Fig. 6(a).

In all three cases the targets are cylinders with a negligible shear speed

and a longitudinal sound speed of 1.0 km/s. The densities are Ps

1.0, 1.5, and 20 g/cm

3

, respectively. The incident pulse is centered at 

a

dimensionless frequency 

k0a 

- 12, where ko 

-

2 rfolcw. The echo

responses seen in Figs. 6(b) and 6(d) show a specular reflection beginning at " - - 2 (recall r - 0 when the incident wave reaches the

center of the cylinder) followed by a train of pulses which have taken

internal paths beginning at T 

- 4. The pulse train of equally spaced

echoes beginning at r 

- 4 exists in all three cases. No specular reflection exists for the case p, - 1.5 g/cm3 where pScL equals the specific acoustic impedance of the surrounding water. Even in this case significant energy returns in the backscattered direction, as can be observed

in Fig. 6(c). The specular return is 1800 out of phase with the incident

pulse when p 

- 1.0 g/cm3 and in phase when p, 

- 2.0 g/cm3 (Figs.

6(b) and 6(d), respectively). The form function is a steady-state

parameter, which reflects the long pulse interference of the individual

returns isolated in Fig. 6. Most consideration is given to the cases

when psCL 

- 1.5, since specular backscattered reflection is zero, and

for the uses discussed in the introduction (waveguides and lenses)

specular backscatter is not desirable. In addition to the calculation seen

in Fig. 6(c) at 

k0a 

- 12, backscattered echoes were computed at 

k0

a

values of 2.0, 3.0, 6.0, 7.5, and 9.0 for the case Ps 

- 1.5 g/cm

3

, CL 

-

j

1.0 km/s. These calculations showed a buildup of the pulse amplitude

at r 

- 4 at the rate of 0.08 pokoa up to a maximum of approximately

0.6 P0. This accounts for the magnitude of If. 1, in the region from 

0

< 

k

0a < 7. The oscillation in If**Ic are caused by the interference

between this initial return and the rest of train of echoes. The first

echo in Fig. 6(c) had an amplitude of 0.5 P0. The echoes which follow

in the train have reached amplitudes of 0.16 Po, 0.09 P0, and 0.05 P0

and occur at r 

- 12.1, 20.2, and 28.3, respectively.

If the sequence of equally spaced echoes seen in Fig. 6(c) is assumed to be the result of the continuous repetition of the same travel

path, within the cylinder a circumferential wave speed may be ascribed

to the return. The ratio of the circumferential speed, c*, to c. is given

by

c -lc, 2r/Ar, (5)
RUBBER CYLINDERS AND SPHERES 231

50[

40

30,

20

2-

-10

-20 (b)

30

20l

"'Il

-3C

2

10

*0 -

I0-

-302 4 66(0 141662022242628303234

20-[

Fig. 6 -- A computation of the echoes backscattered when a five

cycle pulse (a) is incident on cylindrical targets with C, - 1.0 I~ k/s (b) p, - 1.0 g/cm', (c) P, " 1.5 s/cm3, and (d) P5 2.0

s/cm-!

.

ito

0
232 CHAPTER 12

where AT is the time between echoes as discussed earlier. Peaks in the

form function occur when the individual echoes add in phase as the

incident pulse is lengthened. This occurs when the circumference of

the cylinder is an integer number of wavelengths. Leading to a second

relation [151:

c*/c, - (Aka)mk; (6)

therefore, if one starts with If. Ic vs ka, the circumferential speed may

be determined from the spacing of the successive equally spaced peaks

of If.1, For the target described in Fig. 6(b), the form function gives

(Aka)puk - 0.7, and Fig. 6(c) gives AT = 8.1. The speed ratio c*/c.

obtained from Eqs. (5) and (6) both give c"/cw = 0.7.

The relationship between surface (or circumferential) wave

characteristics [171 and guided wave propagation has been discussed

previously [16,17]. If the case Ps = 1.5 g/cm3 , CL - 1.0 km/s is taken

as typical of the speeds and densities considered here, the refracted

internal ray paths are as shown in Fig. 7. Seven typical paths are followed with incident angles 0 of 100, 200, 300, 400, 500, 80', and 900.

INCIDENT PLANE WAVE

100

oo

Fig. 7 - A cross section of a cylindrical target with

p, - 1.5 g/cmj, CL - 1.0 km/s, showing the internally

refracted ray paths.
RUBBER CYLINDERS AND SPHERES 233

The converging lens effect is seen in the bending of the incident wave

towards the normal, and all internal rays generated from normal to

tangential incidence exit the cylinder at angles within 140 of the

forwar--.- ittering direction. At incidence angles up to approximately

,, tih eflection coefficient curves [181 predict from 90% to 100% of

the energy is transmitted into and through the cylinder back into the

water continuing approximately in the forward direction. Thus, energy

in the ray paths beginning at 10° through 500 incidence shown in the

figure is almost entirely transmitted back into the water. For rays

beginning at 650 to 850 incidence angles, reflection-coefficient calculations predict both significant transmission and significant internal

reflection. Two legs of the ray path beginning at 800 are seen in Fig. 7,

and multiple reflections and transmission from rays generated at angles

greater than 650 incidence angle give the backscattered echoes seen in

Fig. 6(b). The path of the limiting internal ray, generated at tangential

incidence (0 - 90) is also seen in Fig. 7. Repetition of the same circumferential path gives the apparent circumferential behavior. A plot

of If. 1, vs angle at a ka of 12.0 for the material considered above is

given in Fig. 8. This shows the convergence of energy in the forward

scattering direction (Os - 0°) with the steady-state pressure amplitude

scattered in the forward direction 20 dB greater than that backscattered

(es = 1800) and 14 dB greater than that incident.

CONCLUSIONS

The scattering of sound from silicone rubber cylinders and

spheres was found to be equal to that of an identically shaped liquid

exhibiting the same density and longitudinal velocity. The resultant cw

scattering can be considered in two regions: (1) a low ka region defined

as that over which the first (or primary) lens wave is growing to its

equilibrium value and (2) the region of higher ka. In the low ka

region the scattering can be explained almost entirely in terms of

interference between this primary lens wave and specular reflection. In

the case where PZcL - pwc., no specular reflection occurs but there is

nevertheless a return which at low ka is due essentially to the primary

lens wave. The corresponding low ka region of If..I reveals the

manner in which the primary lens wave builds up. In the cylindrical

case this buildup is observed to be linear with ka, reaching an amplitude approximately one-half that of the incident wave for lossless

materials. For spherical configurations the build up is quadratic in ka,

and in the lossless case the amplitude approaches four times that of the

incident wave. In the higher ka region scattering is due to the interference between various lens waves and (when psCL p,c,) the specular

return.

L f' dU14~..~~ I
~7

234 CHAPTER 12

3300 3400 350* 0 10" 200 30'

320' 40'

310 V 50

300 ° 0 60.

2900 700

200 co

2 70' W 0

260* o00n

250" I 10"

240. 120

2302 130

2)20* 140"

210" 2000 1900 180" 170"0. 150

Fig. 8 - The bistatic-refraction-function curve (If_1l vs 0, ) for a

cylindrical target with p, - 1.5 g/cmn3, qL - 1.0 km/s. Backscattering

is 180° , forward scattering 0°.

REFERENCES

1. J. Faran, Jr., J. Acoust. Soc. Am. 23, 405-418 (1951).

2. R. Hickling, J. Acoust. Soc. Am. 34, 1582-1592 (1962).

3. A. Rudgers, NRL Report 6551 (1967).

4. L. Hampton and C. McKinney, J. Acoust. Soc. Am. 33, 694

(1961).

IsII mili i I I I ,
RUBBER CYLINDERS AND SPHERES 235

5. W. Neubauer, R. Vogt, and L. Dragonette, J. Acoust. Soc. Am.

55, 1123-1129 (1974).

6. L. Dragonette, R. Vogt, L. Flax, and W. Neubauer, J. Acoust.

Soc. Am. 55, 1130-1137 (1974).

7. R. Vogt, L. Flax, L. Dragonette, and W. Neubauer, J. Acoust.

Soc. Am. 57, 558-561 (1975).

8. R. Vogt, L. Flax, L. Dragonette, and W. Neubauer, J. Acoust.

Soc. Am. 62, 1315 (1977).

9. L. Flax and W. Neubauer, J. Acoust. Soc. Am. 61, 307-312

(1977).

10. L. Scheutz and W. Neubauer, J. Acoust. Soc. Am. 61, 513-517

(1977).

11. D. Folds, J. Acoust. Soc. Am. 56, 1295-1296 (1974).

12. P. Rogers and W. Trott, J. Acoust. Soc. Am. 56, 1111-1117

(1974).

13. D. L. Folds and J. Hanlin, J. Acoust. Soc. Am. 58, 72-77 (1975).

14. W. Mason, "Dispersion and Absorption of Sound in High

Polymers," in Handbuch der Physik, Vol. XI, Acoustics I, edited by

S. Flugge (Springer-Verlag, Berlin, 1961).

15. H. Uberall, L. Dragonette, and L. Flax, J. Acoust. Soc. Am. 61,

711-715 (1977).

16. M. Harrison, U. S. David Taylor Model Basin Report 872 (January 1954).

17. W. Neubauer and L. Dragonette, J. Acoust. Soc. Am. 48, 1135-

1149 (1970).

18. W. Mayer, J. Appl. Phys. 34, 909-911 (1963).

19. W. Mayer, J. Appl. Phys. 34, 3286-3290 (1963).
Chapter 13

ELASTIC CYLINDERS AND SPHERES

AT HIGH ka*

The scattering of sound by elastic cylinders and spheres immersed

in fluids has been discussed by a number of authors [1-41. In these

papers, and in others mentioned in their references, it is assumed that

the elastic body has incident on it a plane sound wave of pressure

amplitude Po and circular frequency w - kc, where k is the wave

number and c is the speed of sound in the fluid medium of density p.

The density and compressional and shear-wave speeds in the solid

material of the cylinder or sphere are assumed to be pl, cL, and c,

respectively. The radii of these bodies are denoted by a.

In general, one obtains the exact normalized scattered steady-state

pressure amplitude per unit incident pressure (form function) by means

of a partial wave expansion or normal mode solution. The solution

contains all creeping, as well as through, waves [5]. The form function

is usually expressed in terms of a nondimensional frequency parameter

ka. Agreement between partial wave theory and experiment is

presented in Refs. 2-4. To obtain much higher frequency values usually required elaborate expansions of asymptotic series [61 or a

creeping-wave formalism (71.

The present investigation shows that even at a ka of 950.0, no

limiting value of the form function is approached. Calculated results

are obtained for cylinders and spheres made of aluminum, for which

the material parameters are given in Table 1.

Table I - Material Properties

pxI0' CLxl0' c×10O'

(kWm) (m/sec) (m/sec)

Aluminum 2.7118 6.370 3.136 Water 0.998 1.482 ..

*Theue results first appeared in: Lawrence Flax, J. Acoust. Soc. Am. 62, 1502-1503

(1977)

237

. , A ......... n,. . -
238 CHAPTER 13

The mathematical solution for the scattered sound field is given in

Refs. 1-4. Essentially, the scattered monostatic pressure at large distance R from the sphere can be represented by the expression

2R . k, P CLC

where J is a complex function of the nondimensional frequency ka

and other nondimensional parameters related to the elastic constants

and geometric properties of spheres. Similarly, the farfield monostatic

reflection for cylinders is

po(a/2R),/2'f=jka, ¢L El]. (2)

The evaluation of f- entails computations of sums of Bessel

functions of the first and second kind and their derivatives. Integerorder Bessel functions are required for a cylindrical geometry and halfinteger order for a spherical geometry. These Bessel functions must be

evaluated for high orders and large arguments. The ability to compute

the scattered pressure depends upon two factors: the range of the

Bessel functions and the convergence properties of the form-function

summations. The calculations have been checked by single- as well as

double-precision techniques, experimental verification has been

obtained to a ka of 50 [1-41, and series convergence has been checked

by numerous methods. For a detailed discussion of validation procedures, see Refs. 8 and 9.

The form function for a sphere and a cylinder are shown in Figs.

I and 2, respectively. Values of Ifoo were calculated for a range of ka

io

9-

6

4-

2

0

900 9 940 95 geo 95 930 935 940 945 950

ko

F. I - Theoretical calculations of the monostatic

reflection (I. I) for an aluminum sphere
HIGH ka 239

30

2?

24

21

Is

t2

09

06

03

900 905 910 915 920 925 930 935 940 945 950

kO

Fig. 2 - Theoretical calculations of the monostatic

reflection (If.. I) for an aluminum cylinder

from 900.0 to 950.0 at intervals of 0.05. These calculations demonstrate that the form function does not settle down to a limiting value.

It is also apparent that the magnitude of If- I for a sphere in some

ranges has an amplitude over three times that of a cylinder. In these

calculations, no absorption was included.

REFERENCES

i. R. Hickling, J. Acoust. Soc. Am. 34, 1582-1592 (1962).

2. R. H. Vogt, L. Flax, L. R. Dragonette, and W. G. Neubauer, J.

Acoust. Soc. Am. 57, 558-561 (1975).

3. L. Flax and W. G. Neubauer, J. Acoust. Soc. Am. 61, 307-312

(1977).

4. W. W. Ryan, Jr., "Acoustic Backscattering from Cylinders," ARL

Report No. TR-73-20, University of Texas, Austin, TX (1973).

5. D. Brill and H. Uberall, J. Acoust. Soc. Am. 50, 921-939 (1971).

6. T. T. Wu, Phys. Rev. 104, 1201-1212 (1956).

7. R. D. Doolittle, H. Uberall, and P. Ugin~ius, J. Acoust. Soc. Am.

43, 1-14 (1968).
240 CHAPTER 13

8. J. P. Mason and R. V. Baler, "FORTRAN Subroutines to Evaluate,

in Single or Double Precision, Bessel Functions of the First and

Second Kinds for Integer or Halt Odd-Integer Orders and Positive

Real Arguments," NRL Memo Report 2493 (July 1972).

9. L. Flax and J. P. Mason, "FORTRAN Subroutines to Evaluate, in

Single or Double Precision, Bessel Functions of the First and

Second Kinds for Complex Arguments," NRL Report No 7997,

Washington, DC (1976).
Chapter 14

REFLECTION AND VIBRATIONAL MODES

OF ELASTIC SPHERES"

INTRODUCTION

Effects of resonances on the scattered sound by solid elastic

spheres immersed in water have been observed by several investigators

11,21 The normalized scattered steady-stale pressure amplitude per unit

incident pressure regarded as a function of frequency, i.e., the form

function, shows features attributable to the free vibrations of the elastic

sphere. For the case of backscattering by lossless metal spheres considered here, the low-order characteristic frequencies of the free sphere

are separated enough for the effects of its resonant behavior to be seen

in the form function. However, when the frequencies of the natural

modes of oscillation of the vibrating sphere lie close to one another,

the form function does not necessarily show the separated effect of

each resonance.

Faran [11 first considered the origin of the near nulls observed in

the backscattered pressure amplitude, but for cylinders in water instead

of spheres. He observed that they occurred close to the frequencies of

natural vibration of the free elastic cylinder. Effects of resonances on

the scattering of sound by a fluid-filled cylindrical cavity in an elastic

solid have also been observed. 13) In Faran's work, the distribution in

angle of the scattering was observed and computed for relatively few

selected values of the frequency. In the present work, only backscattering and free vibration of spheres was considered and the monostatic

form function was computed for a continuous range of frequency

parameter (ka, where k is the wave number and a is the sphere

radius) from 0 to as high as 28, thereby showing the relationship

between several resonant frequencies of the free sphere and the shape

of the form function.

*These results first appeared in: R H Vogt and W ( Neubauer, J Acoust. Soc Am .

15 (1976).

241
242 CHAPTER 14

Expressions which describe the scattering of sound by lossless

elastic spheres in water are reviewed here briefly for purposes of

definition. (1,41 Let a steady-state plane wave traveling in the direction

of the positive z-axis and incident on a sphere of radius a centered at

the origin be given by

poexpli(wi - kz)]. (I)

where w and k are the angular frequency and wave number. The sound

scattered into the far field at a distance of r from the origin may be

expressed in spherical polar coordinates as

p - (poa/2r)exp[i(wt - kr).f., (2)

where ./ is the far field form function,

fe= - -(2/ka) j (2 n + I)P,(cos)sinq, exp(iiqn) (3)

R,-0

and

tanTI, - - (a/b,). (4)

with

a, - ,j,,(ka) - (ka),j,(ka). (5)

b, = 4,,y,(ka) - (ka)y,(ka). (6)

The quantity c,, defined in the Appendix, is a function of the angular

frequency w and the material properties of the sphere. The quantities

i, and yv are spherical Bessel functions of the first and second kind and

the prime denotes the derivative with respect to the argument. When

4,, is set equal to zero in Eqs. (5) and (6), the expressions describe the

steady-state scattering by a rigid sphere. In the case of monostatic

scattering, the value of the Lengendre polynomial P,(cosO) is (-I )".

The quantity ,. becomes infinite at the characteristic frequency of

each normal mode of vibration of the free elastic sphere. Torsional

modes are disregarded since they are not excited in the case under consideration. The equation determining the characteristic frequencies is

given in the Appendix. Associated with each n in Eq. (3) there is a

sequence of modes of vibration of the free elastic sphere. It is of

interest to determine physically how descriptive a single term in Eq. (3)

is of an isolated resonance effect observed in the form function.

DISCUSSION

Frequency Domain

The nth term in the expression for the scattered pressure (Eqs

(2) - (6)] can be separated into a part which would be present if the
MODES OF SPHERES 243

sphere were rigid and a part describing elastic reradiation 151. For the

reradiated part, the elastic sphere is regarded as being driven by the nth

components of the incident pressure and the "rigidly scattered" pressure. When the terms are separated as just described, the expression

for the scattered pressure can be written

I . PoP" (coso)

p - - kriexpti(ojt kr----rhh(ka)- kr)] (2n + 1) M-0O

X k + (P C) (7)

I. ) + (ka)2h(ka)(Z, + z,)

where

zn - ipch, (ka)/h(ka)

is the radiation impedance and

Z, = - ipcka/,

is the modal impedance of the elastic sphere, h,(ka) is the spherical

Hankel function, j,(ka) - iy,(ka), and pc is the specific acoustic

impedance of the fluid medium surrounding the sphere. The modal

impedance Z,, of the sphere becomes zero at the resonant frequencies

of the free sphere and becomes infinite for the rigid sphere. The terms

remaining for the case when all Z,, are set equal to infinity in Eq. (7)

give the part of the pressure which would be scattered by the rigid

sphere. A single term of the radiated part is referred to as a resonant

term in the following and its amplitude, normalized in the same way as

f- in Eqs. (2) and (3), is designated jp,j where the term number n is

not shown explicitly.

The amplitudes 1p, I of the resonant terms of the immersed sphere

labelled by the corresponding modes of vibration of the free elastic

sphere are shown plotted in Fig. I in a range of ka from 0 to 15 for

three materials. Sphere modal notation is described in the appendix.

The size, spacing and overlap of the resonance terms can be seen in the

figure. The width of the resonance is the ka interval between the point

on either side of the maximum where the amplitude is l/l.,2 times the

peak value. The width of the (2, 1) resonance of tungsten carbide is

0.15. The figure shows an increase in the width of the (n, 1) resonances of aluminum and brass with increasing n. The various overtones have smaller widths and amplitudes. It appears that the fundamental resonance terms of a given material have nearly the same peak

amplitudes. In the regions between the peaks of a given resonance, the

nth resonant term varies smoothly in amplitude, as, for example, for

brass in the region between the (4, 1) and (4, 2) resonances. The (1,

I-laI
244 CHAPTER 14

(2,0 (31) (4.1)

TUNAGSTEN

CARBIDE

ALI(N U M ( (1

0 (2J)

(3.0) (4.0 (5.0 61) 1'1

2a

Fig. I - Modulus [p,I of resonant terms as a function

of ka for tungsten carbide, aluminum, and brass.

!) mode corresponds to a linear motion of the sphere as a whole and

does not appear so that the lowest (1, 1) mode is (1, 2). The (2, 2)

resonance in aluminum which is near ka of 10.8 has a negligibly small

amplitude and does not appear in the figure. The (0, 1) and (0, 2)

resonances correspond to pure radial oscillations and are small and

wide.

For small values of the integer n, except 0 and 1, the fundamental

frequencies of the modes expressed as k2a, vary slowly as a function of

poisson's ratio [1,61. The wave number k2 corresponds to the speed of

shear waves in the sphere material. In the particular case of the (2, 1)

mode, k2a lies between 2.6 to 2.7 for a range of poisson's ratio between

0.05 and 0.45. The corresponding frequency in the scattering form

function is the product of k2a and the ratio of the shear speed in the

material to the speed of sound in water. The effect in the form function is an increase in the value of ka corresponding to the lower resonances in proportion to the shear speed. The (2, 1) modes of free
MODES OF SPHERES 245

vibration have ka values of 7.43, 5.57, and 3.78 for tungsten carbide,

aluminum and brass, respectively, which are very nearly equal to the ka

values of the respective (2, 1) resonance peaks in Fig. 1. The

corresponding shear speeds are 4185, 3120, and 2110 rn/sec which are

very nearly in the same ratios as the ka values of the resonance peaks.

Figure 2 shows the monostatic form function of a tungsten carbide sphere. A mode of vibration of the free elastic sphere is indicated

in the figure by a short vertical line above or below the curve with the

integer corresponding to n on the left and to I on the right. The ka

value of the characteristic frequency of the mode is given by the

abscissa of the short vertical line separating n and I on the scale of ka

immediately below it. It can be seen in Fig. 2 that values of ka

corresponding to frequencies of vibration of the free sphere, which are

close to the peaks of the resonant terms of Eq. (7) plotted in Fig. 1, are

also close to minima or other rapid changes in the modulus of the form

function in tungsten carbide. The form function approaches that of the

rigid sphere in the regions of ka lying between the first few widely

spaced resonances. Figure 3 shows the form function for an armco iron

sphere, similar in appearance to tungsten carbide for ka less than 15,

where the resonances are widely separated and appear near minima. In

Fig. 4 the curve shows the form function for an aluminum sphere. The

small open and closed circles near the curve are experimental measurements discussed elsewhere [2). Figure 1 shows the broad width of the

lowest modal peaks (the fundamental set for which I is unity) in aluminum, i.e., at resonances (2, 1), (3, 1), (4, 1), and (5, 1). Broad dips

are also observed in the form function of Fig. 4 at fundamental modal

values. However, narrow peaks and dips are found in Fig. I and Fig. 4,

respectively, for the overtones (1, 2) and (3, 2).

The form function for a brass sphere in Fig. 5 appears to have

maxima near the resonant frequencies with the minima following on

the side of higher ka, apparently the result of the addition of the phase

of the nth elastic term to the total amplitude calculated for rigid scattering near the resonances. The narrow resonances of the overtones again

appear in the form function as sharp breaks, in contrast to the more

slowly varying features which are related to the large broad resonances

at the fundamentals.

The effect of a single mode on the overall form function in the

region of interest can be seen qualitatively by modifying the t corresponding term in the expressions for the form function of a rigid

sphere or of the elastic sphere. If the form function of the rigid sphere

is computed using only the nith term given by Eq. (7) which characterizes the elastic sphere and all other terms appropriate for the rigid
246 CHAPTER 14

TUNGSTEN CARBIDE SPHERE

FAR - FIELD (kr--,) 214 91'

z01

0

Z 200 210 22.0 230 240 250 26.0 270 28,0

UC3101212 411 511 312 113

0

Lw

-J 0 -- -

w 100 110 120 130 140 15.0 160 170 180 190

a0

0

0

10 2.0 3.0 4.0 50 60 70 80 90

SPHERE SIZE - PARAMETER (ka)

Fig. 2 - Computed form function and resonance positions

for a tungsten carbide sphere.

ARMVCO IRON SPHERE

3.0- ~FAR -FIELD (kr-.) 521 1

S150 16 170 418. 2190 200 . 2. 20-

0

20-

X

10.0 80 310 9 220 1 1 I

0

P- 1.

0

w

,0 1 90 2 00 110 .0 0 0 15

Fig 3.0 21optdfrmfnto1adrsnncoiin f oranArcoirn phre

2.
MODES OF SPHERES 247

ALUMINUM SPHERE

FAR- FIELD (kr--.) 012 1.5 612

1.0 -,512 4 513

.5 '911

Isl l21 li 1211 I 13 115 011 2k 111 1211

z 0 I I "0

, 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0

,. I 11 13z3 1

0 13.0 14.0 15,0 16.0 170 18.0 190 0( 0

0.I. l 2112 31111 71 41 213 

LL 0

50 60 70 8 9.0 1.0 1.0 12.0

SPHERE SIZE-PARAMETER (ka)

Fig. 4 - Computed form function and resonance positions

for an aluminum sphere.

BRASS SPHERE

FAR -FIELD (kr-,,-)

3.0 10l 413 214 IIII 1211 314 012 Is 1 l 2 414 1411 216 ---- " 315 912

2.05

0 1311 513 

S .0 I I I I

LL 15.0 160 170 18.0 19.0 20.0 210 22.0

S3.0-FRFED(rw

-51 Oil 312 611 1 412 eli 114 512

0 2 .031430 2 1 8 9 1 2

z

2

. 0 I I I I I

"' 8.0 90 1 00 1 .0 12. 0 13.0 140 150

at/11 1 1 12 ,1 811 114 ' 0. 2o

10 20 3.0 4.0 51 6.0 70

SPHERE SIZE- PARAMETER tka)

Fig. 5 - Computed form function and resonance positions

~for a brass sphere.

?z
248 CHAPTER 14

sphere, the result is a rigid sphere form function showing the effect of a

given fundamental and its overtones. Or if the form function of a

given elastic sphere is computed with Z, infinite in the n th term,

thereby in effect substituting the n th term appropriate for a rigid

sphere, the resulting form function is that of the elastic sphere without

the nth resonances. For example, the solid curve in Fig. 6 is the form

fui -tion for a tungsten carbide sphere. The solid circles which are connected by a curved line show points of the rigid sphere form function

calculated in the range of ka from 5 to 15 with the term for n - 2

replaced by that for tungsten carbide. The circles appear close to the

form function of the rigid sphere (within 2% for ka between 10 and 13,

for example) except near the resonances corresponding to n - 2,

namely, (2, 1) and (2, 2), where they approach the form function of

tungsten carbide. Or if the form function of tungsten carbide is computed with the second term modified by setting Z 2 equal to infinity, the

resonances corresponding to the second mode disappear. The results of

TUNGSTEN CARBIDE SPHERE

'1, ,2FR FIR -FIELo (kr-a) 214 91t

611 14 42 238,,313

g_ 012\ I- II l

z 20.0 21.0 22.0 23.0 24.0 250 26.0 270 280.

0 1120 12 411 51 312 113

Z 1 .0 7L

0: W

L 0 I I 100 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0

U.

0

0*

01/

1.0 2.0 3.0 4.0 5.0 6,0 70 8.0 9.0

SPHERE SIZE- PARAMETER (Na)

Fig. 6 - Computed form function for a tungsten carbide sphere (solid

line). Points on the form function computed from terms appropriate for a

rigid sphere except for n - 2 for which case the appropriate "elastic, term

is used (solid circles connected by a curved line) and computed from

terms appropriate for an elastic sphere except for n - 2 for which case the

t appropriate "rigid" term is used (x's).

,Imm a a m•H i ol•Hi
MODES OF SPHERES 249

such a calculation are shown by the x's in the figure which fall near the

tungsten carbide form function except in the regions of the (2, h resonances. Figure 7 shows a similar calculation for the (2, 0 resonances

in brass. If the points corresponding to the modified rigid sphere,

shown by solid circles connected by a line, are followed starting at

/ca - 3, it can be seen that they approach the form function of brass at

the resonance (2, 1) and they show a dip corresponding to minima at

resonances (2, 2) and (2, 3). A steplike increase can be seen at resonances (2, 4) and (2, 5). At values of ka sufficiently removed from the

(2, h resonances, the circles approach the form function of the rigid

sphere (within 2% for ka between 8 and 13, for example). Also, it can

be seen that the x's are near the form function of brass except at the

resonances (2, 1), (2, 2), (2, 3), (2, 4), and (2, 5).

BRASS SPHERE

FAR-FIELD (kr-e)

oll 113. 413 214 lilt lot 012 0 15 414 1

1.0 1 1

21 1.0 1 IV1)

11- 612 U 0 _ 1 I 1 -1 - I

3 15.0 16.0 1 00 I.0 19.0 20.0 210 22.0

U.

2 6.0 9.20 10 9. 30 9. 5

f2,3

101

U) 211 112 VI '212

1.0 2.0 30 4.0 5.0 6.0 70

SPHERE SIZE- PARAMETER (ka)

Fig. 7 - Computed form function for a brass sphere (solid line). Points

on the form function computed from terms appropriate for a rigid sphere

except for n - 2 for which case the appropriate "elastic" term is used

(solid circles connected by a curved line) and computed from terms appropriate for an elastic sphere except for n - 2 for which case the appropriate "rigid" term is used ('s).
250 CHAPTER 14

The Time Domain

In the case of an incident plane wave, the pulse scattered at a distance r from the center of the sphere of radius a is given by

p(7) - (poa/2r)(1/2t)Re fow g(ka)f(ka)exp(ikai)d(ka), (8)

where 7 - (ct - r)/a is a propagation parameter appropriate to an outgoing spherical wave and Re denotes the real part [7, 81. The normalized complex spectrum of the incident plane wave is given by g(ka),

and f(ka) is the form function of the elastic sphere immersed in a

fluid. The shape of the form function f(ka) within the frequency

bandwidth of the incident pulse determines the behavior of the scattered pulse in time. Hickling [81 pointed out differences in the computed scattered pulse forms for a long truncated sinusoidal incident

pulse. He showed in particular cases that the pulse form of the echo

depends on whether the central frequency of the incident pulse is at the

position of a maximum or an adjacent minimum of the form function.

A simpler situation results when a long truncated sinusoidal pulse has a

central frequency at a value of ka coinciding with the peak of a single

isolated resonance in the form function. For a sufficiently long incident

pulse, the vibrating sphere approaches a steady-state oscillation and at

the termination of the pulse, the amplitude of the oscillations of the

immersed sphere decays with a time constant related to the width of

the resonance. By analogy with the simple oscillator

A(ka)/(ka)o- l/Q,

where (ka) o corresponds to the resonant frequency and as before, A ka

is the full width of the resonance peak at I/" times the maximum

amplitude. Experimentally Q was derived from the expression

Pl/Po - exp(-irN/Q), where P0 is the amplitude near the beginning of

the decay and p, is the amplitude N cycles later. The (2, 1) resonance

of tungsten carbide at (ka)o of 7.37 and width A(ka) of 0.15, as previously mentioned, is a case for which a comparison of calculation and

measurement was made. The scattered pulse form was computed by

Eq. (8) for an incident truncated sinusoidal pulse of 60-cycles duration.

the envelope of the computed pulse is shown in Fig 8. The abscissa is

given in terms of r, which in this case is such that each sinusoidal cycle

corresponds to 0.85 units of 7 and the pulse first appears at v - - 2. It

can be seen from the figure that the amplitude of the pulse envelope

approaches steady-state after the initial transient and that after a duration in 7 of about 51 corresponding to nearly 60 cycles, the duration of

the incident pulse has passed and is followed by the transient oscillations of the radiating sphere. From the decay of the radiated amplitude, the Q is estimated from a semilogarithmic plot to be 49 which
MODES OF SPHERES 251

go--

V

(2,t) TUNIGSTEN CARBIDE,

0 s0 100

Fig. B - Computed echo envelope of a pulse monostatically

reflected from a tungsten carbide sphere. The pulse spectrum is

centered at the (2. 1) resonance dip in the rorm function for

tungsten carbide. The first cycle within the envelope is shown.

agrees with the value obtained from the width of the computed

resonance by the use of Eq. (9), 7.37/0.15 - 49. Figure 9 shows the

record of a scattered pulse observed when a truncated sinusoidal pulse

of 300 itsec duration, very nearly 60 cycles, is incident on a tungsten

carbide sphere at ka of 7.36 corresponding to the (2, 1) resonance.

from the figure, the decay of the second transient gives a value of Q

estimnated to be 40. In the same way for the (3, 1), (4, 1), (5, 1), and

(6, 1) resonances of the tungsten carbide sphere, the values for Q were

found from the widths all to be 60 with corresponding experimentally

measured values of 60, 50, 50, and 60.

A narrow resonance with small 41(ka) has a large Q when compared with a nearby wide resonance with large A (ka). The resulting

difference in decay times sometimes makes it possible to detect the

presence of a small narrow resonance in the vicinity of a large one by

observation of the "long" decay of the echo after the forced excitation

of the sphere is turned off. As an example, the (1, 2) resonance for an

aluminum sphere is narrow and small compared to the nearby (3, 1)

resonance shown in Figs. I and 4. The echo return related to the (1,

2) resonance of aluminum sphere is shown in Fig. 10. The part of the
252 CHAPTER 14

(2.1) TUNGSTEN CARBIDE

Fig. 9 - Observed monostatically reflected

pulse from tungsten carbide sphere

corresponding closely to theoretical conditions of the computation shown in Fig. 8.

" ae time scales are different for Figs. 8 and

9.

1.2)

Fig. 10 - Observed monostatically reflected pulse

whose spectrum is centered at a k at which the (I, 2)

resonance is computed. The reflecting sphere is of

aluminum.

I I •I I I ...I
MODES OF SPHERES 253

pulse with large amplitude corresponds to the steady-state forced oscillation of the sphere and the "tail" following shows the amplitude decay

of the vibrating sphere and gives a Q value of 90 in comparison with

the value 90 as obtained from the width of the (1, 2) peak in Fig. 1.

The echo return from the (3, 1) resonance in Fig. I1I shows first the

transient peak when the sphere is excited by the incident pulse and following, the transient pulse which appears when the incident pulse is

turned off. Comparison with Fig. 10 illustrates the relatively rapid

decay of the wide resonance. A measured Q of 7 was estimated in

agreement with a value of 8 obtained from the width of the resonance

peak. In the same way, the (4, 1) resonance of aluminum was found

experimentally to have a Q of 10 and a computed Q of 9. It appears

then that the experimentally determined decay of the undriven sphere

is in approximate agreement with the value obtained from the computed resonance width in the case of observable isolated resonances or

of narrow resonances situated near broad ones. When the computed

resonance term indicates a sharp peak, the computed results are more

affected by uncertainties in the values of the maximum amplitude and

in the widths of the resonances. When the peak is broad and the decay

rapid, the experimental measurements are difficult. The experimental

and calculated results are, therefore, given with no indication of their

accuracy.

(3.1) ALUMWI)M

Fig. I I -Obser ved monostatically reflected

pulse whose spectrum is centered at a ha at

which the (3, 1) resonance is computed.

The reflecting sphere is or aluminum.
254 CHAPTER 14

APPENDIX

The monostatic far field form function f.. is given in the form [41

2 (-l)1(2n + 1)sine",

with

tan n - " Cj,,(ka) - (ka)j.(ka),,,yn(ka) - (ka)y,(ka) (Al)

in which j 8 (ka) and yn(ka) are nth order spherical bessel functions of

the first and second kind and the primes denote derivatives with respect

to the argument.

The quantity C. in Eq. (A ) is defined by

C.- .P (k 2a) 2

/2][(A./B) - (C.ID)] W)

Pn (E1B.) - (F/D.)

where

A. - (k a)j.(kla)-

B. - (k, a)jn(kla) - jj(kna),

C. - 2n(n + l)j.(k 2a),

D. - (k2a) 2j'(k 2a) + (n - l)(n + 2)jn(k 2a),

E. - (kia )2[I 2 jai(kia) - i'"(ka)I

F. - 2n(n + l)11.(k 2a) - (k 2a)j'(k 2a)],

with double primes denoting second derivatives. In the expression for

E., Poisson's ratio of the sphere material is a given in terms of the

longitudinal and shear sound speeds cl and c2, respectively, by

ar- (c? - 2c?)/2(c2 - cj).

The wave numbers k, and k2 corresponding, respectively, to the longitudinal and shear waves in the sphere material are related to the wave

number k in the surrounding fluid by

kja - (c/c,)ka

and

k2a - (c/c 2)ka.
-

MODES OF SPHERES 255

The characteristic equation for the resonant frequencies of the

spheroidal modes of the free vibrating sphere results from setting the

denominator in Eq. (A2) set equal to zero,

DE. - B.F. - 0.

The modes are denoted by integers (n, I). For each value of n there is

a fundamental frequency and overtones which are given by I with I - 1

representing the fundamental [9]. (The fundamental is sometimes

denoted by I - 0.) The (0, 0 modes are purely radial oscillations. The

(1, 1) mode corresponds to rigid body motion of the sphere as a whole

and does not appear as a resonant frequency of free vibration. When

the sphere is immersed in water, however, the rigid body motion

appears in the overall shape of the form function below about ka - 5

as shown by Hickling and Wang [101. The spheroidal mode (2 1) has

the lowest frequency for a free sphere and corresponds to a motion

with the shape of the sphere alternating between a prolate and an oblate

spheroid.

REFERENCES

1. J.J. Faran, Jr., J. Acoust. Soc. Am. 23, 405-418 (1951).

2. W.G. Neubauer, R.H. Vogt, and L.R. Dragonette, J. Acoust. Soc.

Am. 55, 1123-1129 (1974).

3. Y. Pao and W. Sachse, J. Acoust. Soc. Am. 56, 1478-1486 (1974).

4. A.J. Rudgers, NRL Report 6551 (1967) (unpublished).

5. M.C. Junger and D. Feit, Sound, Structures and Their Interaction

(MIT Press, Cambridge, MA), pp. 333.

6. D.B. Fraser and R.C. Lecraw, Rev. Sci. Instrum. 835, 1113

(1964).

7. A.J. Rudgers, J. Acoust. Soc. Am. 45, 900-910 (1969).

8. R. Hickling, J. Acoust. Soc. Am. 34, 1582-1592 (1962).

9. Y. Sato and T. Usami, Geophys. Mag. 31, 15 (1962-1963).

10. R. Hickling and N.M. Wang, J. Acoust. Soc. Am. 39, 276-279

(1966).
Chapter 15

RESONANCE EXCITATION AND

SOUND SCATTERING*

INTRODUCTION

The formalism of the classical resonaice theory of nuclear reactions (1, 21 is applied to the problem of acoustic scattering from submerged elastic circular cylinders and spheres. This approach demonstrates in a direct fashion that the strongly fluctuating behavior of the

cross section for sound scattering from elastic bodies, as shown, e.g., by

numerical normal-mode calculations of the backscattering amplitude

due to Hickling [31, is caused by a superposition of (generally narrow)

resonances in the individual normal-mode amplitudes (or partial

waves). This was inferred 141 from considerations of the total backscattering amplitude, where the connection of the scattering resonances

with creeping waves (Regge poles*) on the surface of the scatterer, and

with the eigenvibrations of the elastic body had been established

together with a physical condition for excitation of these resonances.

Here the excitation of elastic-body resonances by incident-plane

acoustic waves are discussed by considering the individual partial waves

in the Rayleigh series. The resonance scattering formalism is

developed for solid elastic cylinders and spheres, employing the

"method of linear approximation" in frequency as used in nuclear resonance reaction theory. The scattering amplitudes are obtained as a sum

of resonance terms and of a geometric background amplitude, while the

elastic waves in the interior are found to be given as pure resonance

terms. Numerical values of the partial-wave scattering amplitudes are

presented, which identify the background as the scattering amplitude of

a rigid body, and demonstrate the Rege pole behavior of the elastic

resonances appearing in the amplitudes. Various types of interferences

between resonances and background are analyzed, and the time decay

("ringing") of an excited resonance is shown to be inversely related to

the width of the resonance.

These results were first reported in L Flax. L R Drallonette and H Uberall. J Acoust

Soc Am 63. 723 (1971)

257
258 CHAPTER 15

The phases of the partial waves, demonstrate that phase jumps of

i. take place when one passes through a resonance. An attempt is

made to identify the lowest "Franz-type" creeping-wave resonance in

this way. Angular distributions and resonance effects appearing therein

are also considered. A connection is established between the total

rigid-body scattering amplitude and certain resonance nulls appearing in

the angular distributions.

RESONANCE THEORY OF ACOUSTIC SCATTERING

Scattering From Elastic Cylinders

An infinite plane acoustic wave exp i(kx - wt) with propagation

constant k - o/c, incident along the x axis on a solid elastic cylinder of

radius a and density Pc, produces the following scattered field p,, at a

point P(r, 0) located in the fluid of density p. surrounding the

cylinder [5, 61:

in L. J.(X) - XJ.(X)

,-o L.H,."(X) - XH,)'(X)

x H,,()(kr) cos n4; (la)

here

eI (n-O0), 2 (n > 0),

and

L. - (hp)(D."/f4 2 ) I b)

where D (" and D.(

11 are 2 x 2 determinants:

D 2 : a23 Ia 31 a13 (1)

-j 3j a3311 - )a, a331(c

whose elements are given in Chapter 5. The argument X of the Bessel

and Hankel functions in Eq. (la) is X - ka - wa/c., with c, the speed

of sound in water. The primes denote differentiation with respect to

the arguments. The matrix elements a, of Eq. (Ic) contain Bessel

functions with arguments XL - kLa - wa/cL and XT - kTa - wa/CT,

where CL and c are, respectively, the speeds of the compressional and

transverse waves in the cylindrical material.

For simplicity in future expressions we define

iR.(X) - L.J.(X) - XJ.(X) (Id)

4 L.H.(')(X) - XrH*("(X)
RESONANCE AND SCATTERING 259

The scattered pressure given in Eq. (Ia) may be rewritten as

PSC - I In"(S. - l)H(')(kr) cos n4i, (2a)

2.-0

where we introduced the scattering function S. = exp (2i8.) familiar

from nuclear scattering theory [71, with 8, the so-called scattering

phase shifts. In the present case, one has

S, - 2R, + 1. (2b)

The far field value of p, can be obtained by employing the Hankel

asymptotic form for kr >> n,

Hg'M(kr) _ (2/irikr)/ 2 i-neir, (3)

and we may introduce the far field scattering "form function"

f (o) E f. ('), (4) n-O

consisting of "partial-wave" contributions

fn(4') - (#rika)- 1/2e.(S. - 1) cos n4'. (5)

In terms of f(4), p,, becomes asymptotically

P-. (a/2r)PI2 ekrf(,*). (6)

For a rigid cylinder, one has asymptotically 181 for the case of backscattering (40 - r)

If(')(m) 12 - 1. (7a)

This can be obtained by considering the limit of RM(X) as Pc -" 00:

[R.(X),,,- - J.(X)/H.)'(X). (7b)

(The corresponding soft-cylinder expression is, incidentally, given by

dropping the primes.)

Following Ref. 6, the expression for S., Eq. (2b), may be

represented in either of the two forms

S. - S.:s)(L. - Z2)/(L. - z,) (8a)

- S,(')(L,' - zf')/(L;' - z-' ). (8b)

where

S ) - - H,(2(X)/H,( 1 (X) e 2

V4' * (9a)

and

S W - H (2)'(X)H,( '(X) -- e "

) (9b)

e (9b)
260 CHAPTER 15

represent, respectively, the S functions for scattering from a soft or a

rigid cylinder, with fs)," the corresponding phase shifts. The latter

can be shown to be given by the real quantities:

tan fs) - J,, (X)/Y(X), tan f(r ) - J,(X)/Yn(X). (10)

In Eqs. (8), use was made of

zi- XHg )(X)/Hn(')(X), i - 1, 2. (11)

Equation (1) used with Eqs. (8) shows that the limits of a soft (Pc <<

p,) or rigid (p, >> Pw) cylinder correctly reproduce the corresponding

soft- or rigid-cylinder scattering functions Sns), S "

r , respectively.

The quantities z of Eq. (11) may be separated into real and imaginary parts as follows:

Z1.2 - A ( ± t is ( (12a)

or

Z.2= s "

) (12b)

with

A(S) X J(X)J(X) + Y,(X) Y.(X)

J,2(X) + Y,(X) (13a)

S"(s) 2 1 (>0), (13b) Ij S(2)(X) + Y,2(X)

1 Jn(X)Jn(X) + Yn(X) Y An(')- ' 2(14a) .(X)

X [J.(X)12 + [Y"(X) 12

j2 2 1 A irX 2 [j.(X)12 + [¥y(X)]2 (<0). (14b)

We now employ the linear-approximation method of nuclear resonance

theory, in which we define "resonance frequencies" X s) or X. '

) by the

conditions:

Ln(X.('))- (15a)

L; (Xn(

')) - A() (15b)

which may lead to a multiplicity of solutions for a given n. The quantities (L. - A(3)) or (L - ' - A()) are assumed to be linearly varying

with frequency, and expanded in a Taylor series in X, in the vicinity of

any one of the resonance frequencies

L.(X) A($) + p(3) (X - X"')), (16a)

L;'(X) A (') + - X',); 0',(X (16b)
RESONANCE AND SCATTERING 261

here, we did not distinguish the multiplicity of resonance frequencies

explicitly by a further index. One may also introduce a "resonance

width" r,, by the definitions

(= - 2SS)/Bls), (17a)

-- 2 2Sr)/#g. (17b)

The S function may then be rewritten in either of the two resonance

forms:

x - XP- ir(S

S 2 S _ en.S(S) s) + 2 1 (18a)

S - - (1 8b) 1 i(

2"

Note that because of the reality of the quantities involved one has the

"unitarity relations"

IVs I s")l--I s")lI = 1, (19)

expressing the fact that no energy is lost in the scattering process (for

the elastic scatterer, this is due to our implied assumption of no absorption, i.e., real values of XL, XT). Furthermore, S, is seen to possess

not only a resonance pole at the complex frequency

Xfi -

" = - -2 iFr, (20a)

but also a resonance zero at ___ 1

X - ( " - X + - iFr (20b)

necessary in order to satisfy the unitarity condition. Since the resonance widths F, as defined in Eqs. (17a) and (17b) will be seen below

to be positive quantities, the pole is thus located in the lower half of 1

the complex X plane, -L r. determining its distance from the real axis,

and the zero in the upper half plane at the same distance above the

axis.

The quantity (S. - 1) which appears in the scattering amplitude

or in the partial waves, Eqs. (2a) or (5), and which has the forms

£ l l I I H
262 CHAPTER 15

S. - 1 e " sin 8. (21a)

S)Sn (s) - (S)

- 2ie21f. - + e a sin f(s)I (21 b)

,,,s,, r) W"

2ie : L ' ) iS ( ) + e sin(') (21c)

may be represented by resonance expressions

(S. - 1)/2i -e '

8 sin 8. (22a)

Se2if (

S) X,(s)- 2 X-1 iFr() + -if sin (W (22b)

2 r,

jr(r)

- e 2i ( 

. + e sin , . (22c)

XJr)- x iFr(r)

2

With Eq. (5), one then has the partial-wave form functions

fn(4) = 2ien(irika ) - 1 2 e " sin 6,, cos n. (23a)

From Eqs. (22), these consist of a (soft or rigid) background or

"potential-scattering" contribution:

f,($). - 2iEn(1rika)-1/2ef (

3f sin ( cos n , (23b)

f~rp n -2iE (1rika)-1/2 i

f((r) r cos n 4 (23c)

(it will be seen below that for the example of an aluminum cylinder in

water, the background corresponds to rigid-cylinder scattering) with, in

general, a large number of resonances superimposed.

In the vicinity of one of these, the partial wave is given by (e.g.,

for a rigid background)

f.(40) -2ie, (Vika)-1/2e~ i ('

2 .r() re sin (

r) cosnf. (23d)

2

When calculating the form function as Eq. (4), interferences between

the resonances and the background will appear, which may sometimes

impart striking shapes to the resonance curves, as will be seen below.
RESONANCE AND SCATTERING 263

Several authors have observed [4, 9, 101 that the location in X of

the resonances obtained both theoretically and experimentally in f(ir)

and f.(ir) coincide with the eigenfrequencies of elastic vibrations of the

scattering body. In fact, these eigenfrequencies are obtained from the

condition of a vanishing denominator in Eqs. (8), which leads to real

eigenfrequencies for a free body (p. << Pc), but to complex eigenfrequencies for the fluid-loaded body. (For the example of an aluminum

cylinder in water as discussed below, the real part of the latter agrees

exceedingly closely with the real eigenfrequencies.) From our resonance representation, it is now clear that the complex eigenfrequencies

of the scatterer are precisely the locations of the resonance poles [Eq.

(20a)] in the complex frequency plane, whose real parts determine the

resonance frequencies in the scattering amplitudes. It should be noted,

however, that the scattering body may possess more modes of eigenvibrations than those appearing in the amplitude of acoustic scattering, if

the latter cannot excite the corresponding mode. For example, the n =

0 "breathing modes' have been found to be excited with a very small

amplitude only, both in cylinder and sphere scattering [4, 101.

It will be instructive to consider, in addition to the external

scattering amplitude, also the solution of the scattering problem in the

interior of the scattering body. There, the displacement field u of the

cylinder is represented in terms of a scalar (Wj) and a vector potential

(A):

u--VW7+V x A, (24)

with the solutions 181

'1 - jnCnJ.(kLr) ,i cos no, (25a)

n-0

Az- . ei"EJ.(krr) sin no. (25b)

iw-0

The coefficients in these partial-wave series are given in terms of

preceding notation as

2i (X/ - 2n2)J.(Xr) + 2XTJ.(XT)

SrP.w 2 XHn(1'(X)D,.')(zj l - L.-')

E 2i 2 n[J(XL) - 2XLJ.(XL)(

rp W2 XH.,),(X)D,, t

)(z., - L_') ,(6b)

where only the version leading to the rigid background has been written. The expansion of Eq. (16b) then leads immediately to the resonance expressions (in the vicinity of each resonance):
264 CHAPTER 15

2 0 (XT - 2n 2)J.(XT) + 2XT4 (XT)

1T1 Sa .

' -o . H,'')D ('

x J.(kLr) cos no2a ,(27a)

x- xJ' + 2

A- 2 e i" 2n[J(XL)-XLJ.(XL) J.(kTr)sinn4 A -ripwo~ fi-0 R XH(')'(X) D,() x X(r) 1 Wr.,

" 2"

(27b)

These results are of a pure resonance form, without (rigid or soft)

background terms, which is logical since rigid or soft objects are

impenetrable and admit no fields in their interior.

The above results, together with the scattering amplitudes of Eqs.

(22), indicate the remarkable fact that for scattering at a frequency in

between two eigenfrequencies of vibration of the elastic body, the

scatterer appears as an impenetrable object and scatters sound accordingly ("potential scattering"), while at and near an eigenfrequency, a

scattering resonance is excited by the incident wave and the field

penetrates into the body. In the scattering amplitude, the resonance

scattering interferes with the potential scattering, and this interference

causes the strongly oscillating structure observed previcusly in the total

scattering amplitudes [3, 41.

In some literature, the resonance features appearing in sound

scattering from elastic bodies have been discussed in a qualitative way,

both for steady-state and for transient scattering [111, showing that

resonances should appear where the mechanical plus the radiation

impedance go to zero. The present treatment goes beyond this earlier

work by (a) indicating that the background scattering does not necessarily correspond to that of a rigid body [soft-body (for air bubbles) or

even intermediate-background types (for thin shells) [121 may and do

appear also] and (b) providing mathematically explicit forms for the

resonances (as well as for the background), as the main features of the

present quantitative approach. This together with the explicit eigenvalue equations for the real resonance frequencies (as distinct from the

complex eigenfrequency of vibration of the fluid-loaded scatterer), and

also the explicit expressions of the resonance widths, serves readily for

interpreting and classifying in a quantitative manner the resonances that

appear in the complicated patterns of calculated or measured total

scattering amplitudes. After the elastic sphere is considered a study of

A
RESONANCE AND SCATTERING 265

resonances in the separate partial-wave amplitudes will be done and will

lead to an interpretation of certain families of resonances in terms of

circumferential waves (or egge poles).

Scattering From Elastic Spheres

The case of plane-wave acoustic scattering from an elastic sphere

of radius a can be treated quite analogously to that of a cylinder. With

plane incident wave

pic ':- ea°s*" s (2n + l)i.(kr)P.(cos 0). (28a) X-0

one obtains [3,91 the scattered amplitude

P - -L (2n + l)i(S. - l)h( t(kr)P.(cos 9), (28b)

where we shall again write S, -- exp (2i8.). With the asymptotic form

h(')(kr) - (I/kr)i-'-Iet ', (29)

and introducing the form function

f(9) - f0(o) (30)

"-0

consisting of partial-wave form functions

f,(@) - (2/ka)(2n + l)ea* sin 8bP,(cos 0). (31)

we obtain the asymptotic scattering amp'itude

& - (a/2r)e*rf(9). (32)

For a rigid sphere, one has asymptotically 131 for the case of backscattering () - ar)

12 -f40(W) . (33)

The S function for the sphere is obtained as

S. - S,,()(L, - z 2)/(L, - zj) (34a)

- S(')(L;i - z2 ')/(L - - zi- ), (34b)

where

-S.) M e (35)

s(,) . h("(X)

i
266 CHAPTER 15

Eqs. (35) again represent the S functions for soft- or rigid-sphere

scattering, respectively, with corresponding phase shifts

tan f I) - j.(X)ly.(X), tan.1 ' - j;,(X)ly (X). 5c)

In Eq. (34), use was made of

z,- Xh.,"'(X)/h."(X). i - 1. 2. (36)

For the sphere, the real quantity L. is given by Hickling 131 (who calls

it F.), Vogt et al. 1131 and in Chapter 10; it contains spherical Bessel

functions with arguments XL and XT, and it is proportional to (p.,/p,),

where p is the density of the sphere. As for the cylinder, the correct

soft (p, << p,) and rigid (p, >> p.) limits for the S function are

obtained from Eqs. (34).

Since the preceding Eqs. 1(34) and (35)] for the sphere are formally equivalent to the corresponding Eqs. [(8) and (9)] for the

cylinder, the subsequent development of the resonance formalism, Eqs.

(12) and (15)-(22), can identically be taken over for the sphere, with

only account taken of the different form of z,, which leads to a replacement of Eqs. (13) and (14) by

SX j .(X)j.(X) + y.(X)y,(X)

j,2(X) + y,2(X)

S .i (37b)

X j,2(x) + y.2

(x)

and

& j A(,) i(x)j(x) + y(x)y.(x) , (8a) X I(X)II . + 1y,(X)12

s, I I

- 1 U.(X)P2 + ly 3(X)] 2 (38b)

respectively. The resonance formalism then goes through in terms of

these quantities, and with an expansion of the new function L(X) as

in Eqs. (16), one thus arrives at a partial-wave form function, e.g., for

the case of a rigid background

f.()- (2n + I)e ""

ka

2 I r; -if _,W -Fr +1 e sin J."') P. (Cos 0). (39)

x."- x - &

2
-7-------L--

RESONANCE AND SCATfERING 267

which aguin consists of resonances superimposd on a background

corresponding to scattering from an impenetrable sphere. All the conclusions drawn from this expression for the cylinder apply equally to

the case of a sphere.

RESONANCES IN PARTIAL-WAVE AMPLITUDES

In this section, results of the preceding resonance theory will be

illustrated by numerical calculations for aluminum cylinders and

spheres immersed in water. It is pointed out, however, that these

numerical results were not obtained from resonance formulas such as

Eqs. (I). which are approximations, but from the corresponding exact

equations such as Eqs. (8).

Figure 1(a) shows a plot versus X = ka of the modulus of the

cylinder backscattering form function If(w))l (Eq. (4)) and Fig. I(b)

shows a similar plot for the sphere (Eq. (30)). Features of the rapid

oscillations occurring in these plots have been analyzed previously (41,

where it was shown that successive minima are caused by a coincidence

of the speeds of various elastic-type creeping waves with the phase

velocities of various normal-mode vibrations of the scatterig body,

which happens at the eigenfrequencies of the latter.

I s. (a) I .S

00S.

o .22142.31 3 12.1 4 3.l 0. 1 1.2) 14.21 3.3,

0.. 20 40 4~itti 14.6~i .0

1.3 *(b)

1.2

o.s

L ... 0 v1.31 12 3

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Fig. I - Form function modulus If(w)I plotted versus ka. for

aluminum bodies in water ((a) cylinder, (b) spherel. Features

are labeled by (n, L). See text.
268 CHAPTER 15

A more detailed picture of the situation is obtained if one plots

the individual partial-wave absolute amplitudes If.(ir)l vs ka This is

done in Fig. 2(a) for the cylinder and in Fig. 2(b) for the sphere, for n

- 0 up to n - 5. In each amplitude curve, a number of narrow resonance features can be recognized superimposed on a smooth background, which is found to be given by the scattering amplitude of the

corresponding rigid body. This is seen by a comparison with Figs. 3.

Partial-wave amplitudes for the rigid, soft, and elastic cylinder are

shown in Fig. 3(a) and similar plots for the sphere are given in Fig.

3(b). The figures clearly demonstrate that the elastic case follows the

rigid background except in the regions where resonances occur. A

further demonstration of the rigid-background nature for cylinder

scattering is given in Fig. 4, where the modulus of 1f2(r) - f lid (W,,

from Eqs. (23) is plotted. This procedure leaves the pure resonance

amplitudes only.

A correspondence can be recognized to exist between the partialwave resonances of Fig. 2 and those of the total form function, Fig. 1.

However, Fig. 2 reveals a more detailed picture. One may discern families of resonances labeled L - 1, 2, 3. ... whose members appear in

parentheses in all the partial waves, shifting to higher frequencies from

one partial wave to the next. They may be identified with the elastic

creeping-wave poles discussed in Refs. 4 and 5, L - I representing the

Rayleigh wave and L - 2, 3, ... the whispering gallery modes (141. Trajectories of the positions VL of these modes in the complex v plane

(obtained by making n a complex variable ) as functions of ka were

obtained previously 115, 161, they are all located in the first quadrant of

the v, plane. Every time such a creeping-wave pole moving along its

trajectory passes near an integer n, a resonance will appear at the

corresponding value of ka in the nth partial wave, with a width determined by the distance of the pole from the real axis. The partial-wave

resonance families are therefore successive manifestations of the various creeping-wave modes (known as "Regge poles" in the nuclear physics literature) as they move along their trajectories, and a picture such

as Fig. 2 may serve to make these connections evident, without having

to go into the complex plane 15, 15, 16) (see Chapter 3).

We note that for the breathing mode (n - 0), only the (n, L) -

(0, 3) We note that for the breathing mode (n - 0), resonance

appears, while all members of all Regge pole families appear in the n

I amplitudes, indicating that these resonances will be excited in acoustic scattering. The numbers shown at each resonance in Fig. 2 are the

eipenfrequencies (ka values) of the various free eigenvibrations of the

elastic body (which were calculated by us for the body in a vacuum;

this agrees closely with the values for the water-loaded body).

!
RESONANCE AND SCATTERING 269

N-0

I .0S

0.:

1,.0 I-orl T N-3

2.12 11

0.0"

2.0f

N-L

I..,"

.,

OSm . ..

2.31 N-S

6.0 2.0 4a 0.0 0.0 104 12.0 14.0 14.0 I .0o

(a)

Fig. 2 - Partial-wave amplitude moduli If.(w)I for the

flrst six partial waves (n - 0 - 5). plotted versus ka for

aluminum bodies in waler 1(a) cylinder, (b) sphere). Positions of the eigenfrequencies are labeled by integer L in

each partial wave

Li
270 CHAPTER 15

N-0

IN

08

t~ol N-2

1i 129(20 

-P -02

(-)

0.$

N-4

1.0

05

0O 2.0 0 0 .0 0o 2o 14 1 160 30 o0 

(b)

Fig 2 - Partial-wave amplitude moduli 1/.(WtI for the

first six partial waves (n - 0 - SI. plotted versus ka for I aluminum bodies in water (a) cylinder. (b) spherel Ponttions of the eigenfrequencies are labeled by integer L in

each partial wave
RESONANCE AND SCATTERING 271

3.0!__ RIGID

2.5 n - 2 ---- ELASTIC -- SOFT =2.0f

441. (1) (2) (3)

0.0 )v.JI V, V~\

3.0,

2.5k n=3

-2.0

(1(2)(3

0.5

3.0

25 NN N2

/1 11 2) (3)

150

0

0.5.

0.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 13.0 20.0

ka

(b)

Fig 3 - Comparison of partial-wave amplitude moduli I I.(v ( for

the nt - 2 itnd or - 3 partial waves or aluminum bodies in water IWa

cylinder. (b) spherel with rigid-body and soft-body amplitudes as

indicated
272 CHAPTER IS

- 1.5

1.2

0r.9

I 0.63

0.0 2.0 4.0 6.0 6.0 10.0 12.0 14.0 16.0 18.0 20.0

ka

Fig 4 - Modulus of n - 2 partial-wave scattering amplitude with

rigid background subtracted. IJ 2(r) - f/'wd(wf)l, for aluminum

cylinder in water

Figure 2 reveals mostly narrow resonances with strikingly

different shapes. However, these features may all be understood as follows. Classify as type A a "pure" resonance which shows a peak in

If.(W)I- examples are the resonances (n, L) - (2, 3) or (4, 2) in Fig

2(a), and for n > 3 the (n, 1) Rayleigh resonances (which, incidentally, are seen to have considerably larger widths than the whispering

gallery resonances). This type of resonance nearly coincides in frequency with a null of the rigid background, i.e., where f (' ) - 0, so that

only the resonance contribution in f. [Eq. (23a)] is present here. Type

B are resonances that nearly coincide with a peak of IJ' t (wr)I. examples are (n. L) - (1,3), (3, 3), (5, 2), and the second one of the Rayleigh resonances (2, 1). Here, v so that the background term

2

in Eq. (23d) becomes -i, while the resonance term at X - X'.(" equals

+i, causing a cancellation and thus the strikingly narrow holes in

If.(#)l of Fig. 2(a). Finally, type C resonances are the cases, e.g., (1,

2), in Fig. 2(a) which do not coincide with peaks nor nulls of the background, showing a dip on one side and a peak on the other side of the

resonance. This is explained by the sign change of the resonance term

in Eq. (23d) as X passes through the resonance; it will hence interfere

constructively with the background on one side of X, ' )

, and destructively on the other.

The frequency dependence of the resonance is given by the resonance expression which is the first term in Eqs. (23d) or (39), referred

to in nuclear physics as having the "Breit-Wigner form" [21. If we take

the temporal Fourier transform f.(t) of the resonance amplitude, we

find

I/,'(t l2 e (40a)
RESONANCE AND SCATTERING 273

This indicates that when excited by acoustic scattering or by other

means, the resonance will decay with a lifetime

t- (a/r~cos. (40b)

The quantity iir. thus measures the lifetime of the resonance in units

of the time a/c it takes the acoustic wave to traverse the length of one

radius. For the (n, 1) resonances, this time is fairly short (6/r. < 2)

due to the large width of the Rayleigh poles, but it is considerably

longer for the whispering gallery resonances. Corresponding longlasting excitations ('ringing') of the resonances have been searched for

by Faran 191, and have been round in some cases.

PHASE JUMP AT A RESONANCE

The phase of a resonance term is known to jump through a value

of w' as the frequency passes through the resonance frequency. To verify this the phase of f,(ir) (Eq. (23a)] is examined which is seen to be

given by

- . + 2(4 1a)

The resonance form of Eq. (18b) leads to

8. - +tan-' 2(~ )(41 b)

which indicates that for r.(" > 0, the phase 6,, - f.() jumps from 0 to

w as X moves from -- to +0-, and that this phase jump will occur

very abruptly just around X - X.('" if the width r., is small. Equation

(41b) also shows that a corresponding phase jump of #f will appear

superimposed on the progression of the background phase which

is expected to vary smoothly with 1.

To investigate this quantitatively for an aluminum cylinder, in

Fig. 5 the phase #f (mod 2w) is plotted for n - 2 on top of the absolute amplitude If.(w)I. Apart from a bending over near ka - 0, the

background phase f.( (dashed curve) moves along a straight line.

However, the total phase 8., (solid curve) is seen to carry out a more or

less abrupt jump upward (indicating that r!"~ > 0) by an amount v

every time a resonance is passed, due to the second term in Eq. (41b);

afterwards, it again parallels the background phase. We note that from

a practical standpoint, if a resonance is so narrow that it may be overlooked in a computer plot of If. I due to insufficient resolution the

corresponding phase jump will still appear and will alert us to the presence of the resonance.
274 CHAPTER 15

0 UA -30,

0 2.0 4.0 6.0 8.0 10.0 12.0 WO. 16.0 18.0 20.0

Luu

-Aa

(a)

(2) (3)

3.0 (1)

2.0-

1.0 -

0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 180. 20.0

ka

(b)

Fig 5 - (a) Phase and (b) modulus of n = 2 partialwave scattering amplitude for an aluminum cylinder in

water

The mentioned bending over near X - 0 (actually, between X =

0.5 and 2.5) of -of for n - 2 is present in all phases of- for n > 1, and

its midpoint moves out approximately proportionally with X. This is

exactly where one would expect, using, e.g., the tables of Ref. 5, the

resonance of the lowest Franz pole F, to occur. The Franz waves are

creeping-wave modes which propagate about the scatterer with a speed

close to the sound speed in the ambient fluid, being concentrated in the

fluid rather than inside the scatterer; hence they are present even

around impenetrable bodies (although they are much less intense for

soft than for rigid scatterers). No corresponding resonance is visible in

our plots of Lf(ir) I; however, F, is located close enough to the real

axis in order to cause the mentioned "pseudophase jump,' while F2,3

probably are not. Note that the total form function f(r) for the rigid

sphere shows a succession of such phase jumps (Ref. 3, Fig. 2) which,

in connection with a corresponding structure of (f(ir) given in the

same reference, makes the connection of this feature of q)f with F,

rather suggestive.
RESONANCE AND SCATTERING 275

ANGULAR DISTRIBUTIONS

The angular dependence of If.() )1 (Eq. (23d)) is simply given by

Icos n0I and hence not very informative. It has been noted, however

191, that the total form function If(4)i, Eq. (4a) shows a deep near

null at 0 - 180°at the (2, 1) resonance [see also Fig. l(a)], while similar near-nulls were seen at 6- +120°at t the (3, 1) resonance. In Figs.

6(b) and 7(b), If(0)I is plotted as a solid curve at ka = 4.79 [the (2,

1) resonance] and at ka = 7.47 [the (3, 1) resonance], respectively,

while Figs. 6(a) and 7(a) show If(0) 1 at 3% below, and Figs. 6(c) and

7(c) at 3% above these values of ka. (The dashed curves show the

corresponding angular distributions of the rigid background.) The mentioned holes in the elastic angular distributions at these resonances (not

present for the rigid cylinder) are clearly visible. It is noted that for a

type B resonance such as (2, 1), the nth partial wave is absent since

here, the resonance amplitude has canceled the rigid background.

Hence, from Eqs. (4a) and (23),

Ifr)(.0)l = If(4)) - 2e,(rika)- '

I2 cos n4)I. (42)

Now, from Eq. (7a), one has f r)(1r)l 1. However, at the (2, 1)

resonance frequency X,, ') = 4.79, and with 4 = r, the magnitude of

the second term on the right-hand side of Eq. (42) is 1.03, which is

compatible with f(w) -; 0. The half-width of this hole is expected to

be that of Icos 20 1, i.e., ;- 35, which is compatible with Fig. 6(b).

Similarly, for a type A resonance, one again has Eq. (42) except for a

plus sign, and for the (3, 1) resonance at t =- 7.47, a similar argu- 2

ment applies but now at 4) - 2 7r.

3.0 --- " -30 -3.

2.0=, .Or 2.

ka 4.64\ ka 4., 1 k 493

(a) • .(b (c)

Fig. 6 - Bistatic-scattering amplitude I./(0)I for an aluminum

cylinder in water, (a) 3% below, (b) at, and (c) 3% above the (2, 1)

resonance frequency ka = 4.79 (solid curve): comparison with rigidcylinder amplitude (dashed curve).
276 CHAPTER 15

/ 3.

20 ka 7.25 ka 747 20 ka 7.7

2. ": -/ 2.° 01 ;

// /

1.0 1A,0

a) Ib

Fig. 7 -- Bistatic-scattering amplitude lf(o)j for an aluminum

cylinder in water (a) 311h below, (b) at, and (c) 3% above the (3, 1)

resonance frequency ka = 7.47 (solid curve) comparison with rigid- cylinder amplitude (dashed curve).

CONCLUSIONS

The resonance scattering of acoustic waves from elastic cylinders and spheres, has been analyzed by restcIng the scattering amplitudes

into their individual partial waves, and by following the movement of

familireso nfreuances, each corresponding to a Regge pole of the

scattering amplitude, through the successive partial waves. The adaptation of the resonance formalism of nuclear reaction theory to the

acoustic-scattering problem, together with a numerical evaluation of the

amplitudes, has demonstrated the resonances to be superimposed on a

rigid-body scattering background, and the various types of interferences

resulting therefrom have been analyzed. The merit of this approach

lies primarily in the fact that in contrast to previous theory, the

behavior of both background and resonances is made mathematically

explicit, and this may readily serve for an interpretation of calculated or

measured total scattering amplitude. Further, scattering prolrns with

both rigid and soft background may be treated; and explicit '!xpressions

are available for both the resonance frequencies of the uid-loaded

scattering and of the wid h of the resonances. The resonance width

determines the lifetime of a resonanc, and the phases of the partial

scattering amplitudes increase abruptly by r when passing through a

resonance. Finally, some arguments have been advanced concerning

the existence of near nulls in the total angular distributions at certain

resonances.
RESONANCE AND SCATTERING 277

REFERENCES

1. E.P. Wigner, Phys. Rev. 70, 15 (1946); P.L. Kapur and R.

Peierls, Proc. R. Soc. London Ser. A 166, 277 (1938); H.A. Bethe

and G. Placzek, Phys. Rev. 51, 450 (1937).

2. See also N. Bohr, Nature 137, 344 (1936); G. Breit and E.P.

Wigner, Phys. Rev. 49, 519 (1936); H. Feshbach, D.C. Peaslee,

and V.F. Weisskopf, Phys. Rev. 71, 145 (1947).

3. R. Hickling, J. Acoust. Soc. Am. 34, 1582 (1962).

4. H. lberal[, L.R. Dragonette, and L. Flax, J. Acoust. Soc. Am.

61, 711 (1977).

5. R.D. Doolittle, H. Uberall, and P. Ugin6ius, J. Acoust. Soc. Am.

50, 921 (1971).

6. L. Flax and W.G. Neubauer, J. Acoust. Soc. Am. 61, 307 (1977).

7. See, e.g., S. de Benedetti, Nuclear Interactions (Wiley, New York,

1964).

8. H. Uberall, R.D. Doolittle, and J.V. McNicholas, J. Acoust. Soc.

Am. 39, 564 (1966); R.D. Doolittle and H. Uberall, J. Acoust.

Soc. Am. 39, 272 (1966).

9. J.J. Faran, J. Acoust. Soc. Am. 23, 405 (1951).

10. W.G. Neubauer, R.H. Vogt, and L.R. Dragonette, J. Acoust. Soc.

Am. 55, 1123 (1974); R.H. Vogt and W.G. Neubauer, J. Acoust.

Soc. Am. 60, 15 (1976).

11. M.C. Junger, J. Acoust. Soc. Am. 24, 366-373 (1952); see also

M.C. Junger and D. Feit, Sound Structures and Their Interaction

(MIT, Cambridge, MA, 1972), p. 330; or H. Uberall and H.

Huang, "Acoustical Response of Submerged Elastic Structures

Obtained through Integral Transforms," in Physical Acoustics,

edited by W.P. Mason and R.N. Thurston (Academic, New York,

1976), Vol. 12, p. 217.

12. J.D. Murphy, E.D. Breitenbach, and H. Uberall, J. Acoust. SQc.

Am. 64, 677 (1978).

A uInmm mm a&[ m
278 CHAPTER 15

13. R.H. Vogt, L. Flax, L.R. Dragonette, and W.G. Neubauer, J.

Acoust. Soc. Am. 57, 558 (1975).

14. Note that Fig. 2 uses the letter L to label these families of resonances, while in Ref. 4 the label I was used, which differs from L

for n - 0 and 1.

15. G.V. Frisk, J.W. Dickey, and H. Uberall, J. Acoust. Soc. Am. 58,

996 (1975).

16. J.W. Dickey, G.V. Frisk, and H. Uberall, J. Acoust. Soc. Am. 59,

1339 (1976).
Chapter 16

CALIBRATION OF ACOUSTIC

SCATTERING MEASUREMENTS USING

SPHERE REFLECTIONS*

INTRODUCTION

Scattering measurements, in which the acoustic characteristics of a

submerged target are investigated by comparing the response of the target with that of a standard scatterer, have been considered previously in

connection with underwater measurements [I]. The sphere is a logical

choice for a calibrating object because it is a finite, three-dimensional

target whose response is independent of rotation, and because exact

analytic scattering solutions can be obtained in the spherical geometry

[2-91. Considerations of the sphere as a standard, in comparative target

strength (target strength) measurements depend on the invariance of

the target strength of a rigid sphere with frequency. As given by Urick

1101, the TS of a sphere is

(TS) - 20 log (a2), (1)

where a is the radius expressed in yards. Equation (I) expresses the

well-known fact that for aIk > I the diffracted or creeping wave component of the scattering by a rigid sphere is negligible and the backscattered response is made up almost entirely of specular reflection (X

refers to the wavelength of the incident sound wave). The idea of a

standard sphere in lake or ocean measurements was abandoned for such

practical considerations as size, fabrication, and mounting; additionally,

no sphere in water satisfies rigid boundary conditions, and in fact, the

computed and measured responses for submerged solid and hollow

metal spheres show steady-state variations from Eq. (1) of as much as

20 dB [4-91. As will be demonstrated here, despite these large steadystate variations from the rigid body solution, spherical targets can be

accurately and conveniently utilized as standard targets in laboratory

measurements which utilize broadband techniques and a single

source/receiver transducer.

*Description of this procedure first appeared in: L. R. Dragonette, S. K. Numrich, and L.

J. Frank, J. Acoust. Soc. Am. 69, 1186 (1981).

279

I
280 CHAPTER 16

Differential measurement techniques, which enable absolute

scattering measurements to be made without the necessity of a calibrated receiver, are standardly used in the scattering literature [7, 8,

11-16J. Absolute calibration can be avoided by using the same transducer to measure both the reflected and incident pressure amplitudes,

and expressing both the analytic and measured results as a ratio of

these amplitudes. A typical scattering experiment is seen schematically

in Fig. 1. The target is centered at Po(xo, yo, zo) and the backscattered

return is measured by a receiver at PI(xi, Yo, z0). The incident pressure amplitude can be obtained with relative ease when the source and

receiver are separate transducers [7, 8, 11-15]. For cases in which

source and receiver are the same, a replica of the incident pulse may be

obtained from the water surface or from a flat plate [16] interposed

between source/receiver and target. In the latter procedure, the plate

must be large enough to avoid illumination of its edges, thick enough

to provide separation between front and rear surface echoes, and carefully positioned for normal incidence; in addition, amplitude corrections

based on the reflection coefficient of the plate must be made. A more

convenient and accurate alternate method is given below.

V

V P(lx °'V¥°0

Fig. I - Schematic of a typical laboratory

scattering experiment.

MEASUREMENT METHODS

The steady-state, backscattered form function of the target seen

schematically in Fig. I can be obtained empirically as

If~. - (2r/a)(p,/po), (2)

where r is the range, a is the characteristic dimension of the target, and 2

p, and P0 are, respectively, the reflected pressure amplitude measured at

P, (xi, Yo, zo) and the incident pressure amplitude measured at

Po(xo, Yo, zo). Steady-state measurements of If. when the target is

an elastic sphere have been made previously and the results agreed, to
[I

CALIBRATION WITH SPHERES 281

within a few percent, with exact normal mode series computations

given in Chapter 8. Measurements of the steady-state form function

can also be made utilizing transient techniques, with the advantage that

If. I is obtained over a broad range of frequencies from a single measurement. The form function of a three-dimensional target illuminated

by a short pulse, p (t), is given by

If0. (ka)l - (2r/a)[Ig,(ka)i/1g(ka)1, (3)

where gi(ka) is the transform of the incident pulse, po(t), and g,(ka)

is the transform of the reflected echo, p,(t). Comparisons between

transient experiments and exact theory for elastic spheres in Chapter 9,

absorbing spheres in Chapter 10, and infinite elastic cylinders given in

Chapter 6, have demonstrated excellent agreement.

Solid tungsten carbide spheres are a convenient, accurate means

of obtaining a quantitative measure of gj(ka), in scattering experiments

in which a single transducer is used as both source and receiver.

Tungsten carbide was chosen both for its high specific acoustic

impedance and because precisely fabricated sizing balls made of

tungsten carbide are available as on-the-shelf items.

CALIBRATION BY A RIGID SPHERE

For a rigid sphere of radius a, positioned at a range r, from a

source/receiver transducer, the backscattered form function would be

given as a function of frequency v as

If '. ( ) I - (2 rl a) [IIg () I/ Ig (0,)1, (4a)

where gi and g, are, respectively, the Fourier transforms of the incident

pulse and of the pulse reflected from the rigid sphere. For frequencies

at which the Franz wave does not contribute significantly to backscattering I.P. (P) I - l and

IgG')I - (2r5/a,)Ig,(v)1. (4b)

The form function of any three-dimensional target at a range rr from a

source/receiver is given in terms of its characteristic dimension aT by

IfV (v) I - (2rr/ar) [1gr(,)1/ 1,.(0 D. (4c)

Here gT is the Fourier transform of the backscattered return from the

target under investigation and g/ is the transform of the wave incident

on the target. Assuming lhr spreading, the transforms of the wave

incident at the reference sphere and at the target are related by

g - grs/ rr, (4d)

__ i -l l
282 CHAPTER 16

and the far field form function of the target, fA can be obtained by

comparing its reflection to that of the rigid sphere from Eqs. (4b) and

(4c) as

fLf W I - (rlr2) (ajar)[Igr(y)Il/g9(1) H. (5)

APPROXIMATION OF A RIGID SPHERE

BY TUNGSTEN CARBIDE

No sphere in water is rigid, even one made of a material with as

high a specific acoustic impedance as tungsten carbide. This can be

verified by comparing calculations of the form functions for tungsten

carbide and for a rigid sphere. The exact normal mode series solutions,

as formulated in Chapter 8, were used to analytically obtain the elastic

and rigid form functions and these are plotted over the range 0 < ka <

75 in Figs. 2(a) and 2(b). Significant differences between the steadystate solutions for the rigid and tungsten carbide curves exist over the

range ka > 8, and these differences will continue until acoustic absorption in the material becomes significant (see Chapter 13). Acoustic

absorption in tungsten carbide is insignificant over the ka ranges discussed in this paper.

1.6

0.8

(a)

0.8

i 0

0 10 20 30) 40 so so

KA

(b)

Fig. 2 - The form function for a rigid (a) and a tungsten

carbide (b) sphere over the range 0 < ka < 75.
CALIBRATION WITH SPHERES 283

The elastic responses of submerged spheres have been described

in terms of free-body resonances as in Chapters 4, 14, and 15. Elastic

effects dominate the form function curve for all ka values greater than

(ka) 2,

1 which is the ka position at which the (2, 1) free body resonance

is excited [18-211. (The initial number in the label is the normal mode

number, the second number is the eigenfrequency.) This is true not

only of tungsten carbide but of all materials with wave speeds and densities greater than the velocity of sound and density of water. A cursory look at the curves shown in Fig. 2 would seem to indicate that a

tungsten carbide sphere could be used as a standard reflector only in

the region ka < 7.5, since this is the region over which rigid and elastic responses are most nearly the same, but this is not the case. A significant finding in Chapters 4, 14, and 15 was the demonstration that

the scattering from metal spheres and cylinders, with the material

properties discussed above, could be described in terms of a background of rigid-body scattering onto which the resonance or elastic

behavior is superimposed. The rigid body and elastic responses cannot

be conveniently separated in a steady-state measurement, but they can

be isolated in short pulse or broadband measurements. This isolation

can be achieved even in the ka range over which the deep nulls and

large peaks occur in the form function.

Figure 3 shows the response of a 2.54-cm-diam tungsten carbide

sphere to a short incident pulse. The source/receiver transducer was

driven by a square wave from a Panametrics Model 5055

Pulser/Receiver, and was located at a range of 73 cm from the sphere.

The reflected echoes were obtained in digital form by a Biomation

model 8100 analog-to-digital converter interfaced to a PDP 11/34 A

computer. The center frequency of the echoes seen in Fig. 3 is 0.5

MHz, and the sampling rate used was 0.5 gs. The response seen in

Fig. 3 is made up of several separated echoes. The first echo is labeled

rigid reflection and as will be shown below, this echo is an excellent

replica of the return that a truly rigid sphere would give. As discussed

earlier, it has been demonstrated that the steady-state reflection from

spheres and cylinders is made up of a rigid body return on to which the

elastic response is superimposed. In Fig. 3 the rigid reflection is

separated in time from the elastic echoes; moreover, the acoustic

energy which generated the series of elastic echoes enters the sphere at

critical angles off of normal incidence so that the backscattered specular

reflection replicates a rigid sphere return both in amplitude and frequency content.

A simple demonstration that the return labeled rigid reflection in

Fig. 3 is a replica of the return from a rigid sphere can be given by

obtaining the form function for a tungsten carbide sphere from
284 CHAPTER 16

RIGID

REFLECTION

0.1

ELASTIC RESPONSE

0.4

0.2

-0.4 -0.2

-0.I I I I I I I

0.00 0.02 0.04 0.06 0.06 0.10 0,12

TiME (mu)

Fig. 3 - The backscattered response of a

tungsten carbide sphere.

If.*(ka)l = (2r/a)[Ig,(ka)/1g1 ,(ka)IJ. (6)

In Eq. (6), g, is the transform of only the first or rigid echo seen in

Fig. 3, and g, is the transform of the entire return given in Fig. 3. The

Fourier transforms of the initial or rigid return, and of the entire

reflected echo were computed, and the operations described in Eq. (6)

were carried out on the computer. A comparison of the form function

empirically obtained by this method and a computation made from

exact theory is given in Fig. 4, and quantitative agreement is obtained

over the range 14 < ka < 40. A similar, higher frequency experiment

was performed on a 2.46-cm-diam sphere at a range of 110 cm. The

center frequency of the echoes in this latter experiment was 1.0 MHz,

and the sampling rate was 0.2 /&s. A comparison of the analytic and

measured results over the range 40 < ka < 70 is seen in Fig. 5. The

broad ka range covered by the two experiments, whose results are

given in Figs. 4 and 5, also evinces the efficiency and accuracy of using

the broadband, short pulse technique to obtain the steady-state

response of a target.
CALIBRATION WITH SPHERES 285

3.2

2.61

2.4

Z 2.0

U.)1.6 0

2.0

Up

.4

3.

2.6-

'I..

2.44

214 1 22 26 30 34 3

ka

SFig. 4 - Experimentally obtained function (the frm 

points) and analytic computation (the solid line) for a

tungsten carbide sphere over the range 14 ( ka 40.

2.6

2.4

"0Fig. 5 - Comparison of experi-

-u"1.6 L " mentally (the points) obtained and analytic form computa- function =!! i tion (the solid line) f'or a tungsten

L 1.2 * carbide sphere over the range 14

; < ka < 70.

40 46 50 56 60 65 70
286 CHAPTER 16

SUMMARY

The results described in Figs. 3-5 demonstrate that tungsten carbide spheres can be used as standard targets in differential broadband

measurements. The steady-state response, or form function of the

sphere, is made up of the long pulse interference of echoes such as

seen in Fig. 3, and differs greatly from the response of a rigid sphere.

The calibration method works because the rigid body and elastic scattering mechanisms which make up the steady-state response of the

spheres can be separated, using short pulses, over almost the entire frequency range. In the case of backscattering, the acoustic energy, which

generates the elastic echoes, penetrates the sphere at angles which do

not significantly effect the backscattered specular response, and this initial backscattered echo closely replicates the response of a rigid sphere.

REFERENCES

1. "Physics of Sound in the Sea," National Defense Research Committee Division 6, Summary Tech. Rep. 8 (1946).

2. J.J. Faran, ONR Tech. Memo. 22, 15 March (1951).

3. J.J. Faran, J. Acoust. Soc. Am. 23, 405-418 (1951).

4. R. Hickling, J. Acoust. Soc. Am. 34, 1582-1592 (1962).

5. R. Hickling, J. Acoust. Soc. Am. 36, 1124-1137 (1964).

6. A.J. Rudgers, Nav. Res. Lab. Rep. 6551 (1967).

7. W.G. Neubauer, R.H. Vogt, and L.R. Dragonette, J. Acoust. Soc.

Am. 55, 1123-1129 (1974).

8. L.R. Dragonette, R.H. Vogt, L. Flax, and W.G. Neubauer, J.

Acoust. Soc. An. 5, 1130-1137 (1974).

9. R.H. Vogt, L. Flax, L.R. Dragonette, and W.G. Neubauer, J.

Acoust. Soc. Am. 57, 558-561 (1975).

10. R.J. Urick, Principles of Underwater Sound for Engineers,

(McGraw-Hill, New York, 1967), pp. 244-245.

11. L.D. Hampton and C. McKinney, J. Acoust. Soc. Am. 33, 664-

673 (1961).
CALIBRATION WITH SPHERES 287

12. G.R. Barnard and C.M. McKinney, J. Acoust. Soc. Am. 33, 226-

238 (1961).

13. K.J. Dierks, T.G. Goldsberry, and C.W. Horton, J. Acoust. Soc.

Am. 35, 59-64 (1963).

14. C.W. Horton and M.V. Mechler, J. Acoust. Soc. Am. 51, 295-303

(1972).

15. R.E. Bunney, R.R. Goodman, and S.W. Marshall, J. Acoust. Soc.

Am. 46, 1223-1233 (1969).

16. H.D. Dardy, J.A. Bucaro, L.S. Scheutz, and L.R. Dragonette, J.

Acoust. Soc. Am. 62, 1373-1376 (1977).

17. L. Flax, J. Acoust. Soc. Am. 62, 1502-1503 (1977).

18. R.H. Vogt and W.G. Neubauer, J. Acoust. Soc. Am. 60, 15-22

(1976).

19. H. Uberall, L. R. Dragonette, and L. Flax, J. Acoust. Soc. Am.

61, 711-715 (1977).

20. L. Flax, L.R. Dragonette, and H. Uberall, J. Acoust. Soc. Am.

63, 723-731 (1978).

21. L.R. Dragonette, "Evaluation of the Relative Importance of Circumferential or Creeping Waves in the Acoustical Scattering from

Rigid and Elastic Solid Cylinders and from Cylindrical Shells,"

NRL Report 8216, September (1978).

Il m ln ll lm llnll m
Chapter 17

EVALUATION OF A

REFLECTION-REDUCTION COATING*

INTRODUCTION

The ratio of reflected acoustic pressure &, to incident acoustic

pressure A1 is called the reflection coefficient R and is alternately

expressed as I R I exp i0, where 0 is the phase shift caused at the

reflecting interface. The reflection-coefficient amplitude IR I affords an

insufficient description of a lossy material intended to modify, usually

reduce, the reflection from a body. In addition, the phase 0 is desired

and often measurements at various angles of incidence are desired as

well. Usual methods of evaluating such reflection-reduction materials

are the measurement of pressure in a wave reflected from a sample of

the material attached to a flat backing plate either in a free field or at

the end of a tube. That measurement is compared to the reflection

from an uncoated flat plate or surface. Tube measurements have the

disadvantage of relatively small sample sizes and are limited to normal

wave incidence. Problems of limited sample size and diffraction, resulting from the edges of plates, are experienced in the reflected field.

Diffraction problems, particularly troublesome for non-normal

incidence determinations, limit conventional and proposed methods

involving finite plates. In addition, such methods result in well-defined

reflection coefficients only for certain relations among material constants such as their wave speeds (see Ref. 1). Those relations may not

be known in advance. The procedure described here has no such limitations. However, quantities descriptive of reflection that require

material parameter values that can be derived from the methods

described here can be evaluated after the parameters have been determined. Such a quantity is the reflection coefficient given by Brekhovskikh [2J as

Z,Cos 2 2y+ Z,sin2 2y- Z V Z, cos 2 2v+ Z, sin 2 2vtj+ Z

This treatment was first presented in: W. G. Neubauer, J. Acoust. Soc. Am. 62. 1024

(1977).

289
290 CIIAPTER 17 

where the impedances of the incident wave in the fluid, the longitudinal 

and transverse waves in the solid are, respectively, 

Z = pc/cos 0, Z1 = p1cLJ/cos 0 1 

and 

Z, = PtCs1/cos 'Yl· 

The incident angle to the surface normal is 0, and the refracted angles 

of the longitudinal and shear waves are 0 1 and 'YI> respectively. The 

ambient fluid density is p and that of the solid is p 1• The wave speed in 

the ambient fluid is c, cL 1 and cs 1 are the longitudinal and shear wave 

speeds, respectively, in the solid. 

A quantity such as the total impedance of the liquid-solid 

boundary given by Bre-khovskikh [3] ~-; 

2 101 = 2 1 cos2 2y 1 + 2 1 sin2 2y 1 

may be evaluated as well. Here, as in the expression for the reflection 

coeffici~nt, the attenuation of longitudinal apd transverse waves can be 

included by considering 2 1 and 2 1 to be complex resulting from complex wave speeds. 

BACKGROUND 

As a point of departure in describing the new method that follows, descriptions of pertinent results will be briefly stated. ThP classical problem of the reflection from a two-layer shell has been reported 

[4, 5] (see Chapter 5). The IT1')nostatic or bistatic field can be calculated as a function of frequency. The solutions are not limited to a far 

field approximation. The medium inside the cylindrical shell can be 

any fluid or a vacuum and the outside fluid, in which the shell is 

immersed, is arbitrary. Both layers of the shell can be arbitrarily 

.::huscn solids that may cause wave attenuations that can be described in 

the theory !Jy complex wave nur.-~bers (see Fig. l). 

Other results given in Chapter 6 show that accurate measurements 

can be made with nonspecialized acoustic sources, of reflections fwrn 

long cylinders and the results are found to be in reasonably close agreement with tlelcl descriptions derived from exact solutions with one 

dimension (leng!h) infinite. To demonstrate this, a least-squares t1t of 

the exact theoretical calculations of the form function 1/oo (ka, r/J) I for a 

solid lucite cylinder is shown versus bistatic angle t/J (180° is backscattering) in Fig. 2. The incident wave number is k and a is the 

' ' 

I 
COATING EVALUATION 291

rr 

SOL 

0

/ I.MrCIDENTLAYER 

I

WAVE 

SOLID

U ZFL I D I 

j- 

- L 

UI D 2

Fig. I - Geometry for the solution of the problem

of the reflection from a two-layer shell in a fluid

.6

--4

.2

S0 100 120 6 140 160 180

Fig. 2 - The reflected form functions f_.(ka, j) for a lucite

cylinder for a ka of 10.90 versus 0, the angle subtended from

the cylinder axis by the source and receiver (180 is backscattering). Exact theory (-) least-squares fit to the experimental

data (0).

cylinder radius. The points are derived from a short-pulsed wave

experiment. Additional similar results for backscattering form function

versus ka appear in Chapter 6.

Experiments have demonstrated [6, 71 reasonable agreement

between cylindrical solutions assuming an infinite length and a short

cylinder whose entire length is insonified within the major lobe of an

acoustic source. It has been shown in Chapter 6 that exact theory can

be fit by least-squares methods to measured data derived from shortpulse (and therefore broadband) reflection measurements. The experiments result in the reflected pressure )Pr1 or form function lf.(ka)I
292 CHAPTER 17

over a frequency range. For a cylindrical geometry these two are

related by the expression I pr/1i I= (a/2r) 1

2 if (ka) 1, where r is the

distance from the cylinder axis to the receiver. In addition, the exact

wave-harmonic solutions for a cylindrical geometry has been extended

as described in Chapter 13, to values of ka as high as 1000. It has been

adequately demonstrated in Chapters 7 and 10 that "Ica scaling" is a

valid concept in reflection problems dealing with simple shapes. That

means that a reflected pressure determination at one frequency f, for a

cylinder radius a, is the same as the reflected pressure at another frequency f2 v- f, from a cylinder of radius a2 d a,, as long as ka is held

constant.

PROCEDURE

In the following discussion, specific materials will be chosen for

describing the method, but it is applicable in general to the same extent

that the classical shell problem described earlier applies to arbitrary

media. Consider a two-layer shell shown in Fig. I with air or water in

the interior (fluid 2), water (fluid 1) on the exterior of the shell, a

solid layer 2-a relatively well-defined solid such as a metal, and a solid

layer 1-an acoustically lossy material. In an experiment, the incident

and reflected pressure amplitudes 1p, I and 1P, 1, respectively, can be

measured with demonstrated accuracy [121 in a controlled environment

as shown in Chapter 9. The parameters considered as known quantities

are the densities of all media and longitudinal and shear-wave speeds

CL2 and CS2, respectively, in the inner (known) shell material. The loss

parameters of the inner layer would probably not be significant [81, if it

is a metal, below MHz frequencies. However, if that layer is composed * of a material that does have significant longitudinal and shear-wave

* losses, those can be determined separately and included in the theoretical calculation. The longitudinal wave speed in the external fluid (1) is

known. Values for the longitudinal and shear-wave speeds cL 1 and cs,

and their associated attenuation factors a LI and as, can be thus derived

from pressure amplitude Ip, p il measurements as a function of frequency, and least-squares fitting procedure provides a measure of the

relative sensitivity to the fit of each of the parameters such as wave

speeds and attenuations. Therefore this process can also be used as a

reflection-reduction design tool to indicate the potential worth, in terms

of acoustic reflection, of varying, or attempting to vary, wave speeds or

losses to achieve a specific result.

It is possible, too, to describe a reflected wave resulting from

incidence of, say, an infinite plane wave or any configuration of a

different geometry than a cylinder such as an infinite plane interface.
COATING EVALUATION 293

This can be readily derived, with the appropriate phase, by using the

material constants obtained from the cylindrical shell theory and associated acoustical field measurements.

It should bementioned that all ofthe solutions referred to here

for cylinders are in a plane normal to the axis of the cylinder. In spite

of that limitation, because of the curvature of a cylindrical shell, if the

radius is not larger than the incident-wave beamwidth, the constants

derived are a result of acoustic incidence at all angles of incidence

around the cylinder surface. In addition, this method of determining

the echo-reduction-coating material constants is not limited to such a

determination in only the far field since the theory is exact and can

accommodate the near field as well. This could be a vital consideration

in practical cases such as measurements in an enclosed vessel or a tank

limited in size as all tanks are. Also, insonification of the cylindrical

shell by a close approximation to a spherical or plane wave, although a

considerable conceptional simplification, is not an absolute requirement,

since the incident beam function can be included in the theory, making

it no less exact.

SOME LIMITATIONS

Two shortcomings of the method will be mentioned even though

one of them is common to existing methods and a means of overcoming the second is foreseen. First, the reflection-reduction material must

be suitable for consideration as a homogeneous isotropic material even

though it may not be, even on a macroscopic scale. If the scale of

inhomogeneities in the absorbing layer is too large the difficulty will

probably appear at the stage of the attempt of least-squares fitting of the

exact theory to measured data. The second potential shortcoming of

the method is that the reflection-reduction material cannot have waveloss properties different from those assumed in the exact theory of the

two-layer shell. For the cases considered here, na is constant where X

is the wavelength in the material, and a is the absorption coefficient

expressed as attenuation per wavelength. So how does one detect

whether the variation of reflected pressure with frequency is caused by

the shell solution itself or a deviation of the coating material behavior

from that assumed in the theory? If a greatly different material

behavior is suspected, it is possible to resolve this potential ambiguity

with additional experiments. A possible choice is the measurement of

the bistatic field pressure radially around the cylinder (see Chapter 6).

These observations can be calculated and will vary in a way consistent

with the exact reflection theory of the two layer shell. Consistent deviations of the theory from those experiments must be attributed to an
294 CHAPTER 17

unexpected behavior of the reflection-reduction coating. The theory

could then be modified to incorporate the frequency-dependent

behavior of the losses in the material and the entire process repeated

with those losses.

VARIATIONS ON THE METHOD

It should be noted that the reflection-reduction material can be

evaluated as the innermost of the two layers of the shell even though in

practice it is intended to be used as an external coating. In that case

again the sound is incident on the outside of the two-layer cylinder and

the outer layer must be sufficiently thin that the external reflected field

be sensitive to different internal layer (solid layer 2 in Fig. 1) wave

speeds and attenuations. Also, the shell can be filled with a different

fluid so that the material is essentially evaluated while in contact with a

different fluid without filling an entire acoustic range, such as a tank,

with that fluid. Similarly, with the echo-reduction material as the inner

shedl layer the interior of the cylinder can be pressurized and the resultant constants for the reflection-reduction materials derived from the

measurements will be gained without the requirement of a pressurized

free-field facility.

SUMMARY

An acoustic reflection-reduction material can be evaluated by a

combination of acoustic field measurements and theoretical computation. The material of unknown echo-reduction effectiveness is made to

be one layer of a two-layer shell the interior of which may be filled with

either a gas or a liquid like or unlike that Surrounding the cylinder.

The two-layer shell is placed in a free acoustic field and the amplitude

of the resulting scattered acoustic pressure wave is measured. The

amplitudes of the incident and reflected acoustic pressure wave are considered known quantities in a computer solution of the exact theory of

a two-layer shell. The solution can be formulated so that it will yield

values of longitudinal- and shear-wave speeds and attenuations in one

of the shell layers if all other parameters of the problem are given.

These include all material densities and wave speeds and attenuations in

the second shell layer as well as the acoustic wave speeds in the external and internal fluids. Experimental measurements of the ratio of

reflected to incident acoustic pressure amplitudes normalized by

(a/2r) 1

/

2 to express reflection form functions versus either ka or

scattering angle are least-squares fit to the exact theory by computer.
I

COATING EVALUATION 295

Such fitting programs indicate correlations between the various parameters as well as the sensitivity of the fit to the variation of each parameter. Thus the procedure would yield quantitative information about

whether a given parameter, such as, say, shear-wave attenuation is a

reasonable candidate parameter to attempt to vary to achieve a different

more favorable scattering solution, whether this is achieved by anechoic

material design or material selection. The procedures or method lends

itself to the evaluation of materials under various conditions at the

anechoic material-fluid interface such as different fluids or increased or

decreased pressure, by capping the ends of the cylinder and changing

the internal conditions. Insonification of those ends by the incident

wave can be avoided by use of a limited acoustic beam and no different

experimental facilities are needed for those conditions. Computer solutions actually yield the acoustic bulk property of the reflection-reduction

material in terms of which the exact theory of the two-layer cylindrical

shell is written. A specific possible such set is the complex wave

number or alternatively the complex wave speed for both longitudinal

and shear waves, either of which allow for acoustical wave attenuation

in the material. Derived quantities such as reflection coefficients or

impedances can then be evaluated.

REFERENCES

1. L. Brekhovskikh, "Waves in Layered Media," translated by D.

Lieberman (Academic, New York, 1960), pp. 28-36.

2. Ref. 1, p. 31.

3. Ref. 1, p. 33.

4. L. Flax and W.G. Neubauer, J. Acoust. Soc. Am. 61, 307-312

(1977).

5. G.C. Gaunaurd, J. Acoust. Soc. Am. 61, 360-368 (1977).

6. H.D. Dardy, J.A. Bucaro, and L. Flax, J. Acoust. Soc. Am. 57,

$59(A) (1975).

7. I.B. Andreeva and Samovol'kin, Soy. Phys. Acoust. 22, 361-364

(1976).

8. L. Schuetz and W.G. Neubauer, J. Acoust. Soc. Am. 62, 513-517

(1977).

i i _ n j n I I I II

w ° lUI NIIIININI __un___n__nnunn__nn__uuu an B 
296 CHAPTER 17

9. L. Flax, J. Acoust. Soc. Am. 61, S37(A) (1977).

10. W.G. Neubauer, R.H. Vogt, and L.R. Dragonette, J. Acoust. Soc.

Am. 55, 1123-1129 (1974).

11. R.H. Vogt, L. Flax, L.R. Dragonette, and W.G. Neubauer, J.

Acoust. Soc. Am. 57, 558-561 (1975).

12. L.R. Dragonette, R.H. Vogt, L. Flax, and W.G. Neubauer, J.

Acoust. Soc. Am. 55, 1130-1137 (1974).
Chapter 18

REFLECTION OF A BOUNDED BEAM*

INTRODUCTION

An analytical formulation of the reflection of a scalar wave beam

with bounded cross section incident from a fluid on a plane interface

between the fluid and an elastic medium was first given by Schoch [11. That theory resulted in the formulation of a lateral displacement of the

reflected beam. This emergent beam was regarded as constituting the

entire reflected field similar to that hypothesized and demonstrated by

Goos and fHfnchen [21 for light waves. An extensive treatise of this

effect has been carried out by Lotsch [31. In each case beam displacement is expressed in terms of the first derivative of the angledependent phase shift of the reflection. A second-order approximation in terms of the second derivative of the reflected phase was given by

Brekhovskikh [41 to express the field distribution in the cross section of

the reflection for a defined incident beam. In the ultrasonic case,

experiments at the Rayleigh angle have indicated [51 that the so-called

lateral displacement along the interface occurs at the Rayleigh wave

speed. Beam-displacement measurements at the Rayleigh angle have

been reported in the literature only for a limited number of materials,

namely, aluminum and its alloys [6,71, stainless steel [51, and beryllium

[7]. Also, the beam-displacement concept was used in reflection measurements on stainless steel [8-10). Experimenters have observed [7- 11 by means of hydrophone reflectivity measurements a pronounced

minimum in the reflection from plane material surfaces that is unaccounted for by the so-called classical theory [121. That discrepancy had

been accounted for in the case of aluminum [71 by imposing a lateral

displacement on the hydrophone predicted by Schoch's formulation.

Later results, to be given here, have shown it to be clearly evident that

such a procedure does not take the entire reflected or reradiated field

into account. In general, for other materials and even for the case of

aluminum, a significant acoustic field was shown to be present outside

of the region that would be intercepted by a receiving crystal of the

same size as the incident beam. For a plane hydrophone, oriented at

These results first appeared in: J. Appl. Phys. 44, 48 (1973).

297
298 CHAPTER 18

the Rayleigh angle, with a diameter of 1 cm or larger, a uniform

unique-phase signal is intercepted neither at the position of the specular

beam nor at the Schoch displacement distance for a range of frequency.

A further modification to the classical theory that included the

effects of attenuating waves was carried out by Becker and Richardson

18,91 whose results were confirmed with experiments 1101. That theory

was written for a plane-wave assumption and the analysis of the experiment allowed for the finite beams employed by imposing the Schoch

displacement on the receiving transducer. The magnitude of the

minimum in reflection at the Rayleigh angle was found in that theory to

be a function of frequency. This was confirmed by reflection measurements at and near the Rayleigh angle on stainless steel. The magnitude

of the reflection at the Rayleigh angle was found to decrease up to 15

MHz (for stainless steel) and then increase with increased frequency

above 15 MHz. That frequency was therefore called the "frequency of

least reflection." At that frequency the frequency-dependent phase shift

upon reflection was found to change sign with respect to the incident

wave. Support of this aspect of the theory was also given by experimental observation [8-101 under the same condition as previously

stated, viz., source and receiving transducer many wavelengths large of

equal size and positioned at equal angles, and receiving transducer centered on the Schoch displacement distance. Although the prediction by

the theory of a frequency-dependent phase is indicated to be accurate

by later observations, the interpretation of a resulting "frequency of

least reflection" is misleading if it is regarded that a hydrophone of the

same size as the source beam is used to observe it, since significant

energy can be radiated beyond the hydrophone surface. Also, the

energy incident on the hydrophone would not have a unique phase over

the entire face.

The fact that previous measurements validating beamdisplacement theory were made with a limited range of equally sized

sources and receivers, or even larger receivers, and a limited range of

materials accounts for the apparent validation of a theory that is not

adequate to describe the total reflected or radiated field. Previous

observations have been rationalized in terms of the results that disclose

that the energy redistribution can be characterized by a specular

reflection and surface-wave radiation, and the entire field is not

represented by the portion incident on a receiving transducer placed in

the reflected field and displaced laterally according to the formulation

by Schoch. Subsequent to the proposal of the model of reflections at

the Rayleigh angle based on experimental observations, Bartoni and

*Tamir (131 presented theory that substantiated the observed behavior

*and model.
BOUNDED BEAM 299

EXPERIMENTAL OBSERVATIONS OF REFLECTION

AT THE RAYLEIGH ANGLE

For a 1.1-cm-wide beam of 7 MHz incident at the Rayleigh angle

on an aluminum (1100)-water interface Fig. 1 (a) shows a diagram of

the cross section of the reflected beam containing all of the reradiation

from the surface laterally displaced to the right by a distance A - 4.5

mm as the Schoch theory would predict. The schlieren photograph of a

circular beam (diameter - 1.1 cm) incident on an aluminum-water

interface is shown in Fig. 1(b). The schlieren system is pulsed (see

Chapter 22). The source transducer is seen at the left and the leading

edge of the pulse at the right of the photograph. Slight divergence is

shown in the diagram since it is unavoidable in the experiment. The

fairly well delineated right edge of the reflected or reradiated energy

appears out to a distance of 6 mm to the right of the right edge of the

incident beam which is 1.5 mm beyond the Schoch prediction. In Fig.

1 (b) energy is radiated into the specular region with no displacement.

Also present is a dark strip in the reflected or reradiated energy. This

occurs at a position within the radiation where specular reflection would

normally be expected to be found, i.e., between the lateral limits over

which the energy is incident. Equivalent observations have been made

with transducers that had square faces. In the case of aluminum the

differences between the experimental observation and the description of

the result demanded by the theory, and depicted in the diagram of Fig.

1 (a), are significant but not startling. In that case the lateral displacement was relatively small, and the distance measured in the schlieren

photograph at which energy is seen to emerge from the interface, in

excess of that predicted, is not large either. However, the theory is not

limited to cases for which displacements are small. Aluminum oxide is

a material for which both longitudinal and shear wave speeds are

significantly larger than those in aluminum and whose density is almost

50% greater than aluminum. These factors combine to cause a prediction of a lateral displacement of an incident beam for aluminum oxide

of 118.1 wavelengths at the Rayleigh angle of 14.80 as depicted in Fig.

2(a). That displacement is more than five times that predicted for

aluminum. A schlieren photograph of an actual beam reflected from

aluminum oxide at a frequency of 7 MHz is shown in Fig. 2(b). It can

be seen that no separation between the incident and emergent beam

occurs. A dark strip is again evident in the reradiation as it was in the

case of aluminum and the brightness decreases significantly toward the

right indicating a decay in pressure amplitude in that direction. Specular reflection in the normal position is evident.

_____________________________________________________.__._.__............_____.
300 CHAPTER 18

(a)

(b)

Fig. I - Finite beam of 7-MHz ultrasound incident from water on a plane aluminum (1100)

interface at the Rayleigh angle. (a) Diagram

showing the entire reflected field on the right

predicted by beam-displacement theory that

should result from the incident beam on the left.

(b) Schlieren photograph of an actual incident

beam the same size as that shown in (a) and the

actual resulting reflected and radiated field.

(a)

(b)

Fig. 2 - Finite beam of 7-MHz ultrasound incident

from water on a plane aluminum oxide interface at the

Rkayleigh angle, (a) Diagram showing the entire

reflected field on the right predicted by beamdisplacement theory that should result from the incident beam on the left. (b) Schlteren photographs of

an actual incident beam the Same size as that shown in

(a) and the actual resulting reflected and radiated field.
BOUNDED BEAM 301

A PHYSICAL MODEL OF REFLECTION AT

THE RAYLEIGH ANGLE

A relatively simple model is put forward here that explains the

observed results. It is supported by specific measurements and is consistent with the previous experimental results of others. Consider a

limited beam of sound incident from a liquid on a solid-plane elastic

surface. For all angles, even at and near the Rayleigh angle, energy is

specularly reflected. The angle of reflection is equal to the angle of

incidence and the size of the specularly reflected beam is the same as

the incident beam. In addition, over the area on which energy is

incident, and at the same time that specular reflection occurs, a Rayleigh wave is generated by the phase-matching mechanism for incidence

at and near the Rayleigh angle. However, when energy is incident near

but not exactly at the Rayleigh angle, the Rayleigh wave is still generated but to a lesser degree and if permitted to radiate, as it may in the

case of an adjacent fluid, it will be radiated at precisely the Rayleigh

angle.

In the case of aluminum in water, for incidence at precisely the

Rayleigh angle, Fig. 3(b) shows the relative phase of the energy in the

specular region to be 1800 from that in the region of the Rayleigh wave

radiation to the right of specular reflection. Between the two regions is

a narrow dark region where the equal amplitude coincidence of energy

resulting from both causes mutually cancel. Figures 3(a) and 3(c) show

reflection at angles smaller and larger than the Rayleigh angle, respectively. It can be seen in Fig. 3(b) that at a frequency of 833 kHz the

phase fronts may reasonably be considered to be planar in both regions.

The observation of individual wavefronts and the determination of their

relative phases for reflection at the Rayleigh angle was first reported in

Ref. 14.

A slightly diverging incident beam was produced at 5 MHz with a

transducer having a Gaussian-shaped beam [151. For incidence at the

Rayleigh angle for aluminum (1100), the resulting reflection and

reradiation into water from the surface is shown in Fig. 4. No

significant effect was observed that could be attributed to beam divergence. The hydrophone is shown in place with the sensitive area in the

specular reflection. The hydrophone has a plane round active area with

a diameter of 0.32 cm. In addition to a significant amount of specular

reflection and the cancellation area along the right edge of the specular

reflection apparent in the schlieren picture of Fig. 4, a bright highamplitude region is seen with subsequent amplitude oscillations to its

right. The precise cause of amplitude oscillations is not known but it
302 CHAPTER 18

Fig. 3 - Schlieren photograph of an incident beam on

the left and a reflected and reradiated beam on the

right. The acoustic frequency is 833 kHz, the fluid is

water, and the solid is aluminum (6061). Individual

cycles are made visible by using a short light pulse.

(a) Beam incidence angle 0 - 25.9, (b) 9 - 30.4

(the Rayleigh angle), and (c) 9 - 35. 9 .

Fig. 4 - Narrow Gaussian incident beam incident on aluminum (1100) at the Rayleigh angle. A receiving

transducer is shown at the left with the sensing area in

the specularly reflected field. (a) labels the specular

region and (b) labels the Rayleigh radiation region.
BOUNDED BEAM 303

has been observed that the pattern changes periodicity inversely with

source-beam width.

Representative hydrophone output voltage signals are shown in

Fig. 5. The top oscilloscope trace is received in the region of the specular reflection labeled A in Fig. 4 and the lower trace is that measured

in the Rayleigh-wave-radiation region labeled B in Fig. 4. The two

traces have had their amplitudes adjusted so that their respective first

cycles are equalized in amplitude and occur at the same time marker so

that the pulses may be compared. The pulses are very much alike and

can reasonably be called out of phase. The slight change in pulse shape

is caused by the fact that amplitude variations exist in the region

labeled B and the hydrophone is approximately the size of the bright

specular reflection in Fig. 4, and since the amplitude variations to the

right are narrower than that, a single and uniform wavefront is not

incident on the hydrophone.

A

Fig. 5 - Voltage output signal of the receiving

transducer, corresponding respectively to the

specular region labeled (a) in Fig. 4 and in the

Rayleigh radiation region labeled (b) in Fig. 4.
304 CHAPTER 18

The much quoted formulation of Schoch [11 for incidence at the

Rayleigh angle describes the lateral displacement A along the boundary

of a limited beam as

A ' 2 "I (c,/c') 2[(cjc)')2 

- ](c/c,) 2 1)/2

Au 1 (cd1c,) 2[(p,/c,)2 - 11 1

I + 6(;,/c,)4

[1 -(cld) 21 - 2(cIc,)2 [3 - 2(clcd)21

(cIc,)2 - (c/cd)2

where p is the density, X the wavelength, and c the wave speed.

Primed parameters refer to the liquid, and unprimed parameters refer

to the solid. The subscripts s, d, and r refer to shear, dilatational, and

Rayleigh, respectively. This displacement is, therefore, directly proportional to the density p of the solid as well as the wavelength X' of the

dilatational wave in the liquid and inversely proportional to the density

p' of the liquid. It is also a complicated function of the various wave

speeds in both the liquid and the solid.

The shear and dilatational wave speeds, the densities, and the

anticipated displacements A calculated from Eq. (1) are given in Table

I for seven materials. The various materials were chosen for specific

reasons for the observations reported here. Aluminum is the material

on which beam displacement was first demonstrated and later extensive

work has been done with 304 stainless steel. One is not at liberty to

adjust material parameters such as density or wave speed at will, but it

was possible to choose materials for which large displacements would be

predicted and for which density and wave speed had a desirable relationship to those of aluminum. Stainless steel, molybdenum, and

Kennametal K-91 all have wave speeds that are within 24% of the

corresponding wave speeds in aluminum and their densities are within

6% of being three, four and five times the density of aluminum, respectively. The last three materials in Table I were chosen because they

have significantly higher wave speeds for a range of densities. All displacements are predicted to be more than twice that of aluminum.

Radiation from the Rayleigh wave generated by the incident wave was

observed beyond the distance A predicted by the Schoch formula and

was observed on materials other than those listed in Table I.

A broad well-collimated beam of 7 MHz is shown just incident on

Kennametal K-8 in Fig. 6(a) and the resultant reradiation caused by

that beam is shown in Fig. 6(b). The actual maximum distance to

which radiation could be observed was not measured since the end of

the limited sample caused a sharp termination of the radiated wave at

the far right edge. The displacement formula predicts a termination of
BOUNDED BEAM 305

Table I - Relevant Values for Seven Materials

p C, #I Cs Os G0 Material (g/Cr 3 ) (m/sec) (des) (m/see) (deg) (d)

Al 2.7 6296 13.6 3106 28.5 30.5 21.3

(1100)

SS 7.9 5840 14.7 3130 28.3 30.6 58.4

(304)

Mb 10.1 6400 13.4 3369 26.1 28.3 90.0

K910 13.5 6542 13.1 3847 22.7 24.8 139.8

K8" 14.9 6900 12.4 4200 20.7 22.6 174.5

K165a 5.7 8850 9.6 5410 15.9 17.5 115.0

AI20 3 4.0 10700 8.0 6360 13.5 14.8 118.2

*Material is a sintered product whose main constituent is tungsten

and is made by Kennametal Inc.

(a)

(b)

Fit. 6 - Schlieren photograph of (a) an incident ultrasonic beam of 7 MHz on a plane Kennametal K-S interface in water and the resultant reflection and radiation at the Rayleigh angle of 22.6". The radiation energy terminates with a sharp edge on the right at the

edge of the material. The sample is 6.35 cm wide.
306 CHAPTER 18

the radiated energy at a position about four-fifths of the sample width

measured from the left edge of the sample. For the last five materials

listed in Table 1, reradiated energy was observed beyond a distance of 5

cm to the right of the left edge of the incident beam for an experimental arrangement similar to that shown by Fig. 6. The characteristic cancellation strip can be seen in Fig. 6(b) as it was in all materials at this

frequency. The distance along the surface from which radiation was

observed was a function of the incident amplitude. The rate of amplitude diminution along the direction parallel to the boundary was

observed from schlieren photographs not to be the same for all materials and was quantitatively measured with a hydrophone. Results agreed

well with loaded Rayleigh wave attenuation reported by Viktorov [161

for a totally different method.

COMPARISON OF REFLECTION AT THE RAYLEIGH

ANGLE AND THE MODEL

Some previous reflection measurements at the Rayleigh angle on

a water-stainless-steel interface have presumed that the location of the

reflected finite beam would be at a lateral displacement along the interface predicted by Schoch and therefore the transducer was placed at this

position. Also, the reflected beam was supposed to have the same size

as the incident beam, i.e., the source and receiving transducer were the

same size and shape. Under those conditions it was observed that there

was a frequency at which the reflection amplitude at the Rayleigh angle

was minimum when measured at the displacement distance A. The frequency has been called the frequency of least reflection. The observations presented here indicate that reflected (reradiated) energy is distributed over a region larger than the incident beam and is not of uniform

amplitude or phase within that region over which it is reradiated. The

observation of a frequency of least reflection can be rationalized by

showing that it is a result of the experimental method.

Presume a 1-cm-diam incident beam, without divergence, impinging on a plane water-solid interface. Such a beam is shown diagrammatically in Fig. 7 with the hydrophone placed in the field to the right

of the incident beam at an angle equal to the angle of the source

transducer (Rayleigh angle for stainless steel) and at a distance A

predicted by the Schoch formula. A schlieren photograph in Fig. 8

shows the actual radiation at 5 MHz from stainless steel that is idealized

in the diagram of Fig. 7. The regions labeled A, B, and C constitute

the radiated field. The regions A and B combined normally constitute

most of the specular reflection and C is the region in which the Rayleigh wave radiation predominates. Specular radiation and Rayleigh
BOUNDED BEAM 307

HIGH FREQUENCY

HYDROPHONE LOW FREQUENCY

TIN \HYOROPHONE

P0 ITIONPOSITION

.............. .......... ~i~iiii}Tiii

* *0*1.

X : ... iiiiiiiiiiiii!iiiiiii ..: ii.. 

HIGH FREQUENCY A -

"I _ I-LOW FREQUENCY,&

Fig. 7 - A diagram corresponding to the schlieren photograph in Fig.

8 that depicts the various regions of the radiated field: (a) is the specular reflection, (b) is the null region, and (c) is the Rayleigh radiation

region.

Fig. 8 - Schl.'zren photograph of a 5-MHz beam in

water incident from the left on a plane stainless steel

(304) interface at the Rayleigh angle (30.6).

radiation coexist in the region B. Computation of the specular

reflection discloses that specular reflection from stainless steel (304) is

1800 out of phase with the incident wave at exactly the Rayleigh angle

up to a frequency f. (for stainless steel f. lies between 15 and 20

MHz). Above . the phase in the specular reflection is modified as a

result of bulk wave attenuation to be in phase with the incident beam.

Thus at and below f. in region B a nullification would occur when the

specular and Rayleigh radiation are equal in amplitude. As may be seen

in Fig. 8, the delineated regions are, of course, in reality not as sharp

and spatially well defined as in the diagram of Fig. 7 and are drawn as
308 CHAPTER 18

they are for purposes of description. The lateral displacements A,

predicted by Eq. (1), and hydrophone positions at high (i.e., nearly as

large as f,) and low frequency that must be imposed on the receiving

transducer to measure a frequency of least reflection are indicated in

Fig. 7.

At relatively low frequency, say 5 MHz, the hydrophone is

integrating the pressure that is at phase 9 - 0* in region C and part of

the field in the region B where the resultant amplitude is at least diminished and at most cancelled by a signal of opposite phase. At a higher

frequency as A is decreased consistent with Eq. (1), the hydrophone

has incident on it less of the pure Rayleigh radiation because of its

attenuation. This results in a decreased hydrophone output at 10 MHz.

The hydrophone output is a minimum at f,, the highest frequency for

which the signals in regions A and C are of opposite phase. At f., and

for the A prescribed, the energy intercepted in the region A equals or is

very nearly that intercepted in the region C resulting in a minimum

hydrophone output. Above f, the hydrophone output rises significantly

since the pressure in all regions has been observed to have the same

phase.

It must be remembered that the radiation in the region labeled C

decreases to the right as the Rayleigh wave is attenuated along the

interface. Theories of both Becker and Richardson [8,91 and Mott [171

that describe a frequency of minimum reflection for stainless steel also

predict that below a frequency of 5 MHz the reflection increases monotonically to unity at zero frequency when received at the Rayleigh angle

and at the Schoch distance. However, this is inconsistent with the

observation of the generation and continuous radiation of a surface

wave of the type seen in Fig. 6(b). The displacement formula would

cause the hydrophone to be placed at an ever increasing distance as frequency is decreased for a fixed beam size. It would intercept a small

portion of the Rayleigh radiation even at the lower frequencies when

the Rayleigh wave is only slightly attenuated along the surface by radiation or loading of the adjacent liquid. The behavior of the amplitude

with frequency predicted by the attenuative theory 18,9,171 is shown in

Fig. 9. The monotonic dotted portion of the curve below fb is the

result given by attenuative theory. At the frequency fb below f, the

hydrophone would have incident on it only radiation resulting from the

Rayleigh wave because the distance A predicts that the reradiated beam

would be intercepted only there. The frequency fh is a function of

hydrophone size. As frequency is decreased below fb, the predicted

hydrophone distance A increases directly according to Eq. (1) as the

wavelength of the ultrasound. The amplitude of the radiated wave
BOUNDED BEAM 309

W (L

Jo

LW

fb fa

FREOUENCY

Fig. 9 - Qualitative plot of relative reflection amplitudes vs

frequency. The frequency f. indicates the "frequency of

least reflection."

decreases exponentially along the surface inversely as the frequency

[16,18]. Thus, the combined result of these two effects at any frequency below fb is a constant hydrophone output resulting in the flat

portion of the curve in Fig. 9.

No attempt was made to identify exactly the specific frequency f

at which the phase in the specular reflection changed from being 180*

out of phase to being in phase with the incident beam. The manifestation of this change is the disappearance of the dark strip normal to the

radiated wavefront in the radiation. The source transducer outputs for

this range of frequencies changed greatly with frequency which changed

the schlieren-observed result, making it difficult to precisely identify the

frequency at which a strip disappeared. It can be unequivocally stated,

however, that schlieren observations showed f. to be between 15 and 20 MHz for stainless steel.

EXPERIMENTAL OBSERVATIONS AT THE

CRITICAL ANGLES

Evidence of a radiated wavefront similar to that observed at the

Rayleigh angle has been observed at the shear critical angle in the

region outside the specular reflection for some materials on which Rayleigh wave radiation was observed. The effect is difficult to isolate and

distinguish from the Rayleigh Wave radiation that occurs for incidence

angles near the Rayleigh angle. This indicates that at the shear critical

. maJo
310 CHAPTER IS

angles a surface wave is generated on the interface and radiates in a

manner similar to the Rayleigh wave but to a much lesser degree.

Specular reflection at an angle slightly smaller than the shear critical

angle 0r for Mallory 1000 (90% tungsten, 6% nickel, 4% copper) is

shown in Fig. 10(a). At exactly Gr, shown in Fig. 10(b), radiation can

be observed to the right of the specular reflected beam. For an

incidence angle greater than 9S, Fig. 10(c) is obtained. By comparison

of Fig. 10(b) with Figs. 10(a) and 10(c) the presence of the spreading

of the wave to the right can be detected. Actually, the observation was

first made in the process of changing the incident angle through 0,.

The continuous viewing of the change in the appearance of the

reflected pulse makes the effect more visible than the static views of

Fig. 10.

(a)

(bi

(c)

Fig. 10 - Schleren photographs after reflection of a

7-MHz pulse incident from the left on a plane

Mallory- 1000-water interface at (a) the incident angle

# < 32.1. (b) 0 - 32.1° the shear critical angle, and

(c) 9 > 32.1° .
BOUNDED BEAM 311

Evidence of surface wave generation and radiation similar to that

at the shear critical angle has also been found at the longitudinal critical

angle 0 1, and is shown in Fig. I I for Pyrex glass at 5 MHz. This effect

is present to even a lesser degree than that at the shear critical angle.

Figures I 1 (a)-I I (c) show refiected pulses for incidence angles less than

01c, equal to 01, and greater than 0tc, respectively. Again significantly

more energy appears at the right edge of the reflected pulse in Fig.

11 (b) which is taken at 0 c.

(a)

(b)

(c)

Fig. I ! - Schlieren photographs after reflection of a

5-MHz Oiulse incident from the left on a plane Pyrexglass-water interface at (a) the incident angle 9 <

15.2', (b) 9 - 15.20 the longitudinal critical angle, and

(c) e > 15.20.
312 CHAPTER 18

The reader should keep in mind that all of the data, i.e., schlieren

photographs in support of observations or conclusions that are reported

here are not presented here. Conclusions were often derived from continuous observations made in the process of changing the incident angle

or pulse length. The photographs, that are shown were chosen to

demonstrate those observations and to maximize the chances of adequate reproduction.

CONCLUSIONS

The theory of Schoch is inadequate to account for observed radiation from materials for finite beam incidence. Previous apparent agreement between theory and experiment resulted from a reception of only

part of the radiated field by a receiving transducer of the same size as

the source beam. Radiated energy outside of the region predicted by

displacement theory is difficult to measure for materials on which most

measurements were previously made, such as aluminum and stainless

steel. On these materials relatively small displacements are predicted

by the appropriate theory. For materials for which significantly larger

displacements are predicted, separations of the incident and reflected

beam along the plane interface would be expected. Such separations do

not occur for any of the materials examined for any size of source

beam.

For the observations reported here, it is concluded that in addition to specular reflection, a Rayleigh wave is generated by the incident

beam, propagated along the interface, and radiated continuously into

the fluid. Extant theory does not adequately describe this surface wave

generation and radiation. Interference occurs between the specular and

Rayleigh radiations when they are of opposite phase and are of equal

amplitude causing a null field in a portion of the specular region. The

results of previous experimental observations of others can be

accounted for by this description of the reflected and reradiated field of

an incident finite beam. Significant Rayleigh wave generation and radiation also occurs for a range of incidence angles near the Rayleigh angle

for sufficiently low frequencies at which the phase of the specular

reflection is opposite that of the Rayleigh radiation which is always true

[191 when attenuation of bulk waves can be ignored. The theory

describes the observed reradiated field.

The theory of Becker and Richardson has shown the phase for a

plane reflected wave to be frequency dependent as a result of attenuation of the bulk longitudinal and shear waves. That theory predicts a

frequency for incidence at the Rayleigh angle above which the phase in
BOUNDED BEAM 313

the specular reflection is the same as that of the incidence wave and

Rayleigh radiation. That change in phase was detected in schlieren

observations by the disappearance of the dark null strip in the specular

reflection for stainless steel at a frequency between 15 and 20 MHz

indicating the need for considering an attenuative theory in this frequency range.

The attenuative theory also predicts a consequence of the phase

change manifested as a frequency of least reflection. That effect is a

result of the experimental configuration that was used to observe it

which does not include the entire reradiation caused by finite beam

incidence.

The theoretical consideration of the attenuation of bulk waves in

the solid is indicated to be consistent with schlieren observations at frequencies above fa. That same theory is deficient for the explanation of

reflection amplitude at lower frequencies. Theory predicts the monotonic increase of amplitude to unity toward zero frequency. The model

of surface wave generation and attenuation introduced in this paper

demands an approach of the amplitude to a constant value as zero frequency is approached.

REFERENCES

1. A. Schoch, Ergeb. Exakten Naturwiss. 23, 127 (1950).

2. F. Goos and H. Hinchen, Ann. Phys. (Leipz.) 1, 333 (1947).

3. H. Lotsch, Optik (Stuttg.) 32, 299 (1971); Optik (Stuttg.) 32, 553

(1971).

4. L.M. Brekhovskikh, Waves in Layered Media (Academic, New

York, 1960).

5. 0.1. Diachok and W.G. Mayer, IEEE Trans. Sonics Ultrason. 16,

219 (1969).

6. A. Schoch, Acustica 2, 18 (1952).

7. R.F. Smith, M.S. thesis, (University of Tennessee, 1968) (unpublished).

8. F.L. Becker and R.L. Richardson, Research Techniques in Nondestructive Testing, edited by R. S. Sharpe (Academic, London,

1970), Chap. 4.
314 CHAPTER 18

9. F.L. Becker and R.L. Richardson, J. Acoust. Soc. Am. 51, 1609

(1971).

10. F.L. Becker, J. Appl. Phys. 42, 199 (1971).

11. 0.1. Diachok and W.G. Mayer, J. Acoust. Soc. Am. 47, 155

(1970).

12. W.G. Mayer, J. Appl. Phys. 34, 909 (1963).

13. H.L. Bertoni and T. Tamir, App!. Phys. 2, 157 (1973).

14.' W.G. Neubauer, J. Acoust. Soc. Am. 50, 106 (1971).

15. F.D. Martin and M.A. Breazeale, J. Acoust. Soc. Am. 49, 1668

(1971).

16. [.A. Viktorov, Rayleigh and Lamb Waves (Plenum, New York,

1967).

17. G. Mott, J. Acoust. Soc. Am. 50, 819 (1971).

18. W.G. Neubauer, J. Acoust. Soc. Am. 52, 154 (1972).

19. A.L. Van Buren, J. Acoust. Soc. Am. 45, 341 (1969).

I II I 
Chapter 19

RAYLEIGH AND SHEAR SPEED DETERMINATION

USING SCHLIEREN VISUALIZATION*

INTRODUCTION

The low-MHz region is the frequency range in which significant

internal losses in the material can be neglected and for most metals

would have an upper limit in the neighborhood of 10 MHz. It has been

shown in Chapter 18 that when a finite acoustic beam was incident on

the solid, in the vicinity of the Rayleigh angle, a "head-wave was generated and progressed along the interface. Figure 1(a) shows a

schlieren visualization of the radiation which results from a 1-cm-wide

acoustic beam incident on a molybdenum surface in water. The source

is in the upper left-hand corner of the picture radiating downward, and

the frequency is 5.3 MHz. The beam displacement theory of Schoch

[1 predicts a displacement A shown in the diagram of Fig. 1 (b) and is

clearly inadequate to explain the observation. The observation in Fig.

1 (a) and observations on other metals are explained by the following

model presented in Chapter 18. At and near the Rayleigh angle, in

addition to the normal speculJar reflection, an incident acoustic beam

generates a Rayleigh wave in the solid which radiates into the water as

it progresses along the surface. At exactly the Rayleigh angle the specular reflection and the Rayleigh wave radiation are 180 ° out of phase.

This model, based on experimental observations, is in agreement with a

theoretical model of Rayleigh-angle phenomenon by Bertoni and Tamir

[2]. The dark strip in Fig. I (a) is caused by the mutual cancellation of

specular and Rayleigh radiation where they have equal amplitude.

It is this null strip which makes the Rayleigh angle easily and

accurately observable and thus makes possible accurate measurements

of Rayleigh phase velocity. Specular reflection occurs from the region

directly illuminated by the incident beam, and outside of this region

Rayleigh wave radiation alone is responsible for the observed wave

front.

*These results were first reported in: W. 0. Neubauer and L. R. Dragonette, J. Appl.

Phys. 45, 618-622 (1974).

315
316 CHAPTER 19

(a) (b)

Fig. I - (a) Schlieren visualization of the reflection of a finite

beam from a molybdenum-water interface at the Rayleigh angle.

The incident beam is on the left-hand side. (b) Diagram of a

displaced beam according to the prediction of Schoch theory for

the incident beam in (a).

MEASUREMENTS

The materials were solid circular cylindrical samples with radii of

6.35 cm and with 7.62-cm lengths. The circular ends are flat to within

±0.001 cm, and these flat ends are the metal surfaces used in the

experiments. The sound source is a 1.9-cm-diam lead-zirconiumtitanate transducer. No receiving transducer is needed for the velocity

measurements. A transducer similar to the source with a mask having

a 1-mm wide slit was used as a receiver in attenuation measurements.

The Rayleigh phase velocity c, is related to the Rayleigh angle 0,

by (41

c, - c,J/sin 9, (1)

where c,, is the speed of sound in water. The speed of sound in fresh

water is known to at least 0.01% from Chapter 23. Thus, the limitation

on how accurately c, can be obtained is limited by the measurement of

0,. The generation of a head wave by an incident acoustic beam does

not by itself give an accurate measure of 0,. As shown theoretically in

Viktorov [3] and experimentally by Neubauer [11, Rayleigh waves are

generated at many angles and are strongly generated over a wide range

of angles in the vicinity of the Rayleigh angle. The dark null strip seen

in Fig. 1(a) is, however, a sensitive function of the Rayleigh angle,

since it is at this exact angle that the Rayleigh radiation and specular

reflection are 180" out of phase. Figures 2 and 3, respectively, show

the radiation observed when stainless steel and molybdenum interfaces

are insonified at and near the Rayleigh angle. In both cases the dark

strip in the middle frame at Rayleigh-angle incidence is easily distinguishable from the radiation at angles 0.10 away, With real-time obser-
RAYLEIGH AND SHEAR WAVE SPEED 317

-Fig. 2 - Reflection of a finite beam from a

water-stainless steel interface for incidence at (a)

0.1* less than the Rayleigh angle, (b) at the Rayleigh angle, and (c) 0. 10 greater than the Rayleigh

angle.

Fig. 3 - Reflection of a finite beam from a ________________

water-molybdenum interface for incidence at (a)

0.1* less than the Rayleigh angle, (b) at the Rayleigh angle, and (c) 0.10 greater than the Rayleigh

angle.
318 CHAPTER 19

vation of the radiation using a TV monitor, the Rayleigh-angle measurements could be repeated to 0.100 and the limitation was the

mechanical measurement system and not the method. Table 1 gives

measured Rayleigh angles and calculated phase velocities for several

materials that were measured. The velocity errors given in the table are

based on the ± 0.1 ° limitation in angle measurement imposed by the

angular measurement system used here. This is not necessarily the

limitation of the experimental method.

Table 1 - Rayleigh Phase Velocities at 20'C

Material Experimentally determined Rayleigh phase

0r (deg) velocity (m/sec)

stainless steel 31.62 2827 t 8

aluminum 31.36 2848 t 8

molybdenum 28.61 3095 ± 8

K-91 25.40 3456 :t: 13

K-8 23.00 3794 ± 16

K-165 17.26 4996 ± 28

aluminum oxide 15.10 5691 ± 38

SHEAR VELOCITY ESTIMATES

In addition to the accurate and direct measurement of Rayleigh

phase velocity obtained from this schlieren technique, estimates of the

shear velocity may be made which are accurate to within a few percent.

These estimates can be made by using only the acoustic surface

reflection measurements discussed above. The determination of a bulk

property using a surface property can be very advantageous in materials

where direct shear velocity measurements are quite difficult. The shear

velocity estimates can be made by using the approximation found in

Mason [5]. This approximation is shown graphically in Fig. 4, which

shows the ratio of Rayleigh velocity c, to shear velocity c, plotted vs

Poisson's ratio a,. The velocity ratio is slowly varying with Poisson's

ratio so that a need not be known with great accuracy. For example, a

value cr - 0.25 ± 0.10 on the curve of Fig. 4 corresponds to a range in

c,/c1 of 0.90 < c,/c < 0.935. This is an error of about ±2% in the

estimate of c/c. Over the entire curve, the ratio c,/cs varies only from

0.86 to 0.96, so that a value c,/c = 0.91 can be chosen if nothing is

known about a, and still give a result within usable limits for many

applications.

! _

___ "7__ "__________. ______________ . ..
RAYLEIGH AND SHEAR WAVE SPEED 319

O.96

0.94

.0.92-

U

0.90-

0.86

0 0.05 0.10 0.15 0120 025 0.30 O.35 0.40 0.45 050

POISSON'S RATIO

Fig. 4 - Ratio of Rayleigh velocity to shear velocity plotted

vs Poisson's ratio.

Table 2 gives a comparison between shear velocities estimated

from the schlieren measurements of c, and shear velocities measured

by direct shear wave propagation through the same samples.

Table 2 - Comparison of Shear Velocity Estimates

with Direct Measurements

c. direct

c, estimates propagation Material (m/sec) measurements % difference

(m/sec) _

aluminum 3030 3143 3.6

stainless steel 3040 3148 3.4

molybdenum 3328 3347 0.5

K-91 3757 3853 2.5

K-8 4135 4172 0.8

K-165 5490 5410 1.5

aluminum oxide 6254 6300 0.7

aThe direct propagation measurements were made by the Ocean Materials Criteria Branch at NRL on the same samples used in the schlieren results. These

direct measurements are accurate to within ± 1%.

A similar technique, one of measuring elastic constants by the

identification of the Rayleigh angle through observation of the welldefined dark strip in the reradiated field, can be applied to plates. This

method of estimating shear wave speed on plates is usable for plates

three Rayleigh wavelengths thick, it is probably usable for plates even

____________________________
320 CHAPTER 19

thinner. A finite beam incident on a nickel plate approximately three

Rayleigh wavelengths thick is shown in Fig. 5. The sound pulse has a

center frequency of 5.8 MHz. Values for Rayleigh speed measured on

five different plate materials, as well as the estimated shear speed in

those materials, are given in Table 3. Comparison with handbook

values is also given where these are available [6].

Fig. S - Reflection of a finite beam at the Rayleigh angle

from a nickel plate 3.0 Rayleigh wavelength thick.

Table 3 - Shear Wave Speeds Inferred from

Schlieren Measurements on Plates

Rayleigh Shear Handbook values

Material speed speed of shear speed % difference

(m/sec) (m/sec) (m/sec)

Nickel 2780 2973 3000 0.9

Armco iron 2935 3166 3240 2.3

Mu metal 2749 2988 ......

Iconel 2757 2996 ...

Permalloy 2757 2996 ...

I
RAYLEIGH AND SHEAR WAVE SPEED 321

ATTENUATION RESULTS

In the low-MHz region, in which these experiments were performed, the attenuation of the Rayleigh wave caused by radiation into

the water is given by [7)

a0'Pw4,/PcrA, (2)

where a, is the absorption coefficient of the Rayleigh wave, Pw is the

density of water, p is the density of the solid, and X is the Rayleigh

wavelength. As demonstrated in Fig. 2 for stainless steel, even when

the Rayleigh wave is not transmitted over a large distance along the

interface, the identification of the null strip remains a sensitive

identifier of the Rayleigh angle. Attenuation measurements were made

on the materials listed in Table 1. As mentioned previously, the

receiving transducer used in attenuation measurements was similar to

the source transducer and was masked to have an active face 1 mm

wide. The Rayleigh angle was first determined by using the null strip

method described. A receiving transducer was then placed in a fixed

position and the source transducer was moved parallel to the liquidsolid interface. A typical plot of amplitude normalized to the maximum

value vs distance parallel to the interface for reflection from molybdenum is given in Fig. 6. The initial peak is the specular peak. The dip

10

,906-

Fig. 6 - Relative amplitude of the reflection

of a finite beam from a molybdenum -water

interface at the Rayleigh angle, plotted vs

the relative distance parallel to the interface

-j

W 04 atong the surface.

02-

0 1 2 3 4 5

PICL*TIV DISTANCE
322 CHAPTER 19

in the curve corresponds to the position of the cancellation of specular

and Rayleigh radiation, and the second peak is the Rayleigh peak which

is followed by an exponential decay. Figures 7(a) and 7(b), respectively, show a linear and logarithmic plot of the decay of the Rayleigh

amplitude as a function of distance parallel to the interface in cm for

stainless steel. Measured values of a, are listed in Table 4 and are

compared to values calculated from Eq. (2). This equation predicts that

a, is directly proportional to frequency, and this was verified experimentally on aluminum oxide at frequencies from 4.46 to 6.19 MHz.

The amplitude measurements given in Figs. 6 and 7 show oscillations

about the exponential decay of the radiated wave. Hydrophone measurements of the amplitude of these fluctuations about the exponential

decay vary with receiving hydrophone aperture, and these differences

with aperture are discussed in Ref. 8. However, a consistent value of

a, is obtained from a least-squares fit of a plot of the logarithm of the

amplitude vs the distance.

I0

09-

.06

W

0j4

Q2

0. 0" 1. 0 1 2 3 4 5 0 1 2 3 4 5 6 RELATIVE ISTANCE RELATIVE DISTANCE

(a) (b)

Fig. 7 - (a) Relative amplitude of the reflection of a finite beam from a stainless Steelwater interface at the Rayleigh angle, plotted vs the relative distance parallel to the

liquid-solid interface. (b) Relative amplitude in (a) plotted logarithmically vs relative distance parallel to the liquid-solid interface.
RAYLEIGH AND SHEAR WAVE SPEED 323

Table 4 - Measurement of Rayleigh Attenuation a, in Np/cm

Material Frequency Experimental Calculated _________ (MHz) a, _ _ __

stainless steel 3.8 0.82 0.89

aluminum 5.00 3.43 3.41

molybdenum 3.3 0.58 0.51

K-8 4.72 0.35 0.33

K-91 3.8 0.42 0.35

K-165 6.50 0.67 0.64

aluminum oxide 4.46 0.46 0.52

aluminum oxide 5.00 0.55 0.58

aluminum oxide 6.19 0.71 0.71

CONCLUSION

Measurements of the Rayleigh phase velocity in the low-MHz

region can be directly and easily made by using a schlieren technique.

When a finite acoustic beam is incident from water on a plane solid

interface, head waves are generated at and near the Rayleigh angle.

These head waves are due to the generation of a Rayleigh wave. In

addition to normal specular reflection, the Rayleigh wave radiates into

the water as it progresses along the interface. At exactly the Rayleigh

angle, the specular reflection and the Rayleigh wave radiation are 1800

out of phase. Over a small region where they are of equal amplitude,

the specular and Rayleigh radiations mutually cancel and thus form ian

easily observable dark null strip in the radiation. This measured angle is related to Rayleigh phase velocity by a simple equation. The attenuation of the Rayleigh wave in the frequency range used was shown to be

primarily due to radiation loss into the water. Even for materials with a

high attenuation, the null strip was easily observed and phase velocity

was easily obtained. The direct measurements of Rayleigh phase

velocity were also used to estimate the shear velocities of the materials

by utilizing a theoretical approximation. The least accurate estimate of

shear velocity by this method disagreed with measured shear velocities

by 3.6%. The technique was attempted on plates of the order of three

Rayleigh wavelengths thick, and agreement, when comparisons could

be made, was excellent. Thus, given an arbitrarily shaped body with a

flat surface, a simple method exists for determining its Rayleigh phase

velocity accurately and for estimating its shear velocity to within a useful accuracy.
324 CHAPTER 19

REFERENCES

1. A. Schoch, Ergeb. Exakten Naturwiss 23, 127 (1950).

2. H.L. Bertoni and T. Tamir, Appl. Phys, 2, 157 (1973).

3. I.A. Viktorov, Rayleigh and Lamb Waves (Plenum, New York,

1967).

4. W.G. Neubauer, J. App. Phys. 44, 48 (1973).

5. W.P. Mason, Physical Acoustics and the Properties of Solids (Van

Nostrand, Princeton, N.J., 1958), p. 21.

6. Handbook of Chemistry and Physics, 52nd ed. (Chemical Rubber

Co., Cleveland, Ohio, 1971-1972), p. E-41.

7. K. Dransfeld and E. Salzmann, in Physical Acoustics, Vol. VII,

edited by W.P. Mason and R.N. Thurston (Academic, New York,

1970), pp. 260-261.

8. W.G. Neubauer, in Physical Acoustics, Vol. X, edited by W.P.

Mason and R.N. Thurston (Academic, New York, 1973), p. 61.
Chapter 20

SCHLIEREN VISUALIZATION OF

"HEADWAVES" ON PLATES*

INTRODUCTION

A schlieren technique of measurement and a path trace method of

analysis have been used [I] to identify the wavefronts radiated from

solid and hollow aluminum cylinders illuminated by short acoustic

pulses in water. The schlieren technique was found to be helpful, and

in some cases necessary, towards understanding pressure pulse hydrophone measurements of periodic pulses in the shadow zone of cylinders

and cylindrical shells 11-51. Two distinct guided waves with circumferential properties were found on cylindrical shells [1). The position

and group velocity of the radiated wavefronts resulting from these two

effects were shown to be consistent with appropriate combinations of

multiple shear and longitudinal reflections between the shell surfaces.

Illumination by a plane wave at a single angle of incidence is not possible in the case of the shell because of the cylindrical geometry and the

finite beam width. Methods for determining the angles at which

incident energy contributed to the observed effects are discussed in

Ref. I. Illumination at a single incidence angle can be more nearly

achieved in the case of a flat plate.

Guided modes in aluminum plates are isolated here by illuminating the plates with short acoustic pulses at specific angles over a range

from 00 to 35° . This range includes the longitudinal and shear critical

angles, 13.50 and 28.50, respectively, and the Rayleigh angle, 31.00. In

general, only those guided modes which are not strongly attenuated by

radiation into the water will be isolated and considered. Both the trace

velocity and the group velocity of the modes are easily obtained from

schlieren visualizations. A formula giving the group velocity of the

observed radiated wavefronts or "headwaves" along the plate is derived

independently of the type of wave motion in the plate. The effects generated are compared with previous experimental results on plates in

water (7, 81. The utility of a path trace technique for predicting the

group velocity of the observed "headwaves" is determined for the plate

geometry. Comparisons between the plate and previous [11 cylindrical

shell results are made, and the association of Lamb modes, with the

guided waves seen in shells, is discussed.

"The results were first reported in: Louis R. Dragonette, J. Acoust. Soc. Am. 51 920

(1972).

325
326 CHAPTER 20

EXPERIMENT

The schlieren system utilized here is fully described in Chapter 22

and in Ref. 9. A diagram of the system is given in Fig. 1. The sources

are lead-zirconium-titanate transducers and are, in all cases, above the

plate and radiating downward. The radiated field in the water is visible

above and below the plate. The plate, in some cases, rested at each

end on a 1/4-in.-thick rubber cushion glued to a brass block. In other

cases, the plate rested at each end on a brass rod supported by a device

capable of accurately rotating the plate. The aluminum plates used in

the schlieren measurements were made of 6061 aluminum and had

dimensions of approximately 12.7 x 7.6 cm. The plate thickness

ranged from 0.002 to 0.241 cm. At the frequencies used, these

thicknesses (D), expressed in wavelengths, varied from 0.023xp to

1.89,kp, where X, is the compressional wavelength in aluminum, or

from D - 0.046A, to D - 3.84K,, where X, is the shear wavelength.

The values for shear wavespeed (V, - 3.136 x 10 5 cm/sec) and

compressional wavespeed (Vp - 6.370 x 105 cm/sec) were measured

directly from circular cylindrical samples of 6061 aluminum. These

samples were 2.54 and 10.16 cm thick, and 7.62 cm in diameter. The

speed of sound in water at 200C, 1.4853 x 105 cm/sec, was obtained

from Ref. 10, and is given in Chapter 23.

iFig -A diagram ofthe experimentaI system.

A I IO/I

Fig I-Adar mofteeprmnasye .
HEADWAVES ON PLATES 327

Figure 2 shows a schlieren observation of the radiated field in the

water above and below a 0.130-cm-thick plate illuminated by a 3-jssec

acoustic pulse. At the frequency 7.2 MHz, this thickness corresponds

to 1.47x,. In the successive frames of Fig. 2, specific angles of

incidence, between 0 and 26.50, are shown. Frames (a)-(c) (incidence

angles 00, 2.00, and 4.00) show specular reflection above the plate and

transmission below it. In addition, a definite brightening appears

behind the specular and transmitted pulses which seems to maximize

between 2.00 and 4.00. No detailed interpretations of this brightening

are attempted since the effect does not separate sufficiently from specular reflection to make any useful study of its properties possible; however, it could correspond to a highly attenuating plate mode. The only

effects which will be considered in detail are those which persist far

enough along the plate, so that separation and isolation from specular

reflection are achieved. The first occurrence of such an effect is seen

in frame (e) (8.00). At 10.00, in frame (f), only specular reflection and

transmission are observed. In frames (g) and (i) (12.00 and 13.20), the

appearance of two distinct wavefronts which have progressed along the

plate are observed and in frame (h) (12.50), both are seen simultaneously. From 16.00 through 22.0' [frames (j)-(0)), only specular

reflection and transmission are present. Finally, the occurrence of

another progressing wavefront is observed in frames (n) and (o) (26.00

and 26.50). Only specular reflection and transmission were observed as

rotation of the plate was continued through 35.00. The additional

wavefront, seen in frames (g)-(o), is a reflection from the transducer

of the reflected pulse. The effects seen in Fig. 2 are observed

separately in Figs. 3 through 7.

When the 0.130-cm-thick plate is illuminated by an acoustic pulse

at an incidence angle of 12.50, the result shown in Fig. 3 is obtained.

The driving pulse is approximately 3-psec long at a frequency of 7.2

MHz. The successive frames of Fig. 3 show the sound beam striking

the upper surface of the plate and at later times the formation and progression along the plate of two distinct wavefronts, each with its own

characteristic inclination angle and speed. Separation in time between

the frames is obtained by varying the time delay between the triggering

of the sound source and the strobe-light source. In this way, the sound

pulse and its effect can be frozen at any instant, from the time the

sound pulse leaves the transducer until it and its effects disappear below

the dynamic range of the system. Both of the wavefronts, seen in Fig.

3, are equally well observed above or below the plate. The two wavefronts can be isolated by changes in the incidence angle and are

observed separately in Figs. 4 and 5 (12.00 and 13.2°). The strong

________________________________________________________________________
328 CHAPTER 20

pFig. 2 - Schlieren photographs taken as a 0.130-cm-thick plate is rotated from normal to the direction or propagation of an incoming acoustic pulse through 26.50. Pulse length is 3 gsec; frequency is 7.2 MHz. Frame (a) is 0.0, (b) 2.0%. (c) 4.0, (d) 6.00, (e) 8.00, (f) 10.0, (g) 12.0, (h) 12.5, (i) 13.2 °

, (j) 16.0*, (k) 20.0 °, (I) 22.00, (m) 24.0', (n) 26.00,

and (o) 26.50.
HEADWAVES ON PLATES 329

(a) (b) (c)

(d) (e)

Fig. 3 - A time sequence of schtieren photographs showing the effects of

illuminating a 0.130-cm-thick plate with an incoming acoustic pulse at 0, -

12.5* . Pulse length is 3 jsec; frequency is 7.2 MHz,

Fig. 4 - A time sequence of schlieren photographs showing the effect of illuminating a 0.130-cm-thick plate with an incoming acoustic pulse at 8l - 12.0*. Pulse length is 3

usec; frequency is 7.2 MHz.
330 CHAPTER 20

(a) (b)

(c) (d)

Fig. 5 - A time sequence of schlieren photographs showing the

effect of illuminating a 0.130-cm-thick plate with an incoming

acoustic pulse at 0, - 13.2'. Pulse length is 3 /sec; frequency is

7.2 MHz.

Fig. 6 - A time sequence of schlieren photographs showing the effect of illuminating a 0.130-cm-thick plate with an incoming acoustic pulse at 0,- 8.00. Pulse length is 3

Asec; frequency is 7.2 MHz.
HEADWAVES ON PLATES 331

(a) b)

(c) (d)

Fig. 7 - A time sequence of schtieren photographs showing the effect of illuminating a 0.130-cm-thick plate with an incoming acoustic pulse at 0, = 26.50. Pulse length is 3 /Asec;

frequency is 7.2 MHz.

persistence of each of the radiated wavefronts over a small range of

incidence angles is consistent with the amount of beam spreading

expected with the source transducer. At incidence angles of 8.00 and

26.50, the final two effects observed in Fig. 2 were maximized and

time-sequence schlieren photographs of the progressions of the radiated

wavefronts along the plate are seen in Figs. 6 and 7, respectively. The

angles of incidence of the incoming acoustic wave in Figs. 2-7 are

measured with a possible systematic error of ±0.5'. Measurements

were made from Figs. 3-7 of the angle of inclination, 0, between the

radiated wavefront and the plate surface (i.e., the angle at the intersection of the wavefront and the plate). The measurements show that 0,

is not equal to 0,, the original angle of incidence of the sound pulse. In

all cases, however, the specular reflection makes an angle 0, with the

plate. The radiated wavefront angles are measured when the

"headwave" has progressed far enough along the plate that the specular

reflection is easily avoided. Differences between 0, and 0i are especially obvious in Fig. 6, where sound is incident at 0, - 8.0", and 0, is

60.0", and, in Figs. 4 and 5, where a change of 1.2" in incidence angle leads to generation of two wavefronts, whose inclination angles differ

5,
332 CHAPTER 20

by 10.00. Measurements of 0,, taken from different frames in the

time-sequence photographs, show it to be a constant for each wavefront, as that wavefront progresses along the plate. Theoretically,

Snell's law predicts that energy will leave the plate at an angle equal to

the angle of incidence. This is confirmed by the plate results of Worlton [7, 81. The apparent paradox between the expected result, that

energy should leave the plate at the same angle at which sound was

incident, and the determinations of 0,, made from the schlieren photographs, is in reality only a manifestation that the group velocity and

phase velocity of the effect involved are different.

On the single assumption that energy making up the wavefronts

observed in the schlieren photographs does leave the plate at an angle

equal to the incidence angle, a formula for the group velocity of the

wavefront along the plate can be derived from geometric considerations

alone. The derivation does not depend upon any consideration of the

type of motion within the plate. In Fig. 8, the lines AC and BD

represent positions of a radiated wavefront above a plate OD at times tt

and t2. Positions A and B represent positions of a ray which has left

the plate at an angle 0, and traveled with the speed of sound in water,

V,, to position A in time it, and from position A to B in time t2- ,-

The radiated wavefront moves a distance CD along the plate in the

time, t2 - tt. The group speed V. is related to the speed of sound in

water, V., by Vg/ V. = CD/AB, which can be expressed as

V, - V,, cos (0, - O,)/sin 0,. (1)

N

N

A

t0 c D

Fig. 8 - Diagram, representing the position of a radiated

wavefront at successive times tI and 12, from which group

velocity formula is derived.
HEADWAVES ON PLATES 333

Table I gives values of V which were calculated from the measured

values of 0 , The latter were obtained from the schlieren photographs

presented. The radiated wavefront angle, determined from photographs

shown in Fig. 6 (entry No. 1), was measured with a precision of : 5.00.

The thinner, sharper wavefronts in the schlieren photographs,

corresponding to entries 2 through 9 in Table 1, were measured with a

precision of ± 2.00. Included in Table 1 are results obtained from

schlieren photographs of waves generated on plates thinner compared

to a wavelength. These schlieren photographs are seen in Figs. 9

through 12, and correspond to entries 5 through 10 in Table 1. Figures

9(a) through 9(c) show wavefronts which occur when a 0.130-cm-thick

plate is tilted at 12.00, 12.50, 13.40, and 26.00 with respect to the direction of propagation of an incoming acoustic pulse. The pulse is 5 psec

long and the frequency is 5 MHz. At this frequency, the plate thickness corresponds to 1.02X, The three separate wavefronts isolated in

Figs. 9(a), 9(c), and 9(d) were the only observed progressions along

the plate, in a range of rotation from 00 to 35.00. Figures 10-12 show

the single strongly persistent wavefronts observed on plates of thickness

0.051, 0.025, and 0.002 cm, respectively. In all three cases, the pulse

length is 3 jusec, the frequency is 7.2 MHz, and the angle of incidence

is 13.00. The 0.002-cm plate is not 6061 aluminum. It is made from

commerically available aluminum foil. Entries 8-10, in Table I, show

the results obtained from these three figures. Values for the incidence

angle and wavefront angle for the thin foil are only best estimates,

since the surface of the foil could not be flattened sufficiently for precise measurements to be made.

Some direct measurements of V,, which utilized a receiving

hydrophone, were also made in a separate experiment. A source and

receiving hydrophone were set with angles of incidence and reception

the same. The receiver was moved at a constant height above the plate

and the velocity is given by V - dit, where d is the distance that the

receiver moved and t is the time delay in the received pulse. The

results of these measurements are also included in Table 1. Pulse

shape changes and pulse lengthening effects, noted by others [8, 11J,

made meaningful direct measurements of V, for the situation

described in entry No. I of Table 1 impossible. No hydrophone velocity measurements were attempted in the two thinnest cases because of

the difficulties in obtaining and mounting long plates that thin. A summary of the experimental results, tabulated in Table 1, shows that

almost all of the persistent "headwaves" were generated at angles within

2.00 less than the longitudinal and shear critical angles (13.50 and 28.50,

--- ~-.
334 CHAPTER 20

Table I - Group Velocities of Observed Wavefronts

Incident and Radiated V, x 101

Entry Plate Angles Corresponds t V, x 101 cm/sec

No. Thickness Schlieren Photograph cm/sec Measured Summarized from Eq. (1) with Receiver

0, 0,

1 1.47, 8.00 600 ± 50 1.0 + 0.2 ... -0.1

2 1.47K, 12.5 ° 170 4 20 5.1 ± 0.6 4.8 ±t 0.3

3 1.47)., 12.50 270 - 20 3.2 +0.2 2.9 ±t 0.2

- 0.3

4 1.47x, 26.50 30 ° 0 20 3.0 - 0.2 3.0 ±t 0.2

5 1.02x, 12.00 190 - 20 4.5 - 0.3 4.5 - 0.3

6 1.02L , 13.4 ° 290 - 20 3.0 ± 0.2 3.0 ±t 0.2

7 1.02 , 26.00 290 - 20 3.0 ± 0.2 3.0 ±t 0.2

8 0.58x, 13.00 180 - 20 4.8 + - 0.6 0.5 4.6 ± 0.3

9 0.28\p 13.00 180 20 4.8 + 0.6

-0.5

10 0.023X, 1300 160a 50 a /

Fig. 9 - Schlieren photographs showing effects generated on a 0.130-cm-thick plate -it (a)

@j - 12.0*, (b) 9, - 12.5 ° , (c) 0, - 13.4*, and (d) 0 - 26.00. Pulse length is 5 jusec; frequency is 5 MHz.
HEADWAVES ON PLATES 335

Fig. 10 - A time sequence of schlieren photographs showing the effect of illuminating a 0.051-cm-thick plate with an

incoming acoustic pulse at 0, - 13.0', Pulse length is 3

Msec frequency is 7.2 MHz.

Fig. I1 IA time sequence of schlieren photographs showing the effect of illuminating a

0.025-cm-thick plate with an incoming acoustic pulse at 0, - 13.00. Pulse length is 3

/ssec, frequency is 7.2 MHz.
336 CHAPTER 20

Fig. 12 - A time sequence of schlieren photographs showing the effect of illuminating a 0.002-cm-thick plate with an

incoming acoustic pulse at 0, - 13.00. Pulse length is 3

Asec- frequency is 7.2 MHz.

respectively). Wavefronts with V, - 4.5 x 105 cm/sec were generated

at incidence angles within 1.5* less than the longitudinal critical angle

on plates whose thicknesses ranged from 0.023K, < D < 1.47k,. The

effects are isolated in Figs. 4, 9(a), 10, 11, and 12. Wavefronts with

group velocity, V = 3 x 105 cm/sec, were generated for incidence

angles within 2.50 less than the shear critical angle, on plates of

thicknesses D - 1.47k. and D - 1.02k, [Figs. 7 and 9(d)], but were

not observed on the thinner plates. A second wavefront generated near

the longitudinal critical angle with Vg = 3 x 101 cm/sec was seen on

plates of thicknesses D - 1.02.P and D - 1.47kp [Figs. 5 and 9(c).

No "headwaves" were observed between the shear critical and the Rayleigh angles.

COMPARISON WITH LAMB THEORY

Theoretical [121 and experimental [7, 81 evidence indicates that

Lamb theory for plates with stress-free surfaces can be applied to the

case of water-loaded plates. Lamb [61 derived two transcendental equations, which can be written in terms of the frequency-plate-thickness

product fD and the phase velocity V of the Lamb modes. The first of

these equations,
HEADWAVES ON PLATES 337

tan h{(wjD/ V)( V,2 - V2)/ V'211' 2 7

tan h ((wfD/ V)( V - V2)1 V ]'

I2

4{[(V- V2 2 [(V 2 - V2)1 v 2112

1

[(2 V2 - V2) V,2]2 , (2)

is satisfied if the plate motion is symmetrical with respect to a plane

parallel to, and midway between, the plate surfaces. The second equation, tan h (rfD/ V) [( V'2 - V2

)/ V2

112

)

tan h (fD/(V)[( V 2 - ) 2 1P

V2,- V2 )V 2 ]

2 (3)

=4([(V

2 - V2)1 V2]112 .[( V2 - V2)1/V21112

is satisfied when motion is asymmetrical with respect to this median

plane. The number of propagating symmetrical modes possible at a

given fD is determined by the number of real values of V which satisfy

Eq. (2). Likewise, the number of propagating asymmetrical modes possible is determined by real values of V which satisfy Eq. (3) at a given

fD value. Numerical solutions of Eqs. (2 and 3), in the form of curves

relating phase velocity of Lamb modes to the frequency-plate-thickness

product are presented for various materials in Refs. 7 and 8. A plot of

the first four symmetric and asymmetric modes for 6061 aluminum was

computed and is seen in Fig. 13.

0

W

0

x

-7, \2-\

x ,- ---' -I

254

2 54 508 762 10.1S 12 ?

FREQUENCY x PLATE THICKNESS x IO5CM CYCLES/SEC

Fig. 13 - Lamb phase velocity vs fD for the first four symmetric (-)

and asymmetric modes (--).
338 CHAPTER 20

The angles of incidence at which the effects, seen in the schlieren

photographs, are generated are compared in Table 2 to theoretically calculated incidence angles at which Lamb modes can be generated. The

relationship between the angle of incidence 0, and the phase velocity V,

of a Lamb mode, is given by [71:

V - V,,/sin 0,. (4)

Real values of V, satisfying Eqs. (2 and 3), are computed at the fD

values corresponding to the experimental situations presented, and

these are used in Eq. (4) to obtain the theoretical values of 0, at which

Lamb modes can be generated.

Table 2 - Comparison Between Incidence Angles at Which Effects

are Strongly Observed and Angles Predicted by Lamb Theory

fD x 105 0, Entry cm 0, Lamb Lamb Mode

No. Cycles/ Measured

sec Theory

1 9.4 8.00 9.20 5th symmetric

2 9.4 12.00-12.50 12.70 4th symmetric

3 9.4 12.50-13.20 13.50 4th asymmetric

4 9.4 26.50 27.20 2nd asymmetric

5 6.5 12.0°-12.50 12.70 3rd symmetric

6 6.5 12.50-13.40 14.40 3rd asymmetric

7 6.5 26.00 25.70 2nd asymmetric

8 3.7 13.00 13.50 2nd symmetric

9 1.8 13.00 17.00 1st symmetric

10 0.14 13.00 16.50 1st symmetric

Many more Lamb modes are theoretically possible than were actually seen in the schlieren photographs for a given fD value; however,

with this short pulse schlieren method, the presence of an effect is considered unequivocal only when the radiated wavefront progresses far

enough down the plate so that its separation from specular reflection is

undeniable. In most cases, identification of a mode which is highly

attenuated (that is, radiates a great deal of energy into the water over a

short distance) would be ambiguous using this short pulse method.
HEADWAVES ON PLATES 339

Though no radiation is involved in Lamb's theory, Worlton (71 made

estimates of those modes that would be least attenuated due to radiation, based on the ratio of normal to horizontal particle motion at the

plate surface for a specific mode. By this means, based on Lamb's

theory, it is possible to estimate those modes that would propagate

furthest along the plate at a given fD value. Calculations based on this

method indicate that the wavefronts observed in the schlieren photographs are those expected to be least attenuated.

Table 2 shows good agreement between the experimentally

observed results, and predicted angles of incidence for Lamb modes,

when fD is at least 3.7 x 105 cm cycles/sec. The agreement is poorer

for the two thinnest plates. The poorer agreement between experiment

and prediction on the thinner plates suggests that water loading

becomes a more significant factor for small values of fD where only the

first symmetric and asymmetric Lamb modes are possible. Osborne and

Hart [121 made exact calculations of the effect of water loading on

phase velocity vs fD for the first symmetric Lamb mode on stainless

steel plates. These calculations did not indicate as large a modification

of phase velocity as seen experimentally at the small fD values. Their

observations that, in general, water loading had little effect on Lamb

modes were confirmed for sufficiently large fD.

Of special note are the effects summarized in entries 2, 5, 8, 9,

and 10 of the Tables. These effects are similar in that they are all generated within 1.50 of the longitudinal critical angle, and have the same

group velocity to within experimental error. Despite their similarity,

they do not involve a single Lamb mode, but correspond to the fourth,

third, second, and first symmetric Lamb modes depending on the plate

thickness. Thus for aluminum plates, even as thin as D - 0.02,p, a

strongly persistent wavefront with V, = 4.5 x 105 cm/sec, will be generated by an acoustic pulse incident within 1.5 less than the longitudinal critical angle. This phenomen-n jumps Lamb modes depending on

the plate thickness.

PATH TRACE COMPARISON

Path trace analyses for "headwaves" applied to the effects generated at angles of incidence slightly less than the shear critical angle

[Figs. 7 and 9(d)] show that "headwave" position and angle can be

accounted for by internal reflections of shear waves between the plate

surfaces. Here group velocity is used in the same manner considered

by Brillouin [131 in his discussions of guided waves in pipes. With the

aid of Fig. 14(a), V, can be calculated directly from the formula
340 CHAPTER 20

Vg- V, sin/3, (5)

where 0 is the angle between the refracted, and subsequently reflected,

shear ray and the normal to the plate surface. The formula is derived

by noting from Fig. 14(a) that a refracted ray travels a path AC with

velocity Vs, and that the component of V, along the plate surface is

V1 sinj8. The value obtained for V. at an angle of incidence of 26.50 (/3

- 70.40) is V,, - 3.0 x 105 cm/sec, which is in excellent agreement

with the schlieren observations of Fig. 7, and the direct hydrophone

measurements, both of which gave V. - (3.0 -t 0.2) x 105 cm/sec.

The plate thickness in Fig. 7 was 3.0k , but owing to refraction a shear

ray travels a distance 9 .0k, between surfaces.

In Fig. 14(b), the radiated field above plates of thicknesses d and

2d, illuminated at an incidence angle of 26.50, is constructed. The

upper surfaces of the two plates are colinear in the figure, and

contributions to the radiated field above the plates from a ray incident

at 26.50 at position A are determined. These contributions are calculated by path tracing of the internal shear reflections in the plates and

the radiation into the water at each reflection of a shear ray from the

plate surface. All path lengths, directions, and times of travel in the

plate and in the water were calculated using Snell's law. Each time an

internal shear reflection strikes the surface, sound energy enters the

water at 26.50, the original incidence angle. In the time taken for the

internal shear reflections to reach position B, rays leaving the thinner

plate have reached positions labeled (x). Rays leaving the plate of

thickness 2d have reached positions labeled (0) in the same time. The

fact that internal shear reflections in both plates reach position B at the

same time is expected since the group velocity formula, Eq. (5). is

independent of plate thickness. The radiated wavefronts from the two

plates are observed to be colinear, and to make an angle of 30.00 with

the plate surface. This is in agreement with the schlieren observation

of Fig. 7 that, for 0, - 26.5 , 9, - 30.00. Thus, a ray trace gives

quantitative agreement with both group velocity and position of the

wavefront observed in Fig. 7. From Eq. (5), it can be seen that the

group velocity calculated from a path trace analysis does not depend on

plate thickness. It depends on the refracted angle 3 which is determined from the original angle of incidence. While the group velocity is

not dependent on plate thickness, Fig. 14(b) shows that the continuity

of the radiated wavefront is. As seen in Fig. 14(b), the effect of doubling the plate thickness was to reduce to half the number of contributions to the radiated wavefront of the incident ray, over the same travel

distance along the plate. If this description of the effect, seen at 26.5 ° ,

is physically correct, a combination of a plate sufficiently thick or an
HEADWAVES ON PLATES 341

N

C

i/ 7>

//

16,//9 .

!~ A

(b)

Fig. 14 - (a) Internal path taken by a guided wave consisting of all shear rays, from

which group velocity is calculated. (b) Radiated wavefront above superimposed plates of

thicknesses dand 2d, in which a guided wave consisting of all shear paths is present.

incident plane wave sufficiently narrow, should allow gaps in the radiated wavefront to be observed. In Fig. 15, this effect is demonstrated

by a resulting wavefront, which occurs when a 5-MHz pulse is incident

at 26.50 on a plate 0.241-cm thick. Calculations of the expected spacings agree with the observations in Fig. 15 to within the accuracy with

which the spacing of the gaps can be measured from the photograph.

The wavefront angle and group velocity remain the same as those

observed on the thinner plate at the same incidence angle. This plate is

1.9k, thick.

At angles of incidence where both shear and compressional waves

are possible, the ray trace paths are not readily singled out. Radiated

wavefronts made up of many different combinations of internal

reflections may occur. For sound incident, water to aluminum, at
342 CHAPTER 20

Fig. 15 - A time sequence of schlieren photographs showing the effect of

illuminating a 0.241-cm-thick plate with an incoming acoustic pulse at 0, =

26.50. Pulse length is 5 usec, frequency is 5 MHz.

angles within a few degrees less than the longitudinal critical angle, a

guided wave consisting of an equal number of shear and longitudinal

propagation paths in the plate appears probable from a consideration of

the reflection coefficient curves (14-161. Such a path predicts, to within

experimental error, the group velocity and position of the effects isolated in Figs. 4, 9(a), 10, 11, and 12, which have the same group velocity but are not uniquely determined by a single Lamb mode. The

other wavefront seen in the region of incidence near the longitudinal

critical angle [Figs. 5 and 9(c), entries 3 and 6, Table 21 would have to

be explained by a combination of internal reflections consisting of more

shear than longitudinal paths. A path consisting of three times as many

shear paths as longitudinal paths is needed to give the observed group

velocity. There is, however, no obvious way of predicting a priori such

a path description. The result is not, however, inconsistent with a path

trace explanation. A propagation path consisting of all shear reflections, for the effect generated at 8.0* (Fig. 6), predicts a group velocity

V - 0.9 x 105 cm/sec and a wavefront angle O, - 64.7. This compares favorably with the results seen in Table I (entry No. 1).

COMPARISONS WITH CYLINDRICAL SHELLS

Comparisons between the plate result and results on cylindrical

shells [1l will be limited to shells with b/a - 0.9 for reasons discussed

below. A refracted ray leaving one surface in a plate will strike the

other surface at the refracted angle. For a cylindrical shell this is not
HEADWAVES ON PLATES 343

true, and in fact a refracted ray can miss the inner surface entirely, taking chord paths from outer surface to outer surface. Radiation from

energy taking chord paths predominates for shells thicker than those

with an inner to outer radius ratio, b/a, of 0.9 [11.

A path trace analysis was strongly suggested by the reflection

coefficient curves for a water-aluminum boundary in two regions of

incidence on aluminum plates. The group velocity of the wavefronts

generated near the shear critical angle were described by a path involving only shear reflections (called S-S), and the faster wavefronts generated near the longitudinal critical angle had group velocities described

by a path consisting of an equal number of shear and longitudinal segments (called L-S). The L-S and S-S path descriptions also described

the two radiated wavefronts observed on cylindrical shells [1], which

were also generated near the longitudinal and shear critical angles. Of

the two wavefronts observed on cylindrical shells, only the wavefront

generated near the longitudinal critical angle persisted to shell

thicknesses much smaller than a wavelength, and the L-S path continued to describe the position and group velocity of the observation.

This is in agreement with the results on very thin plates presented in

Figs. 11 and 12, where only one wavefront persists.

A bent plate with both flat and curved portions was used to determine the persistence of the wavefronts generated on plates when they

encountered a curved surface. The effects seen in Figs. 4, 5, and 7

were generated on the flat portion of a bent plate of the same thickness.

The persistence of the wavefronts around the curved portion of the

plate is seen in Fig. 16. Figure 16(a) is included for reference to show

the size and approximate position of the incoming pulse. In Figs. 16(b)

and 16(c), respectively, the effects generated at incidence angles near

the longitudinal and shear critical angles, and described by L-S and S-S

paths, are observed after they have reached the curved portion of the

bent plate. Both are seen to have persisted strongly around the curvature. In Fig. 16(d), the second wavefront generated near the longitudinal critical angle (corresponding to entries 3 and 6, Table 2) is also seen

to persist around the curved surface, but is more highly attenuated as

seen in the fluid above the plate.

Neubauer and Dragonette [11 previously pointed out that the two

predominant guided wavefronts observed on cylindrical shells could not

be related to only the first symmetric or first asymmetric Lamb mode.

Comparisons between the experimental results on cylindrical shells and

the results presented here on plates indicate that the faster of the two

circumferential waves seen on shells corresponds to the plate effect

f-- - . -.
344 CHAPTER 20

Fig. 16 - Schlieren photographs showing the effect of plate curvature on the persistence of three of the waves generated on a 0.130-cm-thick plate. The pulse length

is 3 gsec, the frequency is 7.2 MHz. (a) An incident pulse, (b) The wavefront isolated at 12.0', (c) The wavefront maximized at 26.5°; (d) The wavefront isolated at

13.2'.

which jumps symmetric modes as thickness is increased, and the slower

circumferential wave corresponds most closely to the second asymmetric mode.

CONCLUSION

Radiated wavefronts were observed when aluminum plates were

illuminated at specific incidence angles by short acoustic pulses in

water. The effects least attenuated by radiation into the water occurred

at incidence angles within a few degrees less than the lotgitudinal and

shear critical angles. The propagation of these "headwaves" along a

plate has not been previously reported and would be difficult to

describe with hydrophone measurements alone. With a single schlieren
7---7---- -

HEADWAVES ON PLATES 345

photograph, it is possible to determine both the group velocity along

the plate of the effects generated, and the trace velocity, even when

these are quite different. The measured angle of inclination between

the radiated wavefronts and the plate surface was related to the group

velocity of the wavefront along the plate by a formula derived independently of the type of motion within the plate.

Comparisons between the incidence angles at which the effects

were generated and incidence angles calculated from Lamb theory for

unloaded plates showed good agreement for plates as thin as 0.58Xh.

Agreement between Lamb theory and experiment became poorer when

plate thickness was reduced below 0.3kp, suggesting that water loading

was becoming more significant. There are more Lamb modes theoretically possible than there were effects observed at any given plate thickness. Only those effects which persisted far enough along the plate so

that specular reflection was avoided were considered by this method.

Estimates of the attenuation following a procedure used by Worlton

indicated that the modes observed were those that would be least

attenuated by radiation into the water. Those effects on plates whose

position and velocity were describable by path tracing corresponded to

the two circumferential waves found on cylindrical shells of b/a > 0.9

[1]. The faster circumferential wave on cylindrical shells is similar to

the plate effect, generated near the longitudinal critical angle, which

jumps modes as thickness is increased. The slower cirumferential wave

on shells is most closely associated with the second asymmetric Lamb

mode, generated near the shear critical angle.

REFERENCES

1. W.G. Neubauer and L.R. Dragonette, J. Acoust. Soc. Am. 48,

1135-1149 (1970).

2. C.W. Horton, W.R. King, and K.J. Dierks, J. Acoust. Soc. Am.

34, 1929-1932 (1962).

3. K.J. Dierks, T.G. Goldsberry, and C.W. Horton, J. Acoust. Soc.

Am. 35, 59-64 (1963).

4. W.G. Neubauer, J. Acoust. Soc. Am. 44, 1150-1152 (1968).

5. W.G. Neubauer, J. Acoust. Soc. Am. 45, 1134-1144 (1969).

6. H. Lamb, Proc. Roy. Soc. (London) 93, 114-128 (1917).

Cl IIl l I N-
346 CHAPTER 20

7. D.C. Worlton, J. Appl. Phys. 32, 967-971 (1961).

8. D.C. Worlton, Hanford Atomic Products Operation Rep. No.

HW-60662 (1959).

9. W.G. Neubauer and L.R. Dragonette, J. Acoust. Soc. Am. 49,

410-411 (1971).

10. W.G. Neubauer and L.R. Dragonette, J. Acoust. Soc. Am. 36,

1685-1690 (1964).

11. T.N. Grisby and E.J. Tajchman, IRE Trans. UE-8, 26-33 (1961).

12. M.F.M. Osborne and S.D. Hart, J. Acoust. Soc. Am. 17, 1-18

(1945).

13. L. Brillouin, Wave Propagation and Group Velocity (Academic New

York, 1960), pp. 144-148.

14. W.G. Mayer, J. Appl. Phys. 34, 909-911 (1963).

15. W.G. Mayer, J. Appl. Phy. 34, 3286-3290 (1963).

16. R.E. Bunney, "Circumferential Waves on Cylinders," PhD dissertation, Dept. Phys., Colorado State Univ. (1968).
Chapter 21

SCHLIEREN VISUALIZATION TO

DETECT FLAWS ON PLATES*

INTRODUCTION

The use of Lamb [1) waves for nondestructive testing applications

has been the subject of numerous papers [2-51. Generally these applications involve the hydrophone reception of a Lamb mode or a Lamb

wave reflected from a flaw.

Surface waves may be launched on a plate immersed in a fluid by

illuminating the plate at specific angles of incidence with short acoustic

pulses. A wave thus launched propagates along the plate and reradiates

energy back into the fluid as it propagates. The reradiated energy

leaves the plate at an exit angle equal to the original angle of incidence.

This reradiated wave is called a "headwave," and the detection of a

"headwave," in a region beyond that where normal specular reflection

occurs, serves as an indicator for the presence of a surface wave.

Schlieren visualization of "headwaves" generated when thin aluminum plates in water were illuminated by short acoustic pulses were

presented in Chapter 19. "Headwaves" were observed when the plates

were insonified at specific angles of incidence, and their properties compared with predictions of Lamb theory. In addition, the position and

group velocities of the "headwaves" were described in terms of a path

trace technique. Those waves not strongly attenuated by radiation into

the water are of particular interest in nondestructive test applications.

Such waves are found to be generated by acoustic pulses incident within

20 less than the longitudinal and shear critical angles on plates made of

metals and glasses. The angle at which the "headwaves" are generated,

and their phase and group velocities, can be predicted by the known

elastic properties of the materials, and the distortion of these

"headwaves" when flaws are present in the material is easily observable

in real time by the schlieren method.

*Some of this material first appeared in "Materials Evaluation," Oct. 1974.

347
348 CHAPTER 21

EXPERIMENT

The schlieren system used is fully described in Chapter 22. The

sound source used is a lead zirconium titanate transducer pulsed at a

center frequency of 6.6 MHz. The active face of the transducer has a

diameter of 0.75 in. The plate material in all of the photographs that

will be shown is 6061 aluminum, which has a compressional wave

speed of 6370 m/sec and a shear wave speed of 3136 m/sec. The longitudinal and shear critical angles are 13.5* and 28.20, respectively. The

plates are supported at each end by brass rods. Both the source and the

plates can be accurately rotated. The radiated sound field in the water

above and below the plates will be visible.

Figure 1 shows the "headwave" which results when an aluminum

(6061) plate 0.130-cm thick is illuminated by a 5-/As-long acoustic pulse

at an incident angle of 12.0'. At the center frequency of 6.6 MHz, the

plate thickness, d, is 1.5 Xp, where Ap is the compressional wavelength

in aluminum. The incident pulse is seen in Fig. 2(a), and the successive frames of Fig. I show the generation and propagation of a

"headwave" along the upper and lower surface of the plate at later

times. Separation in time between the frames is obtained by varying

the time delay between the triggering of the sound source and the light

source. In this manner the sound pulse and its effects can be frozen

and observed at any instant, from the time the sound pulse leaves the

transducer until it and its effects attenuate below the dynamic range of

the system. Observations are made in real time on a TV monitor, and

the photographs seen here were made from the TV screen. The wave

seen in Fig. 1 was generated within 2* less than the longitudinal critical

angle on aluminum plates as thin as d - 0.023 XP. The phase velocity,

V, of the plate mode seen in Fig. I is given by:

V -. V/sin Oi (1)

where 0, is the angle of incidence of the sound wave, and V,, is the

velocity of sound in water. The group velocity, Vg, of the wave is

given by [61

VgV. cos (0i - o)

sin 0, (2)

where 0, is the angle of intersection of the plate surface and

"headwave" as seen in the photograph. The group velocity of this

"headwave was measured to be Vs.= 4.5 x 10 cm/sec. This velocity

can be predicted by assuming that the observed radiation results from a

guided wave in the plate which takes an equal number of shear and

longitudinal paths between the plate surfaces. This type of guided wave
FLAWS IN PLATE 349

(a)

(b)

(c)

Fig. 1 - A time sequence showing the generation

and propagation of a "headwave" on a 0.130-cmthick plate when sound is incident near the longitudinal critical angle.

Fig. 2 - The distortion of the "headwave seen in Fig. 2 by

a 0.025-cm (deep and wide) scratch on the upper surface of

a 0.130-cm-thick plate.
350 CHAPTER 21

is consistent with reflection coefficient calculations on aluminum, and

with previously shown results on cylindrical shells [6]. This "headwave"

was generated on all aluminum samples studied. These samples ranged

in thickness from 0.023Kp K d 4 2,p, and the "headwaves" had the

same properties on all the samples, i.e., they were generated within 20

less than the longitudinal critical angle. Their group velocity was

Vg - 4.5 x 10 cm/sec, and their explanation was consistent with a

guided path consisting of an equal number of shear and longitudinal

legs between the plate surfaces. The strong generation of a guided

wave near the longitudinal critical angle is not limited to aluminum.

Such waves were generated on samples of nickel, armco iron, inconel,

stainless steel, and other metals and glasses.

Figure 2 shows the distortion which has occurred in a "headwave"

generated under conditions similar to those in Fig. 1, except that the

plate had a 0.025 cm (deep and wide) flaw on its upper surface. The

radiated wavefront has proceeded past the position of the flaw, and the

distortion the wavefront is easily observed. The distortion obtained

from a scratch on the bottom surface of a 0.050-cm-thick aluminum

plate is seen in Fig. 3. The flaw is 0.012-cm deep and wide. The plate thickness here is d = 0.6Kp, and the "headwave" has the same properties and is generated within 1.5* of the longitudinal critical angle under

the same conditions (except for plate thickness) as that seen in Figs. 1

and 2. The distortion of the wavefront begins when the flaw is initially

encountered in frame (4c) and clearly identifies the presence of the

flaw.

The plate seen in Fig. 4 is 0.130-cm thick and has a 0.040-cmdiam hole drilled 0.254-cm deep into its side between its surfaces. This

size hole was chosen since this was the smallest drill easily available.

The encounter of the "headwave," seen in Figs. 1-3, with a flaw

between the plate surfaces is seen in Fig. 4 to distort the "headwave" in

a similar manner.

Within 2* less than the shear critical angle, a strongly persistent

"headwave" with a group velocity V. = 3.0 x 105 cm/sec was generated

on aluminum plates as thin as d - 1.0KP, and was not observed on

thinner plates. This wave was also generated on the other metals previ- ously mentioned and glasses. The group velocity observed can be

predicted by assuming a guided wave consisting of multiple shear

reflections between the surfaces. The generation of this "shear

headwave" on a 0.130-cm plate is seen in Fig. 5. Thd interactions of

the wavefront, seen in Fig. 5, with flaws are similar to the effects seen

in Figs. 2-4.
F

FLAWS IN PLATE 351

(a) (a)

(b) (b)

(c) (c)

(d) (d)

- i(e)

Fig. 3 - The distortion of a Fig. 4 - The interaction of a

"headwave" by a 0.012-cm (deep and "headwave" with a flaw between the

wide) scratch on the bottom surface of surfaces of a 0.130-cm-thick plate.

a 0.051-cm-thick plate.

I
352 CHAPTER 21

(a)

Fig. 5 - The generation of a

"headwave" by a sound pulse incident (b)

near the shear critical angle on a 0.130-

cm-thick plate.

(c)

When a flaw is contained within the incident sound beam, detection by schlieren visualization is immediately obvious. Figure 6 shows

the illumination of a 0.005-cm-deep scratch on the top surface of a

0.130-cm-thick plate. The angle of incidence is 27.50. Frame (a)

shows the incident pulse. In frame (b) the near shear angle "headwave"

has been generated, and its progression along the plate as well as its

reflection by the flaw in the opposite direction are observed. A semicircular pattern is seen in frames (c) and (d). This is caused by the flaw

which is at the center of the semicircle. With a flaw on the bottom surface of a plate, the semicircular pattern appears below the bottom surface, as seen in Fig. 7. Here the flaw is a 0.025-cm (deep and wide)

scratch on the bottom surface of a 0.130-cm-thick plate. The center

frequency is 7.2 MHz; the angle of incidence is 27.50. Frame (a) shows

the generation and reflection of the near shear angle "headwave," and

the subsequent frames show clearly the semicircular pattern formed
FLAWS IN PLATE 353

(a)

(b)

-- (a)

(c)

r , r* d (b)

(d) (c)

Fig. 6 - The observation of effects, Fig. 7 - An observation under condicaused by a surface flaw, contained tions similar to Fig. 7, with the flaw on

within the incident beam. The plate is the nonilluminated side of the plate.

0.130-cm thick the scratch is 0.025-cm

deep and wide.
354 CHAPTER 21

below the bottom plate surface. This effect is not limited to angles of

incidence at which "headwaves" are generated. The semicircular patterns seen in Figs. 6 and 7 are easily observed and separated from specular reflection at all angles of incidence from 5° through the shear critical angle, the only necessary criterion being that the flaw is contained

within the physical dimensions of the incident beam.

The distorted "headwave" seen in Fig. 8 is characteristic of results

obtained when a flaw between the surfaces of the plate is within the

dimensions of the incident beam. The flaw in this case is a 0.040-cmdiam hole drilled into the side of the plate and centered between the

plate surfaces.

Fig. 8 - The pattern obtained when a flaw

between the plate surfaces is contained within the

incident beam.

CONCLUSION

A schlieren visualization technique can be :sed to perform flaw

detection inspections on flat samples of various thicknesses.

"Headwaves" are generated by acoustic pulses incident at specific angles

on plates in water. Distortion of these "headwaves" in the presence of

flaws is directly observed by schlieren visualization.

The advantages of a schlieren technique are:

1. Detection of energy scattered by a flaw is not limited in

direction by a receiver. Scattered energy in all directions

iperpendicular to the light source can be accomplished with

the schlieren.

I

4
FLAWS IN PLATE 355

2. The existence of the flaw and its position along the surface

are visually observed in real time.

3. A flaw not large enough to generate a significant reflected

echo may still distort the radiated "headwave" and be

detectable.

REFERENCES

1. Horace Lamb, "On Waves in an Elastic Plate," Proceedings of the

Royal Society (London), 93, 114 (1917).

2. D.C. Worlton, J. Appl. Phys. 32, 967 (1961).

3. T.N. Grisby and E.J. Tajchmam, IRE Transactions on Ultrasonics

Engineering, UE-8, 26 (1961).

4. C.L. Frederick and D.C. Worton, Nondestructive Testing (Materials

Evaluation), 20, 51 (1962).

5. M. Luukkala and P. Merilinen, Ultrasonics, 11, 218 (1973).

6. W.G. Neubauer and L.R. Dragonette, J. Acoust. Soc. Am. 48, 1135

(1970).

4t

i . | I llll~mll Il i I I I ~ m roll i I in l m . l
Chapter 22

EXPERIMENTAL FACILITIES

INTRODUCTION

Throughout this volume various experimental measurements have

been presented; mostly acoustic field measurements of pulsed sine

waves in water and air, and schlieren visualization of pulsed acoustic

fields. In some cases, detailed experimental procedures were described,

but there has been only little physical description of the facilities suitable for performing the experiments. Of course, a variety of equipment

had been used over the period of almost 20 years that this work spans.

It would serve little purpose to describe old outdated equipment and

facilities, so mostly recent facilities will be described. In the case of

acoustic pool facilities the new ones are far more elaborate and capable

than earlier facilities because of the direct connection to a Computer.

However, this advanced system is also suitable for measurements that

were carried out in earlier simpler systems.

GENERAL CONSIDERATIONS

The measurement of acoustic reflection in a closed body of water

or an air enclosure, such as a tank or room, is always encumbered with

one ever present factor: reflection from the boundaries (tank walls and

water-air interface or room walls). Additionally, in a water volume,

difficulties, sometimes unexplainable, with electrical grounding seem

omnipresent. The latter becomes a constant enemy conspiring to

obscure measurement values. It often seemed that the greatest care

had been taken to eliminate ground loops one day so that signals were

adequately noise free and the next day a signal is found walking

through the trace on a bed of grass offering a new grounding challenge.

It has happened that a hydrophone immersed in a water tank was pick- ing up aircraft pilot conversations from the airport across the river.

The lesson, surely a common rule in other areas of physical measurement is - while the system is working well, take all the data possible.

Some experimenters have sought to make enclosures that absorb

the energy incident on them. Such anechoic treatments of boundaries

357
358 CHAPTER 22

are very common and quite successful in air. In water, in addition to

the significant expense, anechoic linings are relatively narrowband and

often have undesirable directional properties. Reflection measurements

such as those for spheres in Chapters 7 and 8 demand a very low electrical background for the signal. In that case uniform (plane wave)

insonification is required and a far field measurement is sought. To

achieve a reasonably plane incident wave, usually a large distance is

required between source and reflector, so that a hr diminution with

distance of acoustic intensity is suffered. On reflection, the same or a

similar distance is required resulting in an additional hlr diminution of

the intensity with range. The reflection results from the interception of

only a small portion of the source field. For a sphere and most other

double curved objects the absolute reflection is not large. A representative example will demonstrate the magnitude of signal involved. For

a sphere (see Chapter 7),

Pr P0 Iof. (1)

where po is the incident pressure at the position of the center of the

sphere. Take an average of unity for Ifr.c which is the high-frequency

rigid-sphere value. Consider a sphere with radius a, and impose a 10-

diam range r = WOO2a) for the receiver to satisfy far field conditions.

Now

Pr _ a 0.2,_2

P0 2 (20a) -005 2

or the reflection is 32 dB below the incident pressure level. It can be

seen that a 30 dB echo reduction tank-wall coating would cause serious

competition for reflection from a sphere. Of course, some reflectors

return larger signals and some smaller signals and even an elastic

sphere can return significantly higher signals at some frequencies.

However, to achieve the measurement of small magnitudes of If.. I

such as those in Fig. 4 of Chapter 7, every dB of receiver preamplifier

gain and reduction in background noise that can be garnered are

required.

It may seem that as far as the enclosure (tank, pool, etc.) is concerned, bigger is better. As a practical matter this turns out not to be

the case. With larger size several things become a great deal more

difficult. First, there is the matter of maintenance of the water volume.

That medium is ideally made to be of uniform temperature, still, quiet,

and constant in these and other properties. If optical sighting is

required, such as for distance measurements, optical clarity is also

necessary. Of course, the frequency range and pulse lengths used in
FACILITIES 359

the experiments are a vital consideration, with larger volumes required

for lower frequencies. At all frequencies the shortest achievable pulse

lengths in water or the actual physical size of reflector are usually the

major limiting factors for tank size.

Estimates of tank size required for specific conditions result from

a complicated interaction of many parameters. Limitations of tank size

resulting from reflection interference from walls can be generalized to

some extent and used as a basis for judgment about tank size. When

source and receiver are equidistant from a reflecting surface, the

modification of the received, directly transmitted pulse by energy which

has suffered a reflection and is at least in part received simultaneously

with the direct pulse, may be predicted by geometrical considerations.

Consider a sound source located at S1 (Fig. 1) and a receiver at S2,

both at a distance d from the nearest reflecting surface, i.e., the waterair interface or a tank wall. The interference with the direct pulse takes

place when acoustic energy simultaneously arrives via the path S1 aS2

and the path S, bS2. This is the "Lloyd mirror effect." Let

(I= transit time of pulse via path S, BS2,

t2= transit time of pulse via path SlaS2, and

tP= duration of pulse.

To avoid interference,.

t 2 , + P

Since

t= rnc

and

t2 -2q/c, where q - .d2= 14

2 v/Id 2 + r

214 = r + ctp

or

r 2 d2 CtP J

ci,, 2

Thus r is the maximum separation of S, and S2 for a given depth and

pulse duration where no interference will occur.

For the case where S, and S2 are not equidistant from the nearest

surface, an interference relation can be derived in which the point of
360 CHAPTER 22

REFLECTING SURFACE

d Id

Fig. I - Geometry for reflection interference from side-wall with

relcinmust be experimentally determined. This becomes unnecessarily complicated and makes it advisable in most cases to maintain

both S, and S2 as close to equidistant as possible from a single surface

so that this distance is smaller than that to any other reflecting surface

parallel to it.

Reflection from the plane surface behind the source and normal

to the line between source and receiver can interfere with the directly

transmitted pulse at the receiver. To avoid this interference, the following relationship between the distance hi from the source to the surface should be maintained

2

Reflection from the surface behind the receiver is also capable of

interfering with the directly transmitted pulse; so the above relationship

should also be maikitained where ht is the distance from receiver to the

surface behind it. These considerations will be true for a perfectly

square pulse emitted by the source, reflected, and propagated in the

water and also for the case of measurement on a pulse at some specific

kfixed time along its length. To assume interference for no part of the

pulse, the effective pulse lengthening due to the finite bandwidth of the

source, medium, reflector, and receiver must be taken into account.

The number of cycles needed to allow the pulse that is introduced into
FACILITIES 361

the water to reach a desired percentage of its steady-state value can be

determined from the quality factor Q of the source. The degree of

interference between the direct and reflected pulse will of course be a

function of the relative magnitudes of energy emitted in the direction

of the source and reflector. For a highly directional source and

receiver, experimentation in an interference region may be permissible

to the extent that the reflected signal is smaller to a desired degree than

the directly received signal.

The transient pulse analysis of reflection data described in Chapter

8 affords a significant advantage over long pulse experiments by using a

pulse as short as possible to maximize bandwidth and maximize also

the free volume in which interference-free measurements can be made.

TANKS AND POOL DESCRIPTIONS

One tank of a size that has proven useful had a capacity of

approximately 250 gals which was built to the normal standards of a fish

aquarium of that size. Its dimensions were 76 cm x 183 cm x 76 cm

deep. A frame was placed on top of the tank which allowed the positioning of acoustic sources, receivers, and reflectors. A side view of

the tank showing the frame and the positioning of transducers and

cylinders in the water is shown in Fig. 2. Smaller glass-walled tanks

have been used for reflected-pulse hydrophone measurements and for

schlieren visualization experiments. The relatively small glass-sided

tank was used for measuring the circumferential-pulse sequences of

Chapter 3. No pulses reflecting from the side walls interfered with the

desired pulses in most cases. If and when such wall pulses did exist,

they were identified and avoided for very limited experimental configurations. The glass tank was too small to allow a sufficient distance

between the cylinder and hydrophone at all angles around the cylinder

so scattering-cross-section experiments were done in a larger tank. A

photograph of that larger tank is shown in Fig. 3. The tank has cypress

walls and bottom normally 7.6-cm thick and contained a water volume

152 cm x 305 cm x 152 cm deep. Location of acoustical elements in a

volume 1-in square by 10-cm deep in the center of the tank's volume

was accomplished optically with microalignment telescopes mounted at

right angles to each other on two adjacent sides of the tank (see Fig. 3)

and a vertical cathetometer that viewed through a porthole in one end

of the tank. Elements were placed in the tank by a system of tracks,

rails, and carriages mounted on top of the tank. The wooden walls of

the tank into which water soaked to a certain depth afforded some

reduction in acoustic reflection from them depending on the degree to

which water soaked into the wood. This occurred presumably because

of the gradual impedance change over the wood thickness.
362 CHAPTER 22

A.A

Fig. 2 -A side view of the glass-walled tank showing the framework

which supported the transducers and diffracting cyhnder.

Fig. 3 -A view of the wooden tank showing placement and

location equipment on top of the tank.
FACILITIES 363

A much larger wooden tank was also used. A photograph of the

work area showing the tank protruding above the deck around the tank

is seen in Fig. 4. This redwood tank is set in a hole in the ground and

has dimensions of 9-rn diam and 6-rn deep. Typical swimming-pool

water- maintenance techniques were used to keep the water pure.

Fig. 4 -The work area around the top of the 30-foot-diam redwood tank.

The pool of a pool-type research nuclear reactor became available

after the reactor was deactivated at NRL in 1970. The pool is approximately 8 m X 10 m X 6 mn deep at the shallowest point. A photograph

of the pool area at NRL is shown in Fig. 5. Sources, receivers, and

reflectors are suspended from bridges that span the pooi (see Fig. 6).

The bridges travel the length of the pool on rails that are flat and level

to better than 1 mmn but, are seldom moved during the course of an

experiment. The pool contains approximately 150,000 gal of filtered

and deionized water. The water purification system was originally

designed and used with the reactor to remove from the water moderator all particles capable of becoming radioactive. Therefore, the water

is very clear and has an extremely low electrical conductivity. This is

often a distinct advantage in that electrical leakage paths are minimized,

especially if transducers lose their watertight integrity. The purity of

the water also helps retardation of corrosion-.of immersed metals.

t The incident acoustic pulse is either measured directly or a replica of the incident pulse is obtained by the sphere calibratioh procedure
364 CHAPTER 22

Fig. 5 - The large pool used for acoustic reflection

and scattering measurments.

Fig 6 - The target suspension platform attached to one bridge

that spans the large acoustic pool.
FACILITIES 365

described in Chapter 16. Alignment and relative aspect of source,

receiver, and target center are usually, but not necessarily, fixed in a

horizontal plane. The target is suspended from a stepping motor

assembly so that it can be rotated about its center. Target aspect angle

is varied by the stepping motor which are controlled by the computer.

The resolution of the motor that rotates the target is 1/30th of 10.

Measurements of reflection amplitude vs target aspect are routinely

acquired in the pool at l/2* intervals over 3600. After the target has

been rotated by the stepping motor, a waiting period is imposed before

data collection to permit damping of any oscillation of the target in its

suspension. Care has been taken to design a suspension system that

minimizes such oscillations. The support members for targets are usually 0.076-mm (0.003-in.) -diam- tungsten wires.

Data collection is performed by the computer through a Biomation model 8100 transient recorder. The entire data collection system is

shown diagrammatically in Fig. 7. The computer-operation area is

shown in Fig. 8. The transient recorder digitizes incoming signals into

2's complement 8 bit data (7 bits plus sign). The computer can adjust

the input gain of the digitizer so that each return to be measured is

digitized with the maximum dynamic range possible for that particular

signal. The range of adjustment is from 0.05 volts to 50 volts for full

scale digitization giving a 40 dB range in which individual echoes can

vary and still be sampled with the same relative accuracy. The recorder

digitizes 2048 samples of the incoming signal at a rate set either by the

operator or the computer. Sampling rates may be varied from one sample every 10 sec to one sample every 10 nsec. The sampling rate used

depends on the frequency content and time duration of the signals.

The digitizing process begins on a trigger signal which can be delayed

by some integral number of sample intervals. This delay can be set by

the computer to optimally position the signal that will be recorded in

the time window. Once the signal has been stored in the recorder's

memory, the computer can read the data and store it on permanent

medium such as a disk Or magnetic tape for later processing.

AIR SYSTEM

An air acoustics system is shown schematically in Fig. 9. The

experimental technique in air is similar to that in water, but the

sources, receivers, and frequencies are different. The air acoustics system allows the simulation of rigid boundary conditions for the target.

For the measurements in Chapter 8, the acoustic system employed LTV

telectrostatic sources 5.08 cm and 15.24 cm in diam. The targets are

hung in a large room 9 m x 30 m x 15 m high. Variable length gated- sine-wave pulses were produced by using a Hewlett Packard model
366 CHAPTER 22

I

4

~C

-1I I)

)C

? U

nr, m m mmm m mm l mal m lm m lmm m . . . ..
t •-7------

FACILITIES 367

Fig. 8 - The computer used to control experimental conditions

and analyze the data for the large pool (Fig. 6).

SOURCE.

ACAUTrRA

Filg. 9 - Schematic diagam of an air-acousti€ system.

L

S L N TELEVISION

APRTR LESC M R
368 CHAPTER 22

214A pulse generator and a Sanders switch to gate the continuous sine

wave output of a Hewlett Packard model 5110 frequency synthesiser.

The gated-sine-wave output was amplified by a Krohn-Hite model DCA

50 or ENI model 240L amplifier whose output drives the electrostatic

speaker. The signal received by the 0.635-cm-diam microphone was filtered by a Krohn-Hite model 312 bandbass filter and amplified with a

Bruel and Kjaer type 2107 frequency analyzer. The microphone was

mounted on a stand and placed by hand at the desired aspect angle. A

positioning of -t2° is possible in the air measurements. The received

signal is amplified and analyzed by the same equipment as described for

the pool facility. The water pool and air facility were adjacent to each

other.

SCHLIEREN SYSTEM

On several occasions [1,21 motion pictures of schlieren visualizations of ultrasonic wave interactions with cylinders have been shown.

These movies demonstrated the great usefulness of the schlieren technique in identifying the individual components of the diffracted field of

the cylinders. In fact, such identification was not possible using the

standard pulse measurement technique of placing a hydrophone in the

shadow zone of the cylinder, and interpreting the received pulse train

displayed on an oscilloscope. The schlieren system that will be

described can be used to examine ultrasonic wave interactions with

bodies of various materials and shapes, with a size limitation of the

viewed field imposed by the diameters of the lenses used.

The great utility of the schlieren system and numerous queries

about it prompts the description of the system and its components and

the method of producing motion pictures with it. Although schlieren

visualization of sound waves is not a new procedure, practical information about it seems not to be readily available. The basic description of

the fundamental phenomena involved is described by others, (31 and

only a brief description will be given here. A diagram of the dark field

schlieren system is shown in Fig. 10.

The following description of the schlieren system used to visualize

sound waves would not satisfy the optical science purist, but is

sufficient for the augembly and operation of a schlieren system for that

purpose. A parallel light beam is established between the lenses B and

C in Fig. 10. This parallel light beam has all rays within it parallel to

each other. The system is a telecentric system that is generated by a

small stop at A, which approximates a point source. The parallel rays

are focused by lens C in the plane D in Fig. 10 to a point. A stop is
FACILITIES 369

Fig 10 - A photograph of the broad-band scbberen system.

placed at the focal point of the lens C that obliterates all of the parallel

rays between lens B and lens C. The purpose of the lens before the

aperture A is merely to focus the light source on the aperture. A

source of pulses in- oses a burst of sine waves on the acoustic source

which is placed so that the acoustic wave travels normal to the direction

of the parallel light rays in the light beam. A time delay is introduced

between the pulse source and the light source. At the delayed time the

stroboscope is driven to illuminate the acoustic pulse in the water tank.

In the absence of sound, no image is seen in the image plane. However, when the acoustic source is turned on, that acoustic source causes

deviation of some of the light rays in the parallel light bundle. After

these rays pans through the lens C they are caused to miss the stop in

the plane D in Fig. 10 and pass through a lens between the plane D and

the image plane. This lens focuses the image plane at the center of the

acoustic pulse. There results an image of the sound that has been emitted from the acoustic source. The stroboscope is timed so that the

pulse is illuminated at a time after it has propagated a desired distance

from the acoustic source or after it has interacted with the diffracting

field. This is done repetitively, faster than the flicker rate of the eye, tgiving an apparent fixed image. If the time delay between the sound

emission and light flah is continuously varied, for instance increased,

the acoustic pulse will appear to be emitted from the acoustic source

and will appear to propagate toward the diffracting cylinder at the rate

of time delay increase and subsequently will appear to iipteract with it

and cause the reradiated field. It is not necessary to use the exact
370 CHAPTER 22

instruments that are described here, as many of them have equivalents

and would produce the same result. The instruments are named here

as an example of ones that could be used to assemble a working

schlieren system.

A useful source of illumination for schlieren visualization is a

laser. A helium-neon laser can be used as a steady-state source or it

may be intensity modulated to some extent. Usually the on-to-off

amplitude ratio needed for good strobed schlieren pictures is high and

not achievable by direct internal modulation. A laser light source obviates the need for achromatic lenses in the schlieren system. However,

it puts greater demands on such elements as the glass walls of the water

tank. Because of the coherence of the laser much glass, especially thin

glass plates as well as plastics, cause the optical field to be striated and

nonuniform. Of course, any visible-light laser is usable for this purpose. An argon-iron laser has been used for experiments at NRL. The

laser beam was diverted by an acousto-optic modulator through an aperture. This aperture must be at a distance from the modulator large

enough so that no residual edge of the undeflected beam passes

through the aperture during the time that the light is off in the

schlieren system. It was found that an extinction ratio of about 10,000

to I was desirable, and at times necessary. The light modulator must

have a fast rise time and a pulse duration that is short compared to the

travel time of a particular point on the acoustic wave if the wave is to

appear stationary. In water, sound travels at approximately 1500

nsec/mm, so a pulse length of less than 100 nsec would be required.

In addition, if so called phase resolution of the acoustic wave or pulse is

desired, a light pulse short compared to the period of the highest acoustic frequency is needed. The period of a 5 MHz wave is 200 nsec. so a

pulse of approximately 20 nsec would be desirable.

A photograph of a broad-band light-source system is shown in

Fig. II. The light source is a General Radio Stroboslave (model

1539-A). Even with this relatively slowly switched light source, phase

resolution of I MHz was possible using it. The condensing lens system

focuses the source image, which is in this case a zenon arc sufficiently

bright that an aperture of 0.66-cm diam can be used at A (Fig. 10) to

limit the parallel light rays to a reasonably high-quality telecentric system between B and C. The zenon light source is L: broadband light

source and produces a schlieren picture with the full spectrum of colors.

In Fig. 9 the lens at B has a 12.7 cm diam with a 62.9-cm focal length. All lenses are high-quality achromatic lenses. The aperture A is at the

focus of the lens at B. The distance from B to C is 38 cm. The tank

between B and C is a stainless-steel-frame lass aquarium (40 cm x

I 26.6 cm x 25.4 cm deep). The original two largest sides of the tank

I mImlli m ml
FACILITIES 371

were replaced with 0.03-cm-thick plate glass in order to minimize the

distortion caused by bowing of the sides. Lens C is a pair of 12.7-cm

lenses that have 27.3-cm combined focal length. The stop at D, which

is at the focus of lens C, is a circle (0.076-cm diam) of black photographic masking tape stuck to the center of a slide-glass cover. India

ink stops (on slide-glass covers) can be used as well. Large black circles photographically reduced in size to the obtained diameter afford

the highest quality stops but can be costly. The cover glass is mounted

on an aluminum plate with a 2-cm hole in it. The stop is centered by

means of a microscope mechanical stage mounted vertically. A television camera (Fairchild model TC-177 with high-resolution RCA 8507A

vidicon) is mounted behind the stop. An f/0.95 lens is used on the

television camera but others are usable. The TV camera or any other

recording camera must be focused at the position in the tank where the

center of the acoustic pulse will appear. This is conveniently done with

a finely graduated scale placed in the plane where the acoustic pulse will

be but with external ambient light. Since the conventional television

scan rate is used, an ordinary television monitor can be used to view

the picture. Of course, at this point one may choose to record directly

from the television camera on a video tape recorder. To take advantage of the high-resolution vidicon, a higher-resolution monitor can be used,

as it was for the picture shown in Fig. 11. To make motion pictures, an

Auricon motion picture camera with a TVT shutter was placed 1 m in

front of the television screen. The camera is synchronized to the television scan rate by means of the 60 Hz power frequency. The movie

was taken at f/0.95 with Eastman Plus-X film with an ASA rating of

80. The television image was adjusted for brightness and contrast to

give the best directly viewed picture. Pictures are taken in a dark room.

The dynamic range of this system is 40 dB measured between maximum voltage across a 6.6-MHz transducer driven by an Arenberg

pulsed oscillator (model PG 650-C) and the voltage across the transducer to produce the minimum detectable wave on the '-ievision

screen. The transducer is a Branson type ZT lead-zirconate-titanate

disk in a 2-cm housing.

Motion of the acoustic pulse is introduced by changing the time

delay between the acoustic pulse output and the strobe synchronization

input (light pulse). Although an internal delay is available on the

Arenberg oscillator driver, an external pulse delay (Hewlett-Packard

214A) was used in most cases to ensure the time-delay range that was

from 0 to 100 usec. The entire apparatus was set up on a Gaertner 6-ft

optical bench. A similar system has also been set up on a smaller 4-ft

Cenco bench with 7.6-cm-diam lenses at B and C. Similar results were

obtained but adjustments were significantly more difficult. Careful initial alignment is important for optimum results. It is necessary to align
372 CHAPTER 22

Fig I I - Schiteren photograph of a 3-,u-wec

pukse reflected from a solid aluminum

cMinder and subsequent radiation

the cylinder elements in the same direction as that of the parallel light

bundle and then it is necessary to align the transducer acoustic beam

axis normal to both the cylinder elements and the parallel light bundle.

After the cylinder is aligned in the parallel light beam, the acoustic

pulse is observed as it is specularly reflected from the cylinder surface

so that is has maximum intensity on the television screen both on

incidence and after reflection. To get a schhieren photograph of a 5-

cm-long (Fig. 12) cylinder, without showing its supports, the cylinder

was glued with its elements perpendicular to a piece of 0.64-cm plate

glass with Eastman 910 adhesive. The plate glass was larger than the

1 2.7 -m lenses and was mounted vertically on a 0.64-cm thick horizontal

brass base. The entire structure was set on a tiltable table that was

placed in the bottom of the tank with adjustments accessible through

the water for both tilt and rotation.

REFERENCES

I. W. G. Neubauer and L. R. Dragonette. J. Acoust. Soc. Am. 47,

62(A) (1970).

2. W. G. Neubauer and L. R. Dragonette. J. Acoust. Soc. Am. 48,

101(A) (1970).

3. L. Bergmann and H. Hatlfield, Ultrasonics (Wiley, New York, 1938).
Chapter 23

SPEED OF SOUND IN FRESH WATER*

INTRODUCTION

In the course of free field measurements of the acoustic scattering

by finite bodies in water, it was felt that the measurement of a fundamental quantity would unearth systematic errors in the placement-andlocation equipment, electronics, and the measurement of ambient conditions. Toward this end, the measurement of the free field sound

speed was undertaken since methods of time, distance, and temperature

measurement were already available in the laboratory. In the experiment, a pulse was transmitted from a fixed source to a single receiver at

two different distances along the same radius from the source. Sound

speed was determined by measuring the time difference related to the

distance difference. The resulting values for sound speed proved to be

the most accurate that then existed. Soon after these results first

appeared, corroborating measurements by different methods were published [I I .

APPARATUS

The water medium was contained in the cypress tank described

earlier in Chapter 22. A diagram of the electronic apparatus is shown

in Fig. 1. The 100-kHz laboratory standard frequency was multiplied

first by 5 and then by 20, causing the resultant 500-kHz and 10-MHz

signals to be synchronized. The 10 MHz was scaled down by a factor of

l0 and triggered a pulse generator, which in turn triggered the

cathode-ray oscilloscope sweep each 10 msec. The pulse generator produced a 0.4-jAsec pulse that, after amplification and series tuning, was

applied to the acoustic source. The acoustic signal was received and

was amplified and displayed on one trace of a 4-trace oscilloscope. The

other three traces displayed on successive sweeps, the 100-, 500-kHz,

and 10-MHz signals, all synchronized with respect to each other. Since

each trace utilized the same electron gun and sweep circuitry in the

*Pari of these result first appeared in: W. G. Neubauer and 1.. R. Draaonette, J. Acoust.

Soc. Am. 36, 1685-1690 (1964).

*3 o i 373
374 CHAPTER 23

OSCILATOR [

IOOKC 50 1

SYNC

FSOURCE RECIVE TANK

Fig. I - Diagram of electronic apparatus

oscilloscope as well as the same external synchronization, no timing

error could be introduced. The pulse-generator jitter was unnoticeable.

The composite signal displayed on the oscilloscope is shown in Fig. 2.

The acoustic source and receiver were fixed on rails that spanned

the tank and moved on tracks that were mounted on opposite sides of

the tank. An accurately graduated bar was slung in a framework just

over the water surface. A vernier fastened to the receiver rod moved

along the scale graduations when the dist-ince be: *,een acoustic source

and receiver was changed. The distance between two positions of the

vernier on the scale represented the actual distance between two respective positions of the pressure-sensing element of the receiver, because

at each position the rail to which the receiver was clamped was leveled

along the line between source and receiver. This was accomplished

with a coincidence level rigidly clamped to the rod as close as possible

to the position on the raji where the 1.905-cm 13 -in.J receiver rod was

clamped. The level had a sensitivity of I sec, causing a predictable

uncertainty in the correspondence between actual distances between

two positions of the sensing element in the water and the respective

distance between two positions of the receiver as indicated on the scatc

above the water.
SOUND SPEED37

DELAYED EXPANDED 11~1 SWEEP (4,000 TIMES ...... EXPANSION OF PORTION

OF MAIN SWEEP

.1^ INDICATED)

(a)

IOOKC -. f~ MAIN SWEEP

S0O KC b -~

10 MC -

RECEIVED

ACOUSTIC SIGNAL

(b)

Fig. 2 -Four-trace oscillograph of 100, 500 kHz, 10 MHz and the

first peak of the acoustic signal. (a) Delayed sweep. (b) Main sweep.

It was necessary to darken with ink some of the 10-MHz peaks to

indicate their existence in (a), since they did not survive the photographic reproduction.

EXPERIMENTAL PROCEDURE

Interference of the pulse at the receiver was avoided in all measurements by time separation of the direct transmission and any reflection. The rail holding the receiver was clamped and leveled at a position along the measuring bar, establishing a distance (d,) between

source and receiver. At each setting of the receiver, the temperature

was read with a thermometer, which is described later. A time position

with respect to the 10-MHz signal was read on the oscilloscope and

identified relative to the 500- and 100-kl-z signals (see Fig. 2). The

time position of the signal was determined at the center of the

cathode-ray tube in order to minimize parallax errors in reading. At

each time determination, a photograph of the oscilloscope face was

taken. The photograph (Fig. 2) shows an expanded portion of the

Fentire sweep obtained by the use of a delayed sweep. The accuracy of
-_7---

376 CHAPTER 23

the delay time of the delayed sweep was not used for timing but only to

allow the convenient observation of the signals. The rail holding the

receiver was then moved to the other end of the bar without touching

the bar, and another distance (d,) was then established in the same way

as was d,. To ensure that the bar had not been accidentally moved

while the receiver was being moved, a graduation of the measuring bar

was viewed through a fixed telescope. To ensure that the receivercarrying rail moved along the tank and was maintained parallel at both

distances d, and di, the distances of the ends of the rail were measured

with respect to a fixed rail.

The delayed sweep on the oscilloscope was then moved with

respect to the same main sweep on which both signal positions were

displayed at different times. The amount that the delay was increased

or that the delayed sweep was moved along the main sweep was determined by counting the number of cycles of 100 kHz on the main

sweep. Each interval of l00-kHz peaks was subdivided by 500-kHz

peaks, each of which were subdivided by 10-MHz peaks. In this way,

the acoustic signal was located with reference to the three displayed frequencies. The time interval At thus established in terms of periods of

100 kHz, plus periods of 500 kHz, plus periods of 10 MHz, is related to

the distance difference (d, - d, Ad) or acoustic-signal path. Sound

speed is thus determined, since c = A/t

DISCUSSION

Since a transient signal is employed in the experiment, an exact

analysis of the measurement would necessitate the application of a

Fourier transform to find the frequency components of the pulse. The

inherent differences between the transmission speed of the components

and of the group would be in evidence. Despite the existence of such

differences in the interpretation of sound speed, on the basis of pulse

similarities, which are discussed, and the use of a difference method,

the results are interpreted in terms of a steady-state solution.

The elementary solution to the scalar wave equation most closely

corresponding to the conditions of the experiment that was done is that

of a radial wave propagating outward from a point source in continuous,

simple harmonic vibration, in an infinite homogeneous medium. Such

a pressure wave would propagate with a phase velocity c and would

diminish in amplitude as the reciprocal of the radial distance (r) from

the point source. The wavelength and phase velocity are related by

c - ftL where f is the frequency of vibration of the source and X is the

wavelength. The phase velocity can also be expressed as c - Ad/At,
SOUND SPEED 377

where A d is the radial distance that a wave is propagated and At is the

time taken for the propagation.

In practice, the conditions that are explicit are not satisfied exactly

nor are some that are implicit. Such an implicit condition is that of

detection of an acoustical quantity such as pressure at a true point

rather than a finite area or volume. No specific quantitative evaluation

of each condition is yet possible that will allow a direct relationship

between the quantity c expressed in the elementary solution to the

wave equation and the experimentally determined quantity c. Therefore, no exact numerical limits of uncertainty resulting from these

experimental conditions can be attributed to the measured sound speed.

Subjective judgment of each condition separately and an estimate of

their composite effect is the only recourse. In the experiment, the following conditions prevailed.

The acoustic field for some of the measurements was produced by

a 290-deg spherical cap 121 that approximates a 1 .27-cm-diam-spherical

source. The radial pressure pattern at the 230-kHz resonance of the

source has a central lobe of approximately 200 deg. Near the axis of

this lobe, the pressure at a constant radius is constant to approximately

:L-2% over an angle of about 30 deg. The broad, single-lobe pattern

implies that the cap was vibrating in its fundamental mode, which

approximates the vibration of a sphere to a desired degree if a

significantly small. region near the axis is considered. Since the pattern

is not truly spherical, a region near the source is present, even over

small angles, in which the pressure behavior is measurably different

from that of a true spherical radiator. This "near field" region has been

measured as extending no further than 5 cm from the center of the

source since, beyond this distance, the pressure has been measured as

decreasing with the inverse of the radial distance to ±t 2%.

In sound-speed determination, the pressure field was sampled

with an acoustic probe (21 that has as its sensing element a 0. 16-cm

ceramic disk. The acoustic probe has a radial pressure pattern with a

300-deg lobe, measured for reception, that approaches the spherical pattern characteristic of a true probe. The angle subtended at the source

by the largest sensing part of the probe at its nearest approach to the

source in these measurements was 0.46 deg. For the greatest distance

between source and receiver, an angle of 0.03 deg was subtended by

the probe. The active disk-element radius is approximately 0. 16 X,

whereas the probe housing is approximately 0.25 X, at the source

resonant frequency.
378 CHAPTER 23

If, because of the finite curvature of the wavefront at the

receiver, there was an inaccuracy of pulse location in time, it would be

due to the difference in onset at d, and d, of the pulse on the sensitive

element. This difference in onset would result in a slower rise time of

the received signal at d, than at dl. In such a case, an observer would

have made a time determination at d, that was later on the sweep than

it should have been and the At associated with the Ad determination

would have been too small, resulting in a velocity derived from the

experimental values, which would have been too large. That is, the

true sound speed would have been below that determined experimentally. No such difference in initial rise time of the pulse was noticed.

and therefore the influence of the effect was considered negligible.

However, had a rise-time effect been present below the voltage sensitivity of the equipment and, therefore, had existed and gone unnoticed,

disagreement between results, which are given, and confined-field

measurements would be greater rather than smaller. A noise limitation

would have resulted in an error of the location of the axis departure or

any other characteristic of the received signal. However during these

measurements noise was observed to be not significant; i.e., the effect

on time location was less than the accuracy of the time reading.

Free-field or free-wave conditions at a field point were satisfied to

a desired degree by using short pulses and causing time separation of

reception at a field point of the pulse from the source and all reflections. Also, a pulse repetition rate was used that was long enough so

that all reflections in the tank were below the noise level of the

receiver-system electronics at the time that the source was pulsed again.

The necessity for the use of pulses destroys the conformity of the

experimental conditions with those assumed in the elementary solution

to the wave equation in two respects. There is a lack of both a truly

continuous wave and a true line-spectrum sinusoid. The bandwidth of

the source causes a problem in determining sound speed since c - fX

and in the actual transmitted pulse no unique value of either f or X can

be measured. A range of frequencies is present at the beginning of the

pulse, where measurements are made. If, however, the medium in

which the pulse is transmitted can be considered nondispersive, i.e.,

c (f,X) is a constant, a unique value of c is subject to definition to the

degree that dispersion may be ignored. If dispersion were present, the

various frequency components of the pulse would be transmitted with

different speeds, and, at different distances from the source, a composite pressure resultant would be detected, and the pulse shape would

necessarily change. A similar effect would be present if the attenuation

of the medium over the entire frequency spectrum of the pulse could

not be ignored as negligible. Since the pulse shape would indicate the
SOUND SPEED 379

presence of both of these factors, pulse shape at the short transmission

was compared with that at the long transmission for each combination

of transducers used in the measurements.

The initial part of received acoustic pulse is shown in Fig. 3 as it

was received at distances d, and di. Within the experimenters' ability

to locate in time any characteristic part of this pulse, e.g., initial axis

departure or first or second peak, the two received pulses are indistinguishable from each other. Therefore, the resultant determination of

sound speed is no further in error than that error which is the result of

the time measurement itself. Pulse-shape comparisons were carried out

with different received pulse shapes as well as with different distances,

with identical results. Pulse-shaped comparisons were carried out with

0.63-cm-radius transducer disks and again the results were the same

even though such a disk has approximately a 25-deg central lobe in the

radial pressure pattern. Figure 4 shows the pulses that were compared

using these larger disks. In each situation used to determine sound

speed, the entire transmitted pulses were also compared by photographically superimposing them. Over the entire pulse, no detectable

differences were observed. What has been measured then can be concluded to be the speed of sound emitted by a true spherical source

detected with a true probe within the experimental accuracy at a frequency represented by f - ck, where f may be any frequency within

the relatively small bandwidth involved in the resonance of the source.

The resonant frequency of the 0.63-cm-radius disk is nominally I MHz.

(a)

(b)

Fig. 3 - Initial portion of received acoustic pulse at

(a) d. and at (b) di, using the spherical-cap source and

probe
380 CHAPTER 23

- Ft! 117

(b)

Fig. 4 - Received acoustic pulse using 0.63-cm-radius

* disk source and receiver at (a) d, and at (b) e11.

RESULTS

Figure 5 is a plot of measurements made with the acoustic sources

and receivers described. The measured values appear in Table 1.

Measurements were carried out by three experimenters independently.

As to whether this sound speed is what is normally called phase, signal,

or group velocity, it is all three to within the accuracy of the experimental results.

These results also represent the plane-wave sound speed since the

elementary plane-wave solution to the one-dimensional wave equation

has the same form as the spherical-wave case, disregarding the amplitude and allowing the distance to the source to be sufficiently large that

the spherical divergence or wavefront curvature is essentially zero over

the receiver sensing area. Again, the pulse comparison supports this

conclusion. Actually, in the experiments that were carried out, the limitations resulting from the rigid definitions of conditions of the simple

solutions were sufficiently overcome to result not only in a spherical

wave but also a plane wave. The plane wave could not be as good an

approximation As the spherical wave, but the difference was indistinguishable at the distances used and within the accuracy of the measured

values.
SOUND SPEED 381

1495

1494,

1493

1492

1491

1490

1489

1488

1487

1486

1485

1484

1483 .J U,

" 1482

0

cn 148 I

1480

-AVERAGE

1479- OF 5 DETERMINATIONS

1478

1477

1476

0

1475

1474 - THIS WORK

1473 GREENSPAN a TSCHIEGG

SBROOKS

1472 -

147116 17 18 19 20 21 22 23 24 25

TEMPERATURE ("C)

Fig. 5 - Measured values of sound speed vs temperature *
382 CHAPTER 23

Table I - Measured Values of Sound Speed vs Temperature

1O (°C) c(m/sec) I0(0C)1 c(m/sec)l O(0C)I c(m/sec)

16.62 1471.51 19.01 1479.22 20.42 1483.35

16.8a 1471.65 19.04 1479.36 20.42 1483.48

17.9a 1475.38 19.11 1479.47 20.51 1483.81

18.26 1476.81 19.14 1479.64 20.55 1483.95

18.31 1476.82 19.47 1480.85 20.56 1483.95

18.55 1477.61 19.51 1480.98 21.22 1486.02

18.59 1477.86 19.57 1481.04 21.97 1488.04

18.62 1477.88 19.63 1481.07 22.01 1488.11

18.64 1478.16 19.64 1481.19 22.30 1489.11

18.72 1478.31 19.75 1481.61 22.72 1490.25

18.72 1478.31 20.06 1482.51 22.98 1490.86

18.81 1478.58 20.07 1482.44 22.99 1490.99

18.81 1478.57 20.12 1482.63 23.07 1491.24

18.85 1478.65 20.18 1482.92 23.09 1491.19

18.85 1478.67 20.24 1482.94 23.09 1491.26

MEASUREMENT ACCURACY

In order to determine the total maximum error due to the uncertainties in the measurement of the independent variables, the total

differential of c(AdAtO) was computed [3]. Factors that were considered as contributing to the error in the determination of Ad were

bar-scale calibration, bar-material coefficient of thermal expansion, scale

and vernier reading error (parallax), source and receiver alignment, and

receiver rail leveling and parallelism. The total error in Ad caused by

these factors was +0.0075, -0.0067 cm. Since the maximum error in c

due to a 0.001-cm error in Ad is 0.015 m/sec, the resulting total maximum error in cis +0.113, -0.101 m/sec.

The laboratory standard frequency of 100 kHz was known to be

accurate and constant to better than one part in 109 , contributing no

signifi.,ant error in At. Since the 500-kHz and 10-MHz signals are

V derived directly from the 100-kHz signal, they also are accurate and

constant to one part in 109. If the scaling of the 10 MHz were in error,

causing a lack of a constant synchronization interval, only the 10-MHz

signal would appear synchronized while the signals on the three other

sweeps would appear to wander across the cathode-ray-tube face. The

only factor that was significant in the contribution to the error of a

determination of At was the experimenter's time-position reading error

I I I l gil nmm I iil Ham im
SOUND SPEED 383

on the scope face, relative to the standard frequencies displayed. The

experimenter could determine the position of the received acoustic signal at each distance position to the nearest peak or axis crossing of the

10-MHz signal. In measuring At, two such determinations were necessary and therefore the maximum error in At was ±0.025 jAsec. Since

the maximum error in c due to a 0 .01-jA sec error in At is 0.022 m/sec,

the maximum total error in c related to time reading was ±0.055

m/sec.

The temperature was read with a fast-response, 61-cm-long

bomb-type fuel calorimeter that was graduated at each 0.01°C between

the temperatures of 18° and 28°C. This thermometer was calibrated by

comparison with a similar thermometer that had been calibrated by the

National Bureau of Standards. The thermometer was always read with

the entire thermometer in the transmission path before and after each

pair of distance readings. Also, in each case the temperature was taken

with the bulb toward the receiver as well as toward the source. Soundspeed determinations were not made when a horizontal gradient as

great as 0.01°C was noticed. The partial derivative of c with respect to

0 was taken as 3.28 m/sec/°C from the report of Greenspan and

Tschiegg [41. Temperature was determined in the transmission path to

an accuracy of ±0.01°C, which uncertainty resulted in a maximum

error in c of :E 0.033 m/sec.

WATER PURITY

Greenspan and Tschiegg [41 report that "Several measurements

made on local tap water give results about 30 ppm higher than for distilled water." Thus, since the 1800-gal water tank used in the measurements reported here was filled with tap water in the same city in which

the difference was measured, an error of c of -0.045 m/sec was

assumed. The results do not represent a single body of water since the

temperature was changed by adding hot or cold water from the tap.

The maximum total error in c as a result of the sum of the errors

associated with all of the independent variables and the impurity of the

water was +0.20, -0.23 m/sec.

SOUND SPEED TABLES

m Sound speed in water plays a vital role in the computation and measurement of refiettion and scattering by bodies and surfaces. A

tabulation of sound speed as a function of temperature for pure water is

therefore given here. Confident values for sound speed for all temperatures within the originally measured range were derived by curve

fitting techniques. Forty-two values of sound speed were measured

1 l
384 CHAPTER 23

directly. To these points a best straight-line fit was made as well as a

quadratic fit.

Polynomial fits, in general, have the form

c- A, T".

n-O

To permit the confident extrapolation of our free field data to temperature values above and below the measured values, we assumed the constants for 0 K< n < 5 that were derived from confined-field soundspeed values measured over a temperature range 0°C to 100°C by

Greenspan and Tschiegg [4]. This assumption is justified on the basis

of the standard deviations derived for all fits of the data.

Within the temperature range (from 18.26 0

C to 23.09 0C) for a

straight-line fit of the measured free field values, Ao = 1.422522 x 103

and AI - 2.9824. The standard deviation o- for this data is 0.14 m/sec

calculated by the formula

M Y, - xd) - ._ N - (m + 1)

where xi - the experimental values of sound speed, y = the values

calculated from the straight-line fit, N - the number of data points,

and m - highest order term of the polynomial. A quadratic fit yielded:

A 0 - 1.398732 x 101

A I - 5.2905

A 2 - -5.5639 x 10- 2

with o- - 0.09 m/sec.

Using the values of Greenspan and Tschiegg [41:

AI - 5.03358

A 2 - -5.79506 x 10- 2

A 3 - 3.31636 x 10- 4

A 4 - -1.45262 x 10-6

As - 3.0449 x 10- 9 .

A new value of Ao was calculated at 1.402330 x 103. This means that

the shape of the Greenspan and Tschiegg curve was retained (see Ref.

4) for our free field data and a new ordinate intersection was derived in
SOUND SPEED 385

the plot of sound speed vs temperature at atmospheric pressure. A calculation of a for our data relative to the resulting curve yielded a value

of :t±0.10 m/sec.

Values of sound speed are given in Table 2 to the nearest 0.1

m/sec from 15.00°C to 30.99°C for temperature intervals of 0.01*C.

This range is intended to be useful for general laboratory use at atmospheric pressure especially in open tanks for free-wave conditions.

The probable error for all values of sound speed are not greater

than ±0.10 rn/sec. Experimental values for sound speed reported by

others [4-7] are within a o- of values for that temperature given in the

table. The values are plotted and joined by a solid line in Fig. 6.

Experimentally measured values are shown on the curve.

Similar values of sound speed expressed in ft/sec over a temperature range from 60.00°F to 91.99°F, for temperature intervals of 0.01°F

are given in Table 3. Here the probable error is not greater than ±0.32

ft/sec. As was done for the data of Table 2, an accompanying plot is

given in Fig. 7.

REFERENCES

I. H.J. McSkimin, J. Acoust. Soc. Am. 37, 325-328 (1965).

2. The characteristics of the source and probe have been described in

W.G. Neubauer, J. Acoust. Soc. Am. 34, 312-318 (1962).

3. A more detailed description of these factors is given in W.G. Neubauer and L.R. Dragonette, "Microacoustic System Analysis by the

Measurement of Free-Field Sound Speed," Naval Res. Lab. Report

No. 6113 (1964).

4. M. Greenspan and C.E. Tschiegg, J. Res. Natl. Bur. Std. (US) 59,

249-254 (1957).

5. R. Brooks, J. Acoust. Soc. Am. 32, 1422-1425 (1960).

6. V.A. Del Grosso, "The Velocity of Sound in Sea Water at Zero

Depth," Naval Res. Lab. Report No. 4002 (1952).

7. W.D. Wilson, J. Acoust. Soc. Am. 31, 1067-1072 (1959).

I -. n
386 CHAPTER 23

1540,

1500-

1490

Z1480-

1470

4 46 47 4B 49 20 21 22 23 24 25 26 27 28 29 30

WATER TEMPERATURE (C)

Fig. 6 - Sound speed in m/sec vs temperature

in degrees centigrade

5000-

. 4950

0

4900

4850

4800 1 J 1 1

60 70 so 90 100

WATER TEMPERATURE tIF)

Fig. 7 - Sound speed in f, sec vs temperature in

degrees Fahrenheit.
Table 2 - The Speed of Sound in Water in Meters/Sec as a Function of Temperature in Degrees Centig

23.00 to 26.99 

rr~P n 1 2 1 4 5 6 7 8 

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INDEX

absorbing cylinders, 139 eigen frequency, 73 absorbing spheres, 195, 199 elastic sphere in solid, 205 absorption, 56, 125, 140, 146, 148, elastic spheres, 161, 177

158, 159, 217, 220 facilities, 236

acoustic impedance, 22, 27 farfield, 139

air facility, 365 finite cylinders, 140

angular frequency, 2 finite plane, 5, 18 antisymmetric modes, 105, 108-115, finite wedge, 8 337 flaws in plates, 247

attenuation, 95-97, 151 flexural modes, 56, 92

attenuation measurements, 316, fluid filled shell, 129 321-323 form function, 56, 60, 102, 139, 177, background contribution, 262 178, 196, 208, 225, 238, 242, "background" integral, 40 259, 266

beam-displacement, 297, 307 "form-function", 163

bistatic form function, 140 Fourier theorem, 178

bistatic reflection, 143, 144, 146, Fourier transform, 60-62

275, 276 "Franz-type" waves, 36, 57, 70, 73, 75 boundary condition, 3, 127 Franz-type zeros, 39, 44

bounded beam, 297 Franz wave, 45, 47, 62, 71, 117

bulk properties determined from Franz wave speed, 65 reflection, 289 frequency of least reflection, 306, 309

calibration using a sphere, 279 Fresnel, I characteristic frequencies, 254 Fresnel integral, 19, 20, 23, 27, 29

circumferential waves, 35, 57, 58, 75, gaussian-shaped beam, 301 92, 98, 109, 117, 230 general shape, 11

circumferential wave speed, 50 graphical solution for finite plane, 7 compressional waves, 95 grounding, 357

creeping wave attenuation, 49, 51, 64 group velocity, 94, 99, 102, 332, 334, creeping wave paths, 185 339, 348 creeping waves ("Regge poles"), 35, 41, group wave-speed, 332, 334 43, 44, 47-51, 57, 62, 180, 257, 265, Hankel asymptotic form, 59, 259

268 headwave, 315, 331, 333, 347 curved plates, 342-344 Huygens' construction, 30-32 curved surface, II hydrophone experiments, 49-51

cylinder, 12, 40, 55, 59, 62, 142, 239, impedance, 290

258, 290 incident shear wave, 153 cylinder at high ka, 239 interference, 102

cylindrical cavity, 151 Kirchhoff approximation, 1, 12, 14, cylindrical shells, 125, 342 17, 21

differential scattering section, 37, 43, Lamb modes (table), 338

47,_207 Lamb theory, 336-339

397
398 INDEX

Lamb waves, 93, 105, 117, 325, resonance decay time, 257

336-339 resonance frequencies, 260

Lami constants, 36 resonance scattering from cylinders, 2,

lateral displacement beam, 299, 304 58

lateral waves, 75 resonance width, 261

layered cylinders, 125, 290 resonant modes, 167

leaky Rayleigh wave, 58, 75, 80 retarded potential, 17

lenses, 225 rigid sphere, 177, 180

longitudinal critical angle, 311, 325 rubber cylinders, 225

longitudinal mode, 56, 92 schileren, 12, 46-48, 140, 141, 247,

longitudinal waves, 36 325, 354, 367-372

material constants, 36, 102, 130, 146 Schoch displacement distance, 298, 304

153, 193, 208, 237, 305, shadow region, 47

mechanical impedance, 264 shear critical angle, 309, 325

modal impedance, 243 Shear wave speed, 315, 319, 320

mode conversion, 158, shells, 56, 91, 94, 104

monostatic reflection, 129 silicone rubber, 225

near-field measurements, 21, 24 slow waveguides, 225

nonstatic reflection, 2, solid cylinder, 130

normal mode, 35, 55, 56, 73, 100, Sommerfeld-Watson, 35, 38, 39, 57,

116, 126, 130, 242, 244 58, 63, 75

normal plane, 3, 5, 12 sound speed measurement, 373-383

partial-wave scattering, 257, 265, 267 sound speed tables, 383-396,

particle velocity, 4, source strength, 3,

phase jump, 273 specular reflection, 331

phase velocity, 105, 107, 337, 348 sphere, 12, 161, 238, 265, 281

plates, 325, 327 sphere as a standard reflector, 283

plates and shells, 342 sphere at high ka, 238

.potential scattering", 262 spheroid, 12, 183-185

quadric surfaces, 12 steady-state reflection, 161, 194, 280

quality factor (Q), 253 summation formula, 2, 9, 15

radiation frequencies, 254 surface wave radiation, 298

radiation impedance, 243, 264 symmetric mode, 105, 108-115, 337

Rayleigh angle, 297 temperature effect, 172

Rayleigh phase velocity, 315 test tank size, 359

Rayleigh radiation, 48 thin plate, 336

Rayleigh resonances, 272 tone burst, 194

Rayleigh series, 55, 57, 58 total impedance, 290

"Rayleigh-type waves, 36, 57, 73, 75 total scattering cross section, 151

Rayleigh-type zeros, 39, 45, 73 transient signals, 177

Rayleigh wave, 47, 76, 268, 297 transmission, 327

Rayleigh wave speed, 315 transverse waves, 36

rectangular piston, 28 velocity potential, 1-3, 12, 17

rectangular plane, 1 wave speed, 22, 56

reflection coefficient, 289 "whispering gallery" modes, 268

reflection reduction, 293 "whispering gallery" resonances, 272

"Regge poles", 269 ."whispering gallery" waves, 75

resonance, 72, 75, 80, 91, 257, 267, 283