I UNITED STATES IEPARTMENT OF JOMMERCE

>UBLICATION NBS TECHNICAL NOTE 651

SCATTERING-MATRIX DESCRIPTION AND

NEAR-FIELD MEASUREMENTS OF

ELECTROACOUSTIC TRANSDUCERS

QC

100

15155

K.65I

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SCATTERING-MATRIX DESCRIPTION AND

NEAR-FIELD MEASUREMENTS OF

ELECTROACOUSTIC TRANSDUCERS

Natpial Bureau oi Standards MAY 6 1974 D.M. Kerns

1 00

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS. Richard W. Roberts. Director Issued March 1974
National Bureau of Standards Technical Note 651

Nat. Hur. Stand. (I.S.). Tech,'Nole 651. 40 najjes jMarch 1974)

CODEN: NBTNAE

For -air b) ili< Superintendenl of Documents, I .S. GovernmenI Printing Office, Washington, !).<:. 20402

(Order bj >l> Catalog No. CI 3.46: 651). 80.50
CONTENTS

Page

ABSTRACT 1

1. INTRODUCTION 2

2. SCATTERING -MATRIX DESCRIPTION OF TRANSDUCERS 8

3. SCATTERING -MATRIX ANALYSIS OF COUPLED TRANSDUCERS-- 16

4. TRANSVERSE SCANNING AND DECONVOLUTION 2 3

5. RECEIVED SIGNAL AS A FUNCTION OF DISTANCE AND THE

EXTRAPOLATION TECHNIQUE 25

APPENDIX: EXPANSION OF bg(d)/a 30

REFERENCES 3 7

FIGURES

Figure 1. Transmission system -- schematic 4

Figure 2. Far electric-field amplitude determined from

near-field data for JPL microwave horn antenna 26

Figure 3. Character of data and extrapolation 28

Figure 4. (a) Domains of Q and Q' a

; (b) composition of

the vector p 33

in

SCATTERING -MATRIX DESCRIPTION AND NEAR-FIELD MEASUREMENTS

OF ELECTROACOUSTIC TRANSDUCERS

David M. Kerns

ABSTRACT

Recently developed and successfully applied

analytical techniques for the measurement of microwave antennas at reduced distances are "translated"

into corresponding techniques for the measurement

of electroacoustic transducers in fluids. The basic

theory is formulated in scattering-matrix form and

emphasizes the use of plane-wave spectra for the

representation of sound fields. This theory, in

contrast to those based on asymptotic description

of transducer characteristics, is suitable for the

formulation and solution of problems involving inter

actions at arbitrary distances. Two new techniques

(in particular) are described: One, utilizing deconvolution of planar scanning data, taken with a known transducer at distances d which may be much

2 less than the Rayleigh distance dD (= D /2A), provides a means of obtaining complete effective

directivity functions, corrected for the effects

of the measuring transducer . Applicability of a Xtwo- dimensional, spatial) sampling theorem and the

"fast Fourier transform" algorithm, which greatly

facilitate the necessary computations, is shown.

The second technique provides a means of extrapolating received signal as a function of distance

(observed with d ^ d D ) to obtain on-axis values of

K

effective directivity. Other possible applications

are indicated. These techniques rigorously utilize

observed output of non-ideal (but linear) measuring

transducers

Key words : Electroacoustic transducer measurement

techniques ; near-field measurement techniques;

scattering matrix description of electroacoustic

transducers
1. INTRODUCTION

Recently, a considerable amount of experimental,

computational, and theoretical work in the development of

new techniques for measurement of microwave antennas at reduced distances has been successfully completed [1-6]* at

the National Bureau of Standards (and the work is continuing)

It seems more than likely that analogous techniques may be

found useful in the measurement of electroacoust ic transducers. The main purpose of this paper is to "translate"

some of the new electromagnetic techniques into the corresponding acoustic techniques. This entails reformulation

of some of the scattering-matrix formulation that has been

used in the electromagnetic problems, and it is suggested

that this formulation may be found useful and fruitful in

acoustics --as it has been in electromagnetics. In this new

formulation, "antenna" is replaced by "transducer" and free

"space" is replaced by homogeneous, nonviscous, stationary

"fluid." All devices, both in the original theory and in

this application, must exhibit linear behavior (at least

from an external point of view), and small -amplitude

(linearized) equations of sound propagation are assumed.

The resulting theoretical structure is otherwise essentially

free of restrictive assumptions and built-in approximations.

(A similar statement applies in the electromagnetic case,

with the helpful feature there that Maxwell's equations are

not inherently non-linear.)

For fluids, the governing differential equations

(eqs 2-2) are appreciably simpler than Maxwell's equations

and consequently the scattering matrix formulation in the

acoustics case is appreciably simpler than in the electromagnetics case. On the other hand, the derivation of

^Figures in brackets indicate the literature references at

the end of this paper.
supporting theorems of appreciable generality, such as those

for reciprocity and adjoint reciprocity [13] , is of course

more complicated in the electroacoustics case than in the

purely electromagnetic case. However, the reciprocity relations can easily be stated and applied in the plane-wave

scattering-matrix formalism, and those needed are stated

and applied in this paper.

The new techniques seem likely to be of most interest

for transducers of moderate to high directivity. In discussing transducer separation distances required for measurement purposes, we assume (1) kD >> 1, where k = 2tt/A is

the magnitude of the pertinent propagation vector and D is

a representative transverse dimension of the transducer

aperture, and (2) applicability of the concept of the

2 Rayleigh distance, d

R

, defined as D /(2A). This type of

case appears to present the most severe distance requirements

We next give an introductory discussion of the two

measurement techniques mentioned in the abstract, both of

which enable accurate measurements to be made at reduced

distances

The first of these, the planar scan-deconvolution

technique (including application of a spatial sampling

theorem) was formulated for antennas by Kerns [1] . Experimental results have been given by Baird e_t a_l. [3] and by

Joy and Paris [7]

Thus, in figure 1, we suppose that the radiating

characteristic (i.e., the spectrum S,

n (K)*) of the transducer on the left is to be determined with a known measuring

*Later in this paper, it is shown how the radiating and receiving characteristics of transducers may respectively be

described in terms of plane-wave spectral functions such as

S 10 (K) and Sq,(K), where K is a variable specifying the

direction associated with each spectral component.
z=0 z=z z = z' z=d

Figure 1. Transmission system -- schematic.

transducer (i.e. one with S

ni (K) known) on the right, which

is used to scan the field of the transmitter in a transverse plane (z = const.) a short distance (d << d

R

) away

from the transmitter. If, as is required in this technique,

multiple reflections between the transducers are negligible,

the radiated spectrum is given by the Fourier transform of

the acoustic pressure in the measurement plane. What we may

call an ideal probe would measure this pressure directly,

even in the near field; an actual probe, however, yields a response given by a convolution integral involving both the

structure of the pressure distribution and the characteristics

of the probe itself. In § 4 we give a rigorous analytical

technique for obtaining the true spectrum, S,~(K), of the

unknown transducer, corrected for the effects of an arbitrary,

but known receiving transducer.

A knowledge of S,

n (K) enables one to find the pressure

field that would be produced by the transmitting transducer

in the absence of the receiving transducer. The pressure
distribution in a transverse plane z = z~, say, at any

distance beyond (and within limits, inside) the measurement

plane is given by a Fourier integral depending on S,

n (K)

and on z

? (eq 2-9b); in the limit of large distance, the

far-field (hence far-field pattern) is given almost directly

in terms of S,

n (K) (eq 2-12).

The second analytical technique to be described here is

identifiable as an extrapolation technique. This technique

was introduced for antennas by Wacker and Bowman, has been

described very briefly by Newell and Kerns [4], and, more

fully, in an experimentally oriented paper, by Newell

et al . [5] . The theory and numerical techniques were developed

by Wacker [6]. In the extrapolation technique, one observes

the received signal as a function of separation distance d

(fig. 1), and it is not necessary to assume that multiple

reflections are negligible. For a given axial alignment of

the transducers involved, the technique yields only on-axis

data and requires only on-axis characterization of the receiving transducer. It should permit measurements to be

made with d ^ d„ with accuracy equal to that attainable by

conventional means only with d >> d D

. In the above discussion of the near-field scanning and

the extrapolation techniques, we have assumed for definiteness

that an unknown transmitting transducer was to be measured

with a known receiving one. However (as is necessarily the

case) , what is basically measured is the product of two

factors, one (S-, n

) characteristic of the transmitting

transducer and one (S ni ) characteristic of the receiving

transducer. This means that data (obtained at reduced

distances) can be used in additional ways (which are

familiar in transducer or antenna measurements, or both):
The receiving characteristic may be determined if the transmitting characteristic is known; on-axis (K = 0) characteristics of two identical unknown transducers (subject to

a reciprocity or equivalent relation) can be determined;

three dissimilar unknown transducers can be calibrated,

provided at least one satisfies a reciprocity relation.

The transverse scanning technique offers still another

measurement possibility, which may well be new: Characteristics of two identical transducers obeying reciprocity

and possessing symmetry with respect to an axial plane

(e.g., y = 0) may be determined for all K (corresponding

to real directions with z > 0) from one complete transverse

scan. This last possibility is to some extent peculiar to

the acoustics case; the corresponding achievement using two

identical antennas requires higher symmetry. Equations for

the application of the planar scan-deconvolution technique

to the two- identical transducer measurement problem are

readily derivable with the aid of the formulation given in

this paper. The corresponding antenna measurement problem

is discussed in [2].

The problem of determining far-field characteristics

from near-field data, particularly for large underwater

sound transducers, has received a good deal of attention

in the literature. Apparently two main approaches have

evolved: (1) Far-field simulation by large phased arrays,

which produce (or respond to) a locally approximate planewave in their near-field regions [8, 9] -- Measurements

on devices placed in the substantially uniform portion of

such near fields are essentially direct. (2) Application

of Kirchhoff diffraction theory, using the free-space

Green's function [10, 11] -- This requires both pressure and
normal velocity data over the chosen measurement surface.

Although accuracies of ± 1 dB have been reported, higher

accuracy may be limited by the difficulty of obtaining or

approximating normal velocity data, as well as by inaccuracy

in pressure data yielded by non- ideal measuring transducers

in the near-field.

It should be observed that the use of a Green's function

satisfying homogeneous Dirichlet boundary conditions on the

chosen measurement surface (and vanishing properly at infinity)

would eliminate the need for normal velocity data and thus

solve this particular part of the measurement problem. For

sufficiently simple surfaces (e.g., cylindrical and spherical)

such Green's functions may be found in the form of modal expansions; in the particular case of a plane surface, a suitable Green's function may be constructed (in closed form)

as an antisymmetric combination of free-space Green's

functions

.

Apparently the plane -wave scattering-matrix approach

has not previously been formulated for acoustics applications.* The power of this approach is perhaps most evident

in the ease of formulation and the potential advantages of

the planar scanning-deconvolution technique (above and § 4)

.

This particular technique seems likely to be most applicable

when relatively complete pattern information is needed for

transducers of moderate to high directivity. The potential

advantages include: (1) The ability to make measurements

with separation distances much less than the Rayleigh distance

(or at any convenient distance). (2) The ability to use, and

correct for the effects of, a single non-ideal (but well

characterized) measuring transducer. (3) The applicability

of a (two-dimensional, spatial) sampling theorem and the

*This paper reproduces the substance of an unpublished NBS

report [24]. A talk on the subject was given by the author

at the 84th meeting of the Acoustical Society of America,

Miami Beach, Florida, 28 November - 1 December, 1972.
highly efficient "fast Fourier transform" algorithm in the

required computations.

2. SCATTERING -MATRIX DESCRIPTION OF TRANSDUCERS

Let us consider first the system shown schematically on

the left in figure 1. The transducer we wish to consider

relates phenomena occurring on the reference surfaces S

n and

z = z, . Connection to a (shielded) source or load by means

of electrical waveguide (such as coaxial transmission line)

is perhaps suggested by the figure, but other means of connection (e.g., by wire) are not excluded. The connection

might indeed be acoustic waveguide, but this case will not

be considered explicitly. Whatever the terminal variables

are, we wish to consider them expressed at least formally in

terms of traveling-wave components. In the electrical case,

phasor wave amplitudes a

n and bn for incident and emergent

waves at S

n may be defined in terms of voltage V" n and current

I

fl by the equations V" n = an + b~ and I n

= (a~ - b

n

)ri n

, where

n

n is characteristic admittance. (We remark that Vn and I n could be used directly, but the variables chosen are better

suited to the application of the scattering matrix concept.)

Assuming peak-value normalization, the net time-average

power input to the transducer at S

n is given by

p 7 VKI 2

- l b ol

2 )> v-v

where the vertical bars denote absolute values.

Propagation in the fluid medium surrounding the transducer

is governed by the equations

p Q

c V«u - loop,

Vp = ioop o u, (2-2)
where p is the mean fluid density, c is the characteristic

speed of propagation, u is the acoustic particle velocity, p

is the acoustic pressure, and exp(-icjt) time dependence has

been assumed.

We employ a coordinate system Oxyz with unit vectors

e , e , and e , and let k denote the propagation vector for —

x ' —

y ' —Z

' — r r 6

plane waves in the fluid surrounding the transducer, k will

be regarded as a function of its transverse components k , k (throughout this discussion "transverse" means perpendicular to the z-direction) ; the z-component of k is then

k = ± y, z ' '

2 2 2 2 2 2 2 where y

=

' (kv -k - k ) and k = go /c . The transverse

x y

J ' part r of k is denoted by K, so that K = k e + k e , and — ' _

— x—x y—y

Y = (k 2

- K

2 )^2

,

2 2 where y is taken positive for K < k and positive imaginary

2 2 for K > k . We note that a plane wave solution expfik^r)

of eqs (2-2) for p may be written in the form

(2-3)

iYZ iK'R

e ' e — —

,

where r_ = R + ze , and we also note that both propagated

(homogeneous) and evanescent (inhomogeneous) plane waves

are represented.

For the pressure p and for the z-component of the

particle velocity u , we introduce the plane -wave (Fourier

integral) representations

r \ 1 r T\^rv\ ^YZ r -,r-\ "iYZ n iK'R JTA p(r) = — / [b(K)e r + a(K)e T ] e dK, (2-4a)

2tt

u

z (l) = — / n(K) [b(K)e lYZ

- a(K)e" 1YZ

] e

1 ^'^ dK, ( 2 -4b)

2tt
where n (K) = y/( w P ) is the z -component of wave admittance

for plane waves in the medium. Here and in subsequent

expressions of this type, dK is an abbreviation for dk dk

and integration over the infinite k , k plane is to be

understood. The spectral density functions a(K) and b(K)

for incoming and outgoing waves, respectively, are defined

by eq (4); these are the modal terminal variables, analogous

to a

n and bn

, for incoming and outgoing plane waves, referred

to the plane S-. (z = 0) as a phase reference or "terminal"

surface. Explicit expressions for b(K) and a(K) may be

obtained by inversion of the preceding equations:

b(K) = - / (p + n _1

u ) e"

1 -'- dR (2-5a)

4tt Z

a(K) = - / (p - n _1

u ) e"

1 -'- dR (2-5b)

4tt z

Integrations of this type are to be taken over all values of

x and y in any suitable fixed plane z = const, (here z > z

n ) . -I

l

(n , which is independent of the space coordinates, is

written inside the integral sign merely for convenience.)

The spectral functions b(K) and a(K) are independent of z (although this may not be immediately apparent in eq (5)),

and their phases are automatically referred to the plane

z = 0.

In the absence of incident waves [a(K) = 0] , b(K) may

be related to p or to u ; in terms of p on z = z

1

,

b(K) = / p(R,z 1 ) e - - dR. (2-6)

2tt l

10
Again in the absence of incident waves, time -average power

radiated into z > is given by

P rad

= 2" ^

/ pu

z dR = ^- / n(K)|b(K)| 2 dK, (2-7)

where the superposed bar indicates the complex conjugate and,

as indicated by the notation below the integral sign,

evanescent waves are excluded from the K-integrat ion

The transducer scattering matrix is the detailed

expression of the linear behavior of the device under all

possible excitations by incident waves. That is, a set of

outgoing wave-amplitudes [b

n and b(K)] will be determined

linearly by a set of incident wave -amplitudes [a~ and a(K)],

and we write this in the form

b = S 00 a + f S 01 CK)a(KJ dK, (2-8a)

b(K) = S 1Q (K)a + / S 11 (K,K , )a(K') dK' , (2-8b)

thereby defining the scattering matrix for the transducer

considered. [It is convenient and seems appropriate to use

the term "matrix" here even though one must think of rows

and columns labeled both by discrete indices (0, 1) and by

indices (k , k ) having continuous ranges.] Evidently, the

functions S

ni and S-. n respectively embody the receiving properties and the radiating properties of the transducer; S

nr)

is

essentially an ordinary reflection coefficient expressing

mismatch at the transducer input, and S-. , is the kernel of

the transformation expressing the scattering (into z > 0)

of waves incident on the transducer from the right.*

^Equation (8) provides only a "one-side" description of a transducer. A complete plane -wave scattering-matrix description may be obtained by "enclosing" the transducer between

two reference planes and considering incident and emergent

spectra on both planes. However, in all the problems considered or mentioned in this paper, a one-side (right- or

left-side) description is all that is needed for any one

transducer.

11
Let us examine the definitions of S^, and S,

n (the

quantities that will receive the most attention) and restate

them somewhat more physically.

We note that, according to eq (5), an incident plane

wave with p = A exp (ik • r) /2tt corresponds to the delta- function

spectrum a(K') = A 6(k' - k )6(k' - k ); putting this spectrum

in eq (8a), we see that S

n ,(K) represents the received signal

b

n as a function of direction of incident plane waves,

normalized to A = 1

.

According to eq (8b), S, „ (K) represents the spectrum

tgoing waves u: Hence , from eq (6)

,

of outgoing waves under the conditions a(K) = , a„ = 1

.

10©

iyzi -iK-R

2-rra,

/ p(R,z

1

) e - - dR, (2-9a)

that is, S-, n (K) may be defined in terms of the Fourier trans

form of the pressure field obtaining on the plane z = z, in

the absence of incident waves, normalized to a» = 1. (Here

z, , and in the next paragraph z^ , represent any planes "in

front of" the considered transducer.)

If we multiply eq (2-9a) by exp(iyz-.), take the inverse

Fourier transform, and evaluate for z = z~ , we obtain

p(R,z

2

) = -° / [S 1Q (K)e

2tt

iyz' 2l iK»R Av ] e — — dK . (2-9b)

This exhibits p(R,z

?

) as a functional of S,

n (K) . (To evaluate

p(R,z

7

) by numerical calculation one would naturally use the

fast Fourier transform.)

It is an important convenience to have a more compact

formulation of eq (8) . We can secure this by introducing

function vectors and by making more use of matrix-algebraic

concepts. We first define the column matrices

V Vy ,b j

12
in which a and b may themselves be regarded as column

matrices corresponding to the functions a(K) and b(K).

That is to say, the elements of these column matrices are

labeled or indexed according to the values of K and have the

values a^K) and b (K) , respectively. The transformation from

the set of incident waves to the set of emergent waves is

now written

fb i

b

I J

fs s 1 ^00 ^01 a

o

•\ /v /s s 10 s 11 a

Here S

nf) , Sni , S,n

, and S

1

. may be thought of as point, row

column, and square submatrices , respectively, in a partioning of a grand matrix S representing the whole

transformation. On performing the implied matrix multiplication, we obtain

b S 00 a + S 01 a »

b " S 10 a + S ll a ' (2-10)

The various kinds of products, such as S

ni a and S^a,

appearing here are defined by comparison with eq (8)

Transducers may be classified as reciprocal, anti

reciprocal,* or non-reciprocal, in accordance with the

following definitions: In the reciprocal and the anti

reciprocal cases, the receiving characteristic and the

transmitting characteristic are subject to

n s Q1 (K) = ± n(K)S10 (-K) (2-lla)

*We use the term "antireciprocal" in extension of its use for

2-ports by McMillan [23]. An antireciprocal 2-port is essentially what is identified by the more recent term "gyrator."

13
with the upper and the lower sign, respectively.* In both

these cases the scattering function S, , will be subject to

Y K

S n(K,L) = y LS n(-L,-K), (2-llb)

where y v denotes y as a function of K. If these conditions

do not apply, we may call the transducer non-reciprocal.

[Eqs 11 (with the plus sign in eq 11a) are closely analogous

to the reciprocity relations in the plane-wave scattering

matrix description of antennas [12].]

Yaghjian [13] provides derivations of eqs (H) and a basic discussion of their relationship to the form of

reciprocity relations given in the classical papers of

Foldy and Primakoff [14, 15]. We mention here the following

implications of eqs (11): (1) Eq (11a) implies the electroacoustic reciprocity theorem discussed by Foldy and Primakoff

[14, eq 51] and others. (2) Eq (11a) implies eq (15) (below).

(3) Eqs (11) holding for each of two transducers in a transmission system imply the relation n nM = ± n

n M, ~ for the

elements of the scattering matrix of the system 2-port

(eq 3-10). Except for the possible occurrence of the

minus sign, this is the normal expression of reciprocity for

2-ports with lead characteristic impedances not necessarily

equal [16 ]

.

We note that p(r) is given asymptotically at large

distances r by

p(r) ^ - ik|cos 9 |S 10 (Rk/r)a e lkr /r ; C2-12)

*The occurrence of the argument -K in one side of eq (Ha)

means that the equation relates radiating and receiving charac

teristics in the line of a given propagation vector k (if an

outgoing spectral component has propagation vector k, the

corresponding incoming plane wave has propagation vector -k) .

14
the angle 6 introduced here is the polar angle of r relative to the z-axis. We have written Rk/r for the argument

of S-. n

; if we introduce spherical coordinates for r such

that x = r sin 6 cos <J> , y = r sin 9 sin <j> , z = r cos 9,

we see that S

1 n is expressed as a function of the angular

coordinates of r. Another useful asymptotic quantity, here called the

"effective directivity function" and denoted by D , is

defined as

n = power radiated per steradian as a function of direction

e (net power input to transducer) / (4tt)

(The adjective "effective" is used because in this definition

D is referred to input power, so that D is not merely

relative, but includes effects of internal power losses.)

With the aid of eqs (2) and (12) , D is found to be given

in terms of S

nr)

and S, « by

4^Y k

2 cos 2 9 |S 1n (Rk/r)| 2 D

e

(e,<fO = — — , (2-13)

n (i - Is 00 l 2

)

where Y n is the specific acoustic admittance ( = k/wp = 1/cp ) o o We emphasize that both internal power loss and back radiation

(that into z < 0) may occur. Such losses show up in

diminished magnitude of S,

n

; eq (13) remains valid and

applicable. Further, it can be shown that D is invariant

to the insertion or adjustment of a lossless ("2-port")

component at the transducer input.

The counterpart of D for a transducer functioning in

a receiving mode is the effective area a , defined as P./S,

where P. is the available power at the transducer terminals

and S is the power density per unit area in a plane wave

15
incident on the transducer from a given direction. In the

present framework one finds

a _(6,<D) =

U—^ ; • (2-14)

y n - s

2

* U i

I &

oo ' J

Here 6 and <J) are the direction angles of the direction from

which the incident wave comes. The quantity a has the same

invariance property as does D . In the event that the transducer is reciprocal or antireciprocal , in the sense of

eq (11a), then the relation

D

e

(6,<fr) = 4^a

e (9,(J))/X

2 (2-15)

holds. The analogous relation is well known in antenna

theory.

Thus far we have been concerned with the description

of a single transducer. In a sense, a single transducer

by itself can be of no interest. It is obviously important

to consider the interaction of two transducers coupled

through the acoustic medium, forming a "transmission system."

This is done in the next section, with emphasis upon interaction at arbitrary distances.

3. SCATTERING-MATRIX ANALYSIS OF

COUPLED TRANSDUCERS

We consider a transmission system consisting of a radiating system and a receiving system operating in a given free fluid.

For the description of the active terminal, we apply

eq (2-10) directly:

•\ /s b

o s oo

a

o + s oi a '

•s. /\ b = S 10 a

Q + S-^a. (3-1)

16
These equations are set up with reference to terminal surfaces S

n and S, , the latter being at z = in the coordinate

system Oxyz (fig. 1)

Using primes to distinguish quantities associated with

the passive terminal, we write

b = S 00 a + S 01 a '

b = S 1Q a + S

1;L a . (3-2)

For these equations, the terminal surfaces are S~ and S-, , the

latter being z = d in the coordinate system Oxyz. Note that

these equations describe receiving, transmitting, and

scattering characteristics for directions in the z < hemisphere, with the transducer in its desired orientation.

A minor extension of the definitions used in eq (1) is

implicit.

The third element of the transmission system under

consideration is the section of "waveguide" between the

terminal surfaces S, and S, . We need the matrix description

of this element, too; fortunately, under the simple conditions postulated for the transmission path, we can actually

write this matrix explicitly. We can arrive at the needed

relations by momentarily considering an axial translation

of the terminal surface S, from z = to z = d, , say. From

eq (2-4), the new spectral density functions for the outgoing

and incoming waves would be simply b(K)exp(iyd, ) and

a(K)exp(-iyd, )

, respectively. By taking d, = d and applying

the evident joining equations, we obtain

b(K)e iyd

= a'(K), (3-3)

a(K)e" lyd

= b'(K). (3-4)

17
These equations are intuitively evident once the underlying

conventions are clearly established. Now, with respect to

the transmission path as an element of the system, the set

of incident waves is represented by b and b' and the set of

emergent waves is represented by a and a'. Thus, from

eqs (3) and (4) , the matrix description of this element is

a

I (o T' (b

'

As [t lb 'J where the elements of T are

T(K;L) = 6(k - I) fi(k I ) e^®d

x x' v y yThis is equivalent to the two separate transformations

a = Tb ' ,

a' = Tb. (3-5)

We are now in a position to obtain a complete formal

solution for the behavior of the transmission system under

consideration. That is, we can obtain expressions for both

t b

n and bn

, valid at arbitrary distances and including the

effects of multiple reflections. (We can also formally

determine the field in the transmission path.) The procedure

involves only straightforward combination of eqs (1) , (2)

,

and (5), using rules of matrix algebra. An inspection of

the following intermediate results will make the final

results more intelligible. We assume that the receiving

transducer is terminated with a passive, reflectionless

load. Then we find first

/\. /\ f /\ a = Tb' = TS 11' = TS nTb. (3-6)

18
The operator TS-..T appearing here is the description of the

receiving system, as a passive scattering object, referred

to the reference plane S, of the transmitting transducer.

Since this operator recurs frequently, we denote it briefly

by R' . Substituting eq (6) in eq (1), we obtain

/\ S\ /N. b = S 10 a

Q + S 11 R'b, (3-7)

which (at least when written out more fully) is seen to be

/N. an integral equation determining b. (It may be identified

as an inhomogeneous integral equation of the second kind.)

The solution may be indicated formally by writing

b = (1 - S^n^S^ag. (3-8)

This gives us the spectrum of outgoing waves in the transmission path; it includes both the simple plane waves and

the evanescent waves. (a is now determined by eq 6, and

p(r) and u (j_) are determined by eq 2-4.) Of several "" Li available constructive procedures for the solution of the

basic integral equation, the Liouville -Neumann method of

successive substitutions seems most appropriate in the present

physical context. This leads to a representation of the

inverse operator in eq (8) in a series [analogous to the

geometric series for the algebraic expression 1/(1 - x) ]

the result is

b = [1 + S nR' + (S 11 R») 2 + ---]S

10

a

Q

. (3-9)

The successive terms in this series correspond to successive

round-trip multiple reflections between the transmitting

and the receiving transducers. Convergence must be

expected on physical grounds. The rapidity of convergence

19
depends upon the "smallness" of the product S.. ..R

1 , and it

worth noting that this product depends upon both S n n and

S, (as well as upon the distance between the transducers)

We complete this analysis by calculating the scattering

matrix of the "system 2-port," which has its terminals at

» S

n and Sn and is defined by the equations

b

Q = Mn

a + M

12 aQ,

i (3-10)

b = M

21 a + M

22 a 0'

(The properties of a transmission system are often conveniently embodied in this form) . Inasmuch as we have put

a~ = , solving for b

n /an and for bn /an yields directly

M

ll

= S 00

+ V' C1 " ^iiR'O'^io, (3-11)

M

21

= S^Td - S 11 R')" 1

S 1Q . (3-12)

A similar alternative solution with a

n = and a^. f yields

M22

= S 00

+ ^oxRCl " S^R)' 1^, (3-13)

M12

= hl^1 " S^R)" 1 ^, (3-14)

where R = TS,,T. (Formally identical equations hold in the

purely electromagnetic (antenna) context [17].)

Although the kernel in the basic integral equation

(and iterated in eq 9) may indeed be complicated, we emphasize that the above analysis is not completely formal. It

provides a basis for further useful analysis, some of which

is described in § 5 and in an appendix to this paper; complete analytical solutions can be obtained in interesting

special cases

20
In the measurement technique to described in § 4 , we

assume that the effects of reflections between transducers

have been minimized and may be neglected. (This would not

seem to be an unusual requirement. What may be unusual is

that the measurement technique to be described in § 5 provides a means of coping with proximity effects and with

multiple reflections between transducers.) When such reflec

tions are omitted, eqs (12) and (13) become 1VL, = S

ni TS,n and M22

= S

on , respectively. If the (passive) termination

on the receiving transducer has reflection coefficient I\ , we obtain from eq (10)

b = F,S 01 TS 10 a0»

• -i where F

f = (1 - r

T SnfJ is a circuit "reflection factor."

More explicitly, we have

t b iyd = a F' / S 01 (K)S 10 (K)e^ dK. (3-15a)

The integral appearing in this equation is here named the

(acoustic) "transmission integral" and is given a special

symbol

:

V(d) = / SQ

1

(K)S 10 (K)e lyd dK. (3-15b)

The product S

fil S, „ in the integrand we call the "spectral

coupling product." Equation (15b) is the acoustic analog

of the transmission integral employed, e.g., in [1] . (The

only formal difference is that in the electromagnetic case,

the coupling product is the scalar product of two 2-

component vectors, which contain polarization information.)

In order to exhibit some of the content of eqs (15a)

and (15b) (but not for the present applications) we note

that a well-known type of transmission formula, analogous

21
to the Friis transmission formula in antenna theory, can

be derived from the asymptotic form of eq (15a) . The

asymptotic form in question is

1 ' i Vd

b

Q * -2TrikF l S 01 (0)S 10 (0)a e 1KQ/d. (3-16)

If we now calculate the ratio of the available power at the

receiving transducer terminals to the net input power at

the transmitting transducer terminals, we obtain

P

A De C0)a' CO)

_L = _e e

^ (3-17)

P

Q

4TTd 2

where D (0) pertains to the transmitting transducer, cr'(0)

pertains to the receiving transducer, and we have used

eqs (2-13) and (2-14) . We mention one more important result, contained in

eq (3-11) (and, similarly, in eq 3-13) : The first iterated

integral in the Liouville -Neumann series for M, , is

iy„d , iyT d

$(d) = / S Q1 (K)e K dK / S 11 (K,L)S 10 (L)e L dL, (3-18)

K L

where yK denotes y as a function of K. This is called the

(acoustic) reflection integral. It is the simplest integral

involving a reflection or scattering process. It can be interpreted as a basic monostatic sonar equation, which, apart

from multiple reflections between target and transceiver, is

valid at arbitrary distances. Examples of the use of the

purely electromagnetic analog of eq (18) may be found in [12].

22
4. TRANSVERSE SCANNING AND DECONVOLUTIQN

We can now quite easily give the promised analytical

t basis for determining the coupling product S

n1 S in from

near-field transverse scanning data. The significance of

determining this product was indicated in § 1

.

Let the required relative transverse displacement

between the transmitting and the receiving transducer be

denoted byJ a transverse displacement r P = xe + ye of the — —x ' —

y

receiving transducer (fig. 1) . This requires modification

of eq (3-15a):by considering the phase k*r of the waves in

the spectrum incident upon the receiving transducer, one

readily finds that a phase factor exp(iK»R) must be supplied

in the integrand. Equation (3-15a) thus becomes

I)q(P) = a

Q

F' / e

1 ^^SQ 1

(K)S 10 (K)e lyd

dK. (4-1)

Note that the quantity b (P) is what is observed in the

measurement process; it may or may not be simply related to

the pressure p(r) at the point r = (P,d). The value of the

product S

01 S.n in the integrand is what is sought. Inasmuch

as eq (1) represents a Fourier integral transformation,

its inversion is immediate:

e ' , n

' , n-* -iK>P

s oiWs10 (K) = f h oW e d ^- C 4

" 2 ) 47T

2 F'a,

The inversion of eq (1) is appropriately and usefully termed

"deconvolution . " This is appropriate because both factors,

S

n

-. and S,

n

, can be interpreted as Fourier transforms of

certain physical fields. It is useful because it helps to

emphasize the distinction between eq (2) and the simpler

equation representing the definition of S,

n (eq 2-9a).

23
We note that eq (2) contains the factor b

Q (P)/an

; the

occurrence of this factor is a characteristic of the measurement methods being described. The full exploitation of

t eq (2) requires that b (P)/a

n be measured correctly in

magnitude and relative phase.

Clearly the utility of this result depends upon one's

ability to evaluate the transform of the empirically

observed b~(P). In the electromagnetics work, both a twodimensional form of sampling theorem [1] and least -squares

fitting have been successfully used [3] to evaluate the required

transform of b~(P) from data taken at the points of a rectangular lattice in the measurement plane. The application

of the sampling theorem has become the method of choice,

mainly because of the greater ease of computation, and will

be described very briefly here.

The essential requirement of the sampling theorem is

that the function to be sampled be representable as the

Fourier transform of a band-limited function. That b

n (P)

virtually fulfills this condition may be seen from eq (1)

a band-limit K

R somewhat greater than k, and a distance d,

may be chosen so that evanescent waves for all K > K„ are

D

assuredly virtually zero in the measurement plane (e.g.,

with Kg = 1.05k and d = 15X, attenuation at the band limit

is approximately 260 dB) . Band- limit ing within a smaller

spectral region may result from the behavior of the product

^01^—^10^—^ * n ^dividual cases. If we assign band limits

k, 2tt/ X, and k~ = 2tt/A~ for k and k , respectively,

a straightforward generalization of the usual one -dimens ional

theory [18] enables us to replace the integral in eq (2) by

a sum:

siiCeS10 (|) = e 1Y

, l VP^e-iii-Prs. (4-3)

4k,k

? F'an r,s

24
The vectors P = j T ^i-X

+ J

s ^2-v C^ith r,s, = ...-1,0,

1,2,. . .) define the measurement lattice, the quantities

» b

n (P ) are the (complex) values of probe output directly

observed at the points of the lattice, and the summation goes

over the points of the lattice. According to the sampling

theorem, eq (3) is mathematically exact; that is, if the

data [the t> n (P )] were complete and exact, the result would

be exact. The theorem requires an infinite sum, but in the

electromagnetic application we have found that not even all

values measurable above noise are needed. An important

feature of eq (3) is that the highly efficient algorithm

known as the "fast Fourier transform" is rigorously

applicable to evaluate the sum [19].

Our antenna measurement experience using this technique

indicates that using k., = k» = k is satisfactory as a ruleof-thumb when essentially complete pattern information is

desired. These band limits correspond to a A/2 spacing of

measurement -lattice points and mean that typically a relatively

large number of data are required. Nevertheless, for a measurement lattice comprising a "moderate" number of points,

the fast-Fourier-transform computation time is nominal.

For example, to obtain complex pattern data in 64 x 64 directions from a 64 x 64 input data array, the "in-core" computation

time is approximately 1.5 seconds on a CDC 3800 computer.

Figure 2 gives a qualitative idea of results obtained

in a particular case.

5. RECEIVED SIGNAL AS A FUNCTION OF DISTANCE

AND THE EXTRAPOLATION TECHNIQUE

The extrapolation technique was described qualitatively

in § 1 . This technique requires observation of b

n /a n as a function of transducer separation distance d, which is pret cisely defined by the choice of reference surfaces S-, and S,

25
Figure 2. Far elect ric- field amplitude determined from nearfield data for JPL microwave horn antenna.

Values in 64 x 64 directions specified by the angles

0, (J> are plotted above the points k = k sin 9 cos <J>

,

.A. k = k sin 9 sin cf> and connected by straight- line segments

associated with the respective transducers (fig. 1) . From

the scattering equations for the system two-port, we obtain

the expression

b

Q = a M

21 /(l - M

22 r L ),

which is a precise and complete version of eq (3-15a)

Expressions for the elements M-, and M

22 are given in

(5-1)

26
eqs (3-12) and (3-13). From these expressions, by a rather

lengthy process, one finds for eq (1) a series representation of the form

oo oo b ; - a

Q

F< I exp[i(2P

+ l)kd] j

- q>

U U p = d 2p + 1 q=0 Pq

where F

T is as defined in connection with eq (3-15a) . We

observe that the subseries of terms with a given p represents

the contribution of energy which has experienced 2p reflections (or made 2p + 1 transits) between transducers. In

particular, the series with p = involves no reflections

and is the expansion of the transmission integral (eq 3-15b)

y(d) = 2 1 (A 0Q

+ A

01 d

_1

+ A

Q2 d" 2 + ...). (5-3)

d

(Since this expansion is in powers of 1/d, it is of asymptotic

type; but since it is convergent under mild restrictions, it

is not merely asymptotic. The derivation and convergence of

this expansion are considered in an appendix). Further,

as shown in the appendix (or by comparison with eq 3-16)

,

we see that

A

0Q

= - 27rikS

(

!)1

(0)S 10 (0). (5-4)

Hence determination of the leading coefficient in eq (2)

is tantamount to the determination of the (on-axis) value

of the spectral coupling product. The basic idea of what

we may call the conventional measurement method is simply

to have d large enough to make other terms negligible

compared to the leading term of the series. The basic idea

t of the extrapolation technique is to observe b

n as a function of d and to fit this function with as many terms

of eq (2) as may be significant, and so to determine a good

value for A ~ in particular.

27
A very brief discussion of the application of this

basic idea in antenna measurements may be of interest and

ina> be indicativt for acoustic applications.

03

U.44

0.42

0.40

0.33

0.36

0.34

0.32

0.30

ST

\ ^— d = 00 VALUE

\ - \

\ \ \

\

\

EXTRAPOLATED

MEAN

\

0.350

0.345 I—

0.460 0.470

SEE INSET)

PYRAMIDAL HORN ANTENNAS

APERTURE: 15 cm x 12.5

16GHz, d

R

- 60 cm

0. 1 0. 0.3 0.4 0.5

/d, d - SEPARATION DISTANCE IN METERS

1 igure 3. Character of data and extrapolation

The extrapolation technique has permitted determination

of A

nn

(at a given and high level of precision) from measure

ments made at greatly reduced distances: d ^ d

R

rather

28
than d >> dD

. The general character of data and of the

extrapolation is indicated in figure 3. The inset enlargement shows a sample of the small oscillations of the received

signal. These oscillations are of approximate period A/2

and may be interpreted as due to multiple reflections. In

eq (2) , this is represented by interference between terms

with p = and smaller terms of higher order. In the fitting

process, the values of |b

n (d)d/an | 2 were averaged locally,

and the averaged curve was fitted with a 3 or 4 term polynomial

in 1/d. This procedure determines |A

nf) | (which ordinarily

would be sufficient in acoustic applications) ; the phase of

A „ was determined separately.

29
APPENDIX: EXPANSION OF b

Q

(d)/a

The main result in this appendix is the series expansion (in powers of 1/d) of the transmission integral, which

is the most important part of eq (5-2). The origin of the

complete expression of eq (5-2) will be considered very

briefly.

This work was stimulated by the work of Wacker [6] on

the corresponding purely electromagnetic problem, but the

procedure used here is substantially different from that

of Wacker.

For our purposes the mechanical equations of motion

for small vibrations in an electroacoustic transducer may

be presented briefly in the form

-iwpK u = V* [c : s] + fn m— L = — —

1

(A-l)

Here p is quiescent mass density, u is particle velocity,

c_ is the Hooke's tensor, £ is the strain tensor, and f, is

the total body force other than that due directly to £.

All these quantities are functions of position within the

region V of the transducer. The body force £- in general

includes piezoelectric, piezomagnetic, and Lorentz forces.

The first term on the right in eq (1) is more specifically

I i, j,k,£ -i 7~ ( c ijk£ s k^ » dx J 3

(A-2)

where s. is expressed in terms of u by

'k£

i

2w

fa^ 3u^

3x, dx,

(Our notation is coordinated with that of Yagh j ian [13] and

that of Primakoff and Foldy [15]. For details of the

equations of motion see especially Yaghjian.)

30
Our first task is to reduce the problem represented by

eq (1) formally to that of prescribed sources operating

in a homogeneous fluid and to obtain expressions for the

plane-wave spectra in the reduced problem. To this end we

add and subtract terms in eq (1) to bring it into the form

-iu)P o u - V-fc^s] = F, (A-3)

where p and c are respectively the density and the Hooke's

"tensor" for the ambient medium, and F is a total equivalent

body force given by

F = V«[(c - c ):s] + i(jo(p - p )u + f-, . — L ^= =?cr =J Vl m o^ — —1

The essential features of this rearrangement are that the

constants in the left-hand side are those of the ambient

medium and that F, though unknown in the region V, does

vanish outside V. Now the Hooke's tensor for the ambient

medium is equivalent to the scalar p c

2

, the reciprocal

of the compressibility of the fluid. Taking the divergence

of both sides of eq (3) and simplifying the elastic force

term on the left-hand side we obtain the scalar equation

V

2 V«up c2 -itop o V'U + - = Q, (A-4)

iw

where Q = V»F is the equivalent scalar source distribution.

Equation (4) represents essentially the desired formal reduction to a problem of prescribed sources in a homogeneous medium,

If we set p(r) = p c

2 V*u/ioo, eq (4) takes the simpler form

V

2 p

+ k

2 p = Q, (A-5)

where k

2 = oo 2 /c 2

. Furthermore, for points outside V, the

function p(r) is by eq (2-2) identical to the acoustic pressure p(r). We make essential use of this fact in a moment.

31
The solution of eq (5) is well known [20] to be

p(r) = L_ / QCl<) - =^— dr\. (A-6)

4tt V | r - r '

| Next, we need the plane -wave representation of the spherical

wave [21 ] in eq (6)

,

e lk| -~-

! = i_

f y

-l

e

i}L- (L-L'l dK, (A-7)

| r_ - r_'

| 2 tt where k = +y or -y according as z > z ' or z < z ' , and the

notation for the integral is that established in § 2 . Considering a finite region of integration V, taking z greater

than the maximum of z', substituting eq (7) in (6), and

interchanging the order of integrations, we obtain for p(r)

the representation

p(r) = — / y^e 1 -*- dK / QCr'je" 1 -*-' dr « , (A-8)

8tt

2 K V

wherein k = +y. By comparison with eq (2-4a), we see that

this expression contains the spectrum of p(r) for points to

the right of V.

Let us apply eq (8) to the transmitting transducer,

shown on the left in figure 1 and (even more schematically)

in figure 4. If the field is produced by excitation of the

transducer at its electrical terminals by an incident wave

of amplitude a

n and if we consider points with z > z, , then

the normalized spectrum is just what we have denoted by S,

n

:

S 10

(K) = -i / Q(r)e _i ^I dr. (A-9)

4Tra

n Y V

Here the integration goes over the finite region V, and

the variable of integration, r, is measured from 0.

32
TRANS. SYSTEM

Figure 4. (a) Domains of Q and Q

,a

; (b) composition of the

vector p.

In order to set up the transmission integral, we need

also a representation for the receiving characteristic of

the receiving transducer. We may obtain this readily and

with great generality from the transmitting characteristic

of the (hypothetical) transducer adjoint to the given receiving transducer, using the generalized or adjoint

reciprocity relation for electroacoustic transducers, given

by Yaghjian [13]. The relation in question reads

(A-10)

33
where the primes refer to the transducer on the right and

SJq refe

given by

S' refers to the adjoint of that transducer. Now, S,' is

s 10® = —^~

/ Q'V'je" 1 ^ '^ dr«, (A-ll)

4iTa y V

1 o where the integration goes over the finite volume V

1 and the

variable of integration, r ' , is measured from 0' (the point

(0,0,d)). Equation (11) is similar to eq (9) ; k = K_ - ye_

appears since here we are interested in the spectrum to the

left of the sources. Combining the last two equations, we

obtain

Sq

X (K) = ^

/ Q' a (r')e

+ i ^'

dr'. (A-12)

4 ^ a

o nowpo

We can now turn to the evaluation of the transmission

integral, eq (3-15b). We substitute the expressions from

eqs (9) and (12) into the transmission integral and arrange

the integrations as follows.

¥(d) = C, / / dr dr' (HrJQ'^r') / y^e 1 -'^ dK. (A-13)

V V K

z Here C-. absorbs all constant multipliers and £ = r

1 - r + de_

We further let p = /d 2 - 2d Ar cos a + (Ar) 2

, where

Ar = |r' - r| and a is defined in figure 4. By applying

eq (7), we find that the K-integral in eq (13) is proportional to

g(d) = e ikp

/p. (A-14)

The analytical properties of this expression as a function

of (complex) d are decisive for the behavior of ¥(d) itself.

g(d) has an essential singularity at °°, which we remove by

multiplication by exp(-ikd); it follows from the behavior of

34
p as a function of d that the product exp[ik(p - d)]/p is

analytic in the extended plane for |d| > Ar. This means

that the product is representable by a Laurent series containing only negative powers of d,

oo exp[ik(p - d)]/p = I f

n

d" n

, (A-15)

n = l

convergent & for d > Ar. The f are functions of r and r'; the

n — — ' first of these functions, which is of special interest, is

easily found to be

f

± = exp[ik(z' - z)]

.

(A-16)

To obtain the desired evaluation of the transmission integral,

we substitute eq (15) into (13) and integrate term-by- term.

The result is the expansion shown in eq (5-3). Provided that

d > (Ar) , (A-17)

where (Ar) v ; denotes the maximum value attainable by Ar as max ' r

1 and r range independently over their respective domains,

it can be shown that series in eq (15) will converge uniformly with respect to r and r' for all r in V and r' in V

.

Hence if |Q|

2 and |Q' | 2 are integrable in their respective

finite domains -- as is physically reasonable -- the termby-term integration can be justified. One may incidentally

verify that the contribution of f, to the series is indeed

the result shown in eq (5-4).

The discussion in this appendix thus far pertains

directly only to the transmission integral, the simplest

but most important term in the Liouville -Neumann series for

M?, . The procedure used may be extended to obtain expansions of the iterated integrals. The principal task is to

secure a suitable representation for the scattering functions.

( Cf the purely electromagnetic case [22, § 5].) Convergence

criteria are of the same form as eq (17).

35
The main features involved in obtaining eq [5-2) are

the following. We obtain the Liouville -Neumann series for

ML, and ML~ by expanding the inverse operator in the respective eqs [3-12) and [3-13) . We observe that the terms of ML,

each involve an odd number of transits of energy, the terms

of M

?? involve an even number, and find that the expansion of

an integral involving m transits of energy is of the form

imkd _-. _ ? [C n + C .d" 1 + C n d~ l + ...)• [A-18)

,m

v mO ml m2 J v J

Putting this information into eq [5-1), assuming M

7 ~rT "small"

[ordinarily |M~„r T

| will be much less than unity), expanding,

and rearranging ruthlessly, one finds that all resulting terms

represent odd numbers of transits of energy, as written in

eq [5-2) .

36
REFERENCES

[1] D. M. Kerns, "Correction of near-field antenna measurements made with an arbitrary but known measuring

antenna," Electronics Letters, 6_, pp. 346-347 (1970).

[2] D. M. Kerns, "New method of gain measurement using

two identical antennas," Electronics Letters, 6_,

pp. 348-349 (1970).

[3] R. C. Baird, A. C. Newell, P. F. Wacker, and

D. M. Kerns, "Recent experimental results in nearfield antenna measurements," Electronics Letters, 6_,

pp. 349-351 (1970).

[4] A. C. Newell and D. M. Kerns, "Determination of both

polarisation and power gain of antennas by a generalised 3-antenna measurement method," Electronics

Letters, 7, pp. 68-70 (1971).

[5] A.C. Newell, R.C. Baird, and P.F. Wacker, "Accurate

measurement of antenna gain and polarization at reduced

distances by an extrapolation technique," IEEE Trans.

on Antennas and Propagation, AP - 2

1

, No. 4, pp. 418-431

(July, 1973).

[6] P.F. Wacker, "Theory and numerical techniques for

accurate extrapolation of near-zone antenna and scattering measurements," unpublished NBS Report (April 1972).

[7] E.B. Joy and D.T. Paris, "Spatial sampling and filtering

in near-field measurements," IEEE Trans, on Antennas

and Propagation, AP - 2 , No. 3, pp. 253-261 (1972).

[8] W.J. Trott, "Underwater-sound transducer calibration

from near-field data," Journal of the Acoustical Society

of America, 36_, pp. 1557-1568 (1964).

[9] G. Pida, "Large nearf ield calibration array" (L)

,

Journal of the Acoustical Society of America, 49 , pp. 1683-1686, (1971).

[10] C.W. Horton and G.S. Innis, "Computation of far-field

radiation pattern from measurements made near the

source," Journal of the Acoustical Society of America,

33, pp. 877-880 (1961)

.

37
[11] D.D. Baker, "Determination of far-field characteristics

of large underwater sound transducers from near-field

measurements," Journal of the Acoustical Society of

America, 34, pp. 1737-1744 (1962).

[12] D.M. Kerns and E.S. Dayhoff, "Theory of diffraction in

microwave interferometry ," J. Res. NBS., 64B , No. 1,

pp. 1-13 (Jan. 1960). .

[13] A.D. Yaghjian, "Theory of adjoint reciprocity for

electroacoustic transducers," NBSIR 73-329 (Feb. 1974).

[14] L.L. Foldy and H. Primakoff, "A general theory of

passive linear electroacoustic transducers and the

electroacoustic reciprocity theorem. I," Journal of

the Acoustical Society of America, 1_7, pp. 109-120

(1945)

.

[15] H. Primakoff and L.L. Foldy, "A general theory of

passive linear electroacoustic transducers and the

electroacoustic reciprocity theorem. II," Journal of

the Acoustical Society of America, 19_, pp. 50-58 (1947).

[16] D.M. Kerns and R.W. Beatty, Basic Theory of Wave - guide Junctions and Introductory Microwave Network

Analysis , pT 4~~6 (Pergamon , 1967) . ~

[17] D.M. Kerns, contribution in "High frequency and microwave

field strength precision measurement seminar," unpublished

NBS Report (March 1966).

[18] D.P. Petersen and D. Middleton, "Sampling and

reconstruction of wave -number- limited functions in

N-dimensional Euclidean spaces," Information and Control, 5, pp. 279-323 (1962).

[19] N.M. Brenner, "Three FORTRAN programs that perform

the Cooley-Tukey Fourier transform," Tech. Note 1967-2,

Lincoln Lab., M.I.T., Lexington, Mass. (July 1967).

[20] H. Lamb, Hydrodynamics , p. 501 (Dover, 1945).

[21] L.M. Brekhovskikh, Waves in Layered Media (Trans, from

Russian by David Lieberman) , p. 239, eqn. 17 (Academic

Press, 1960).

38
[22] D.M. Kerns, "Plane-wave spectra and spectral coupling

products for arbitrary radiating and scattering

systems," unpublished NBS Report (March 1972).

[23] E. M. McMillan, "Violation of the reciprocity theorem

in linear passive electromechanical systems," Journal

of the Acoustical Society of America, 1_8, pp. 344-347

(1946).

[24] D.M. Kerns, "Scattering-matrix description and nearfield measurements of electroacoustic transducers,"

unpublished NBS Report (July 1972).

39
NBS-114A (REV. 7-73)

4. TITLE AND SUBTITLE

Scattering-Matrix Description and Near-Field Measurements

of Electroacoustic Transducers

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-651

2. Gov't Accession

No.

3. Recipient's Accession No.

5. Publication Date March 1974

6. Performing Organization Code!

7. AUTHOR(S)

David M. Kerns

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs

DEPARTMENT OF COMMERCE

Boulder, CO 80302

10. Project/Task/Work Unit No. 2725111

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as Item 9

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Recently developed and successfully applied analytical techniques for the

measurement of microwave antennas at reduced distances are "translated" into correspond

ing techniques for the measurement of electroacoustic transducers in fluids. The basic

theory is formulated in scattering-matrix form and emphasizes the use of plane-wave

spectra for the representation of sound fields. This theory, in contrast to those base

on asymptotic description of transducer characteristics, is suitable for the formulatic

and solution of problems involving interactions at arbitrary distances. Two new

techniques (in particular) are described: One, utilizing deconvolution of planar scann ing data, taken with a known transducer at distances d which may be much less than the

2 Rayleigh distance d (= D /2X) , provides a means of obtaining complete effective

R

directivity functions, corrected for the effects of the measuring transducer . Applicability of a (two-dimensional, spatial) sampling theorem and the "fast Fourier

transform" algorithm, which greatly facilitate the necessary computations, is shown. Thi second technique provides a means of extrapolating received signal as a function of

distance (observed with d 'v- d ) to obtain on-axis values of effective directivity. Othe

R

possible applications are indicated. These techniques rigorously utilize observed outp

of non-ideal (but linear) measuring transducers.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

Electroacoustic transducer measurement techniques; near-field measurement techniques;

scattering matrix description of electroacoustic transducers.

18. AVAILABILITY [xT Unlimited

_ For Official Distribution. Do Not Release to NTIS

~2 Order From Sup. of Doc, U.S. Government Printing. Office Washington, D.C. 20402, SD Cat. No. CI 3 4o:b51

~J Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

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40

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