NATIONAL BUREAU OF STANDARDS REPORT

2008

FUNDAMENTAL PROBLEMS

in the

MATHEMATICAL THEORY OF DIFFRACTION

(Steacfy State Processes)

by

V. D. Kupradze

Onti 1935

Translated from the Russian by Curtis D. Benster

October 6, 1952

Editors Gertrude Blanch

<50>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

ized research, development, and engineering in the field indicated by its title. A brief description

of the activities, and of the resultant reports and publications, appears on the inside of the back

cover of this report.

Electricity. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Applied Electricity. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State

Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy.

Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral Products.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Machine

Development.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

Ordnance Development. These three divisions are engaged in a broad program of research

Electromechanical Ordnance, and development in advanced ordnance. Activities include

Electronic Ordnance. basic and applied research, engineering, pilot production, field testing, and evaluation of a wide variety of ordnance materiel. Special skills and facilities of other

NBS divisions also contribute to this program. The activity is sponsored by the Department of

Defense.

Missile Development. Missile research and development: engineering, dynamics, intelligence,

instrumentation, evaluation. Combustion in jet engines. These activities are sponsored by the

Department of Defense.

© Office of Basic Instrumentation 9 Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

ms PROJECT ms REPORT

1101-11-5100 October 6, 1952 2008

FUNDAMENTAL PROBLEMS

in the

MATHEMATICAL THEORY OF DIFFRACTION*

(Steady State Processes)

by

V. D. Kupradze

Translated from the Russian by Curtis D. Benster

Editor : Gertrude Blanch

^Published 193£ in Moscow and Leningrad.

This translation was carried out under the auspices of the Department of Mathematics, University of California, Los Angeles. It was edited by Gertrude Blanch, National Bureau of Standards, Los Angeles.

The publication, repr

unless permission Is (

25, D, C, Such perm

cally prepared If tha

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

In part, Is prohibited

andards, Washington

port has been speclfiport for Its own use,

TRANSLATOR'S ACKNOWLEDGMENT

The translator has had the assistance of

Mr. Jacob Chaitkin, Instructor in Scientific

and Technical Russian, California Institute

of Technology, in construing a refractory but

crucial reading, and to him extends his appreciation.

Curtis D . Benster
..

'
TABLE OF CONTENTS

EDITOR’S PREFACE. v

CHAPTER I

J §

TriE FUNDAMENTAL EQUATIONS

1* Electromagnetic diffraction: statement of the problem.

Fundamental equations. .... 1

2. Diffraction of elastic waves --- statement of the

problem. Fundamental equations . 3

3. The Uniqueness Theorem. The Emission principle ..... 6

CHAPTER II

THE METHOD OF CURVILINEAR COORDINATES

THE SIMPLEST PROBLEMS

U. The diffraction of electromagnetic waves near a cylinder . • lij.

5. The diffraction of electromagnetic waves near a sphere. • .17

6. Diffraction of elastic waves near a clanped circular

cylinder ........... 2h

7. Diffraction near a circular cylinder with a given stress

at the boundary. 26

8. Diffraction near a cylindrical occlusion ....... 28

9. The diffraction of an immovable clamped elliptical

cylinder^ statement of the problem. The differential

equations. 30

10. Mathieu functions . 3U

11. Construction of the Mathieu functions 36

12. The asymptotic behavior of the proper values of the

Mathieu equations Ijl

13. Properties of the Mathieu functions U9

lU. The -Mathieu-Hankel functions 5>3

15. Expansion of the Mathieu-Hankel function 59
.

.

• •

... ,

.... .'
iv

69

75

90

§

16. Diffraction near an ellipse

17 • Proof of the uniqueness

18. Proof of the existence and solution of system (6U)

19. The diffraction of elastic waves near a segment of

finite length ... 101

CHAPTER III

METHODS OF CONSTRUCTION

20. General observations 103

21. Diffraction near a semi-infinite segment 108

22 . Diffraction near an angle 110

23. The method of parabolic coordinates 112

CHAPTER IV

THE INTEGRAL EQUATION METHOD

2l|. General remarks. Auxiliary formulas 116

2^. Generalized logarithmic potentials of a simple and of a double layer 118

26 . Fundamental boundary conditions and integral equations . . 12U

27 . Investigation of the existence and uniqueness of the

solutions .......... 128

28. Exceptional cases 136

29. The integral equations of the stationary theory of

elasticity. lljl

30. Reduction to a regular system . ll|6

31. Proof of the equivalence of systems (33) and (U3) .... lU8

32. Investigation of system (lj.3) l5l

33. Reduction to a regular Fredholm system l5U

EDITOR’S NOTES 157

ERRORS IN RUSSIAN TEXT l59
' ’

’
V

EDITOR’S PREFACE

Kupradze's text came to the attention of the writer during a

1 study of a pamphlet by M. D. Haskind in which Mathieu functions are

used to solve a problem in subsonic compressible flow. Haskind

states that Kupradze's expressions for the Mathieu-Hankel functions

2 are better than those of Bruno Sieger . Since specialists in the field

have heretofore considered Sieger's expansions the best available, the

writer began a search for Kupradze's text. No copy seemed to be avail3

able in this country, until Professor A. M. Ostrowski of Basle,

Switzerland, kindly loaned his personal copy to the National Bureau

of Standards . A photostat of the Russian original is now available

at the National Bureau of Standards, Los Angeles, and the translation

of the text by C. D. Benster is presented here.

The writer read critically the pages relating to Mathieu-function

theory — pages 3h through 69 of the translation. Misprints and errors

found in this section have been corrected in the translation. No

attempt was made to read the remainder of the text with comparable attention to detail, although obvious errors found by the translator,

the proofreaders, and the writer were corrected in the translation.

A list of errors discovered is given on pages 159-360 for the benefit

Haskind, M. D. Oscillations of a wing in a subsonic gas flow.

Translated from the Russian by the Graduate Division of Applied Mathematics, Brown University, Translation A9-T-22.

, 2Sieger, B. Die Beugung einer ebenen elektrischen Welle an einem ' Schirm von elliptischem Ouerschnitt, Ann. d. Physik, s. H,

27(1908), 626-66U. Diss. Wurzburg,

^After the translation was ready for distribution, it was discovered

that a copy of Kupradze's (Russian) text is available in the

Physical Sciences Library, Brown University, Providence, R. I.

vi

of those who may want to read the original Russian text. Readers who

have occasion to examine some sections of the text for accuracy of

detail are requested to communicate with the writer regarding any

other errors they discover.

Regarding Kupradze’s expansions of the Mathieu-Hankel functions —

the chief interest of the writer in this text — it is not at all

clear that his expansions are better or simpler than those of

Bruno Sieger. It is possible that there may be regions of the complex

plane where Kupradze’s expansions converge somewhat better than those

of Sieger — especially for the functions of high order, and relatively

small argument. Conversely it is believed that Sieger’s expansions

are superior over other regions. A further report on the relative

merits of the various expansions now known will be made after experimenting with them.

Kupradze claims that his expansions are valid over the entire complex C -plane, but that those of Sieger are not. This is incorrect.

Sieger’s expansions, too, converge for all finite values of , and

the proof is essentially the same as that given by Kupradze for his

expansions. In fact, for the functions of order zero, Kupradze’s

expansions reduce to Sieger’s, except for the normalization factor.

Kupradze’s functions suffer from one serious defect. He adopted

the normalization suggested by Whittaker^. Tb be specific let

^Whittaker, E. T. and G. N. Watson. A Course in Modem Analysis,

Chapt. XIX, Cambridge, The University Press; American edition, 19U6.
' I i ' s '
vii

be a solution of the even periodic Mathieu function of period 'it' and

order 2n. It is known that when is equal to zero, this periodic

solution reduces to a multiple of cos 2nx. For this reason Whittaker

(2-n)

suggested that the function be so normalized that A^ ' be equal to

unity for all orders n. Such a normalization can lead to serious

difficulty. For Goldstein''" proved in 1927 that every coefficient

for n and s different from zero, must pass through zero for at least

(2-n) one value of For such a value of q^, therefore, setting A^n

equal to unity means making all other coefficients of the series infinite'. It seems that although Kupradze was well acquainted with the

work of Ince on Mathieu functions, he did not know of Goldstein's

contribution

Kupradze' s expressions seem worse than Sieger's in the region

where the asymptotic expansions give good results . Consider, for example, large positive values of £ . An examination of his expression

for Z e

2 n

( % *G]_) equation (^l) on page 6? of the translation shows

that usually the term involving aj^n will be the dominant one in the

approximation to Ze2n

(^, ,q-^) <> In the region where this coefficient

is zero or close to it, adjacent terms of the expansion would have to

be considered. In contrast, the dominant term in the expansion for

large £ in Sieger's expansions is the first one, and the coefficient

associated with this term is always different from zero theoretically.

It would seem that there must be fairly large regions where Sieger's

expansions are not only simpler, but more rapidly convergent

‘'"Goldstein, S„ Mathieu Functions. Camb. Phil e Soc. Trans. 23

(1927) pp. 303-336.
.

' ' -V.

'

'
Vlll

Nevertheless the translator's contribution is very much worth-while,

if only to lay at rest unwarranted claims j and moreover, Kupradze's

expansions are different from Sieger's and from that viewpoint worth

having

The writer wishes to express appreciation for the unusually able

and intelligent work on the part of the translator . He studied doubtful constructions with a diligence that one does not ordinarily expect,

and he made every effort to present, in good English idiom, a faithful

translation of the Russian original. He also proofread part of the

final manuscript. Thanks are also due to Mr. Eugene Levin and

Mr. William Keating, who proofread part of the manuscript. Mr. Levin

noted some errors in the chapters he read, and those were corrected.

The list of credits would be incomplete without mentioning the careful

work of Vendla H. Gordanier and Betty Schoenberg, who typed the

manuscript.

Gertrude Blanch

October 6, 195>2
' ' . # V
CHAPTER I

THE FUNDAMENTAL EQUATIONS

§1 0 Electromagnetic diffraction: the problem stated . Fundamental equations . According to the Maxwell electromagnetic theory, a

light disturbance at a given point in space is describable by the pair

of vectors (£ and & , which are related by the equations

= - c rot <£

div = 0 $

£ = c rot §) 5

div ° 0 ,

( 1 )

where € and c~ are the dialectric constant and the conductivity of

the medium, and c is the velocity of light in a vacuum.

Taking (E and as pure periodic functions of the time:

C s S> =. 5J(He

ic0t

) (2)

equations (l) may be given the following forai:

c rot E = - iu)H c rot H = (iu)€ + <t~)E $

div E = 0 5 div H = 0 .

( 3 )

Eliminating E and H successively from these equations, we obtain for

the determination of these components the Helmholtz vibrational equations :

AE + k

2

! = 0

AH + k

2

H = 0

(U)
'
2

A being the Laplace operator and

,.2 . £ o) K

2

* e

<$)

In case o- = 0, k reduces to the ordinary wave number, which is connected with the wave length X by the equation:

k =- u>\/g~ _ 2if

(6)

We shall be mainly concerned below with the study of cases of reflection

and diffraction of electromagnetic disturbances where there is no dependence of the system On one qf the coordinates, for example z. In this

connection

No. 1. H

z = 0, E

z ^ 0, i.e., the vibrational process is polarized

perpendicularly to the Zr-axis. In this case equations (3) take the

form

2 \

Dj

iu> d E

H

C X

* 2>X

AE + k

2

E„ = 0

- = ihin

c y

’ (7)

No. 2. = 0, H ^ 0, i.e., the vibrational process is polarized

parallel to the z-axis. In this case equations (3) take the fom:

3H i 5 = - (i cJ € + o-)e ; 7>J c x

?

2>H , — - (ico 8 + cr)E ; 3x c J

AH + k H = 0 . z z

( 8 )

It is readily seen that the general case may be regarded as the result

of superimposing the two indicated independent and formally identical

cas es
:

- • ' :

•

...

'

i;.

'

'

' ’

.
3

In addition to the Helmholtz equations that the electric and magnetic vectors satisfy,, certain boundary conditions must be satisfied

on the surface (or on the contour) of the diffracting obstacle. In

the case to be qonsidered here,, these conditions, in conformity with

the Maxwell theory, are expressible in the continuity of the tangential

components of E and H.

Some other boundary conditions will be indicated below.

Lastly, if the mathematical solution of the boundary problem in

question is to be equivalent to a physical picture of the diffraction,

it is still necessary that the conditions for the solution's existence

and uniqueness be formulated. These conditions concern the character

at infinity of the solutions sought

.

In i 3 we shall dwell in detail on an investigation of this question.

S 2 . Diffraction of elastic waves — the problem stated . Fundamental equations

,

In accordance with the general theory of elasticity,

the displacement vector u = u(x, y) , in the plane problem, may be

represented in the form:

or

u = grad $ + rot Y 5 *^=(0, 0, Y) 5

u = 1L + M. ; u = M. - JJE

X ?X Vy 7 ?7 2*

(9)

0 and Y represent the wave potentials of the longitudinal and transverse vibrations respectively, propagated in the given homogeneous

medium with the constant velocities a and b < ^

here the following

wave equations hold:
O',

.

..

'

.

.

•

. I

'
h

(10)

In the simplest diffraction problem or in the simplest external boundary problem of the theory of elasticity, which may be called the

generalized external Neumann problem, the law of propagation of the

displacement u(x, y) has to be determined at an arbitrary point of the

plane, an excision or occlusion of arbitrary form being present in it

Tiiiich distorts the fundamental disturbance and generates additional

waves

.

At the contour of the obstacle values of the displacements, or

definite relations of dependence between them, are ordinarily assigned

j

the picture of the phenomenon as a whole, moreover, ought to be reconstructible in accordance with them.

Finally, for the existence and uniqueness of the solutions, it is

necessary to seek supplementary disturbances in the form of special

waves having the properties of disturbances propagating to infinity and

dying out there. If we call the diffracting contour ( ^ )

,

(the

analytical properties of the curve (^r) will be looked into in detail

below), the simplest problem may be formulated thus: Given on (T)s

3

Jtt)
.

-

£ f rff t, \

' / ^\X:

’

'
find 0 and at an arbitrary point outside of (T ) so that the supplementary disturbances arising from the meeting of the incident

disturbances and the obstacle .(T) have the form of phases which move

in a determinate way off to infinity and there die (the Emission

principle) .

In the boundary problem of the theory of elasticity just considered,

the components of the displacement vector were assigned at the contour

of the obstacle. In other problems relations connecting the components

of the stress tensor with each other are assigned.

The general expression for the stress tensor is given by the

formulas

s

xx

W

- f[a2 ^- + 2b

dx

2 a

2 Y

axdy + ( 2b ) d

2a -i dyc

by 2

a" 0 „ a

2 y <a 2 V

3x3 j 3x3y

g ]

^f(a2 -2b2 )^_ +

2 a

2

0 _ 2b

2 3

, T a

~ 2 axay

B x 2>y ^

( 11)

In spatial problems the formulas for elastic displacements have, as

is known, the following forms

u = grad 0 + rot Ip” $ Y = (A, 0) y

1 2

2

0 “W 'ft" a at

Y -4-^- , b 3t"

( 12 )

and for problems with stresses the basic equations are expressible

in the form?
*

*

.

;

. / §|

'

'

. /; , ’ ;

. - .

' If';.!

:

' /: >

' .. . ;

. - ,

'
6

_

" u - grad 0 + rot j t|T = (A, B, 0) 5

In all these cases the boundary problem is expressed at the diffracting contour or diffracting surface by the right members of formulas

( 11), ( 12 ), ( 13 ), or by definite combinations of them. The general

remarks made above concerning the behavior at infinity of the solutions sought must here also be repeated.

§3. The Uniqueness Theorem . The Emis sion principle . As has already been stated in SI, all problems with which we shall be concerned

below lead to one boundary problem or another of the Helmholtz equation

Au + k u 9

where A is a two- or three-dimensional Laplace operator and f is a

given function of the point, or, for boundary problems, of the system

of equations

Au + k^ u = f

^ ,

2

Av + k

2 v = f2

.

The question arises: Under what conditions can the aforementioned

boundary problems have unique solutions?
.

!

' ' '

'

.

.

1
7

The elucidation of this question is the more necessary in that

the solution of the equation

is, in regions extending to infinity, not uniquely defined with the assignment of the singularity and with the condition of vanishing at infinity, since the infinite region admits proper vibrations for a continuous spectrum of values k.

In contradistinction to harmonic equations, the vibrational equation admits of a solution which is everywhere regular, different from

zero, and vanishing at infinity. Physically, this fact corresponds to

the existence of standing waves which, being solutions of the vibrational

equation, may be superimposed on the sought solution of the given

boundary problem without altering the given boundary conditions. These

parasitic waves which thus destroy the uniqueness of the solution of

the boundary problems, arise as the result of the imposition of the

phases propagated to infinity and coming from infinity. It is perfectly

obvious physically that the last type of wave must not arise.

These considerations led A. Sommerfeld in the year 1912 to the

Emission principle, which is formulated as follows t

Let (T) be a simple (simply connected) contour enclosing a

portion of the infinite region under consideration^ let (2) be a

circumference of sufficiently large radius to contain within itself

the contour (Y)| let r be the distance from a point outside ( 7 )

2 2

(1U)

0X d 7
'

.

;

.

.
8

normal,. Then the solution of a plane vibrational equation (lU) satisfying the conditions

= 0 , (15)

Vru = a finite quantity , n-*oo

u “ 0 or - 0 on (T) (16 ) d n

will be uniquely define

d

e

This conclusion is obviously equivalent to the assertion that

given conditions (l5) - (16) and the conditions of regularity everywhere 9 the equation Au + k

2

u= 0 has only a null solution.

Thus the questions relating to the proof of uniqueness in problems

leading to the consideration of a vibrational equation in infinite

regions are connected in the closest manner with condition (15), which

Sommerfeld called the Emission principle or condition.

The author of the present work has given a proof of this principle

in a special work , which we shall reproduce in basic outlines.

d

2

un 2 THEOREM. A solution of the wave equation -— - c Au^ of the

d v form

e“

int

u(x, y) ,

2 / n where u(x, y) is the solution of the equation Au + k u = 0 Ik = -

satisfying condition (l5) at infinity and one of conditions (l6) on

( T )

,

can only be identically zero for real k

.

^

The ’’Emission Principle” of A. Sommerfeld. (Doklady A. N.

S.S.S.R., 193h, #2)

.

lim

r—>oo

\fr + iku
.

' ''

:

: ’•

. "

:

'

.

.

r
9

We shall prove the following lemma:

2

A solution of the equation Au + k u = 0 satisfying the conditions

lim Vr~ + iku) = 0 , r—^oo

lim / (Vr u) ^ dty = 0 , r-*»oo J

0$)

(17)

is identically zero.

We shall represent the sought solution in form

0° /pN

u = £

; (kr)(an

sin n9 + b

n

cos n9) , n=o

r(2) i where IT '(kr) is the Hankel function of the second type.

Utilizing the asymptotic expressions for these functions, we may

write

Vt

co f

u - E I n=o |_

A + B • 0

|kr |

.

(a sin n9 + b cos n9) , n n

' 9

where A and B are definite bounded quantities.

Now, on the basis of property (17) we have, from Parseval's

theorem,

an

~ 0 5 b

n

“ 0 »

i.e.

,

u = 0

Define the essentially positive form

E = H

d

ax j

' d u^\ 2 ^

/ a u^ 2- 7
*

'

...

.
10

Having in view the familiar physical analogy, let us for brevity call

E the quasi-energy.

Let us determine the vector of the flow of quasi-energy from the

condition

where ^ is the region between (y ) and (2) and n is the ort^ of the

normal. On the other hand, we have

7T ‘ grad TT grad “1 + TT iul

= [4^ grad uj

and by Gauss' formula

div

[ TF ®rad “l] dtO

a"aT^ grad ]« dO~ 9

Let

d \ p = _ “

grad “l •

u = 0 + i if** , (18)

^

Heaviside, 0., Electromagnetic Theory §121: '‘Sometimes it is

convenient to have the variation of a vector exhibited in terms of

the variations of its tensor and unit vector, or its size and ort.”

(lie elsewhere explains that ort is a workmanlike contraction of orientation.) — Translator’s Note.
. : ‘

...

'< C. . ; S'.r;" ' ,
11

then

" 0 cos nt + Y sin nt 9

P = n(0 sin nt - iff cos nt)(cos nt grad 0 + sin nt grad

and therefore

Pn " n (0 sin nt ~ ^ cos nt) cos nt + ^

sin nt

d n

If we take into consideration that, in condition (35), u = 0

+ i

we shall obtain

lim Vr f-TM- “ k f] =0 ; r-*oo ' 0

lim y/r

r-^oo

ii

a n k0 J = 0

Utilizing these equations we may write

Pn = - kn(0 sin nt - iff cos nt)^ + ^ "f) cos

+ ^

+ lc0

^ sin nt (0 sin nt - Y cos nt) * n •

From (l6), (18) and (19) it follows that

/

Pndo-- =0 9 (r)

.(19)

If,

( 20)

( 21)

consequently no change of quasi-energy occurs across (if) into region

-H or from it.

On the other hand, in accordance with (21):

12

/ p5<Js - - / (0 sin nt - ^ cos nt)

2

ds

(Z) (2)

2Tf + n J*

(0 sin nt - ^ cos nt) X

“ k Vj cos nt * siin nt rd«

( 22 )

In view of the fact that as r oo the integrals / Vr 0 d<* and

f d<* are bounded, equations (20) show that the second integral

summand in the right part of (22) may be made as small as one pleases

by taking r sufficiently large.

Hence it follows that if, as r —» oo,

2ir

-kn (0 sin nt - cos nt)

2

ds = -kn / r(0 sin nt - ^ cos nt^d#

(2) o

does not tend to zero, then in accordance with the formula

3

Tl IfEdu) = J

Pnd<r

(/)

we shall have, from (22),

d t

Ed60 < 0 , -nand E tends toward negative infinity, which contradicts its essentially

positive nature.

Hence it follows that
'

:

.

'

'
13

2'rr lim / r(0 sin nt - ^cos nt)^ d<x * 0 ; r->oo <Jo

r-»oo / [L

dcK 0 lim /[**] d«<»0 ,

lim

r-*oo

0 ,

but in this case, in conformity with the lemma, u 3 0, and the fundamental theorem has consequently been proved.

CHAPTER II

THE METHOD OF CURVILINEAR COORDINATES

THE SIMPLEST PROBLEMS

III. The diffraction of electromagnetic waves near a cylinder .

In the following sections we shall give an exposition of some of the

elementary classical cases of the diffraction of electromagnetic waves,

the solution being by the method of curvilinear coordinates. The

essence of this method consists in the introduction of special coordinates

such that the diffracting surface is a coordinatal surface. Once having

expressed all the equations and boundary conditions in these coordinates,

a considerable formal simplification of the former is usually attained.

The actual construction of the fundamental functions is marked, however,

by great difficulties, and an effective solution is obtainable for a very

limited number of special contours and surfaces only.

Let the diffracting contour ( be a circumference of radius f> .

In the plane of ) in the direction Q *= falls a plane, linearlypolarized disturbance u

Q

| together with this, the electrical vector

E ^ 0 and the vibrational process is polarized perpendicularly to

z the axis z (Chap. I, §1, #l)j then

Introducing polar coordinates, we transform equation ( 7 ) (Chap. I, §l)

to:

_ ikr cos(0-9o ) u = E = e

u

e z

( 1)

d

2

u + 1 d u + 1 a

2u 2 + k u = 0
.

'

'

.

. .

-

.

. ,

'

1

'

,

.

'
For fixed r, u is a periodic function of 9 -with period 2M } and

in view of the symmetry, we may seek it in the form of an even Fourier

series in © with coefficients depending on r:

oo u 55 Yf C0£? n^ @ ” @

o

) > ( 2 ) n-o

and for the coefficients, a^( r) , we as usual obtain the Bessel equation;

whence

\ 1 da

n

f ?¥

sl - c J„(kr) + c'l„ (kr) e n n n n n

From the fona of an it follows that u, as determined from (.2), must be

separated into u. and u (u. within nf 9 u outside '/), and in consequence of the obvious regularity of u^, it is necessary to seek it in

the form

oo \ = £ B J (k±

r) cos n(© - © ) . (3) n-o

As regards u , in conformity with the Emission principle, it is necescL sary to seek it in the form;

u

a

ik r cos (9-9 ) a o e

oo / <p\ + £ \i ri ^ k

ar^ cos n ( 9 " 9

o

} n=o(U)

Let us recall that the asymptotic representation of
- '

,

.

-
16

satisfies the requirements of the Emission principle.

The indices i and a on the k are introduced to mark the physical

difference between the media inside and outside of (Y)$ this difference is indeed the cause of there being differing frequencies. On the

boundary (y), in conformity with the general Maxwell theory, the

equalities

aua

* u

a

5 ~

-Tr" 5 1*’ PI •

obtain

.

Let us insert in these conditions the series cited above, first

having represented u too in series form:

©

ik r cos (0-0 ) oo \ = e = J

Q (k

a

r) +2 J 1 Jn

(kar) cos n (0 - 0

Q

) ; n=o

then for the coefficients An

, for example, we have

„.n

A = - 2i 9— — -PT- . (5) W

V

5 '

y

k

i P>-kaHn

< kaP>

Analogous expressions can be found for B

n

as well. A more detailed

inquiry reveals that the series thus obtained converge with a rapidity

that is suitable in practice only in those cases for -which

zp «-Hr ,

i.e., where the diameter of the circle is small in comparison with

the length of the wave. The case considered in this section is
/•‘I. ;
17

obviously identical with the phenomenon of diffraction near an infinitely long circular cylinder, on condition that the phenomenon be

independent of the %- coordinate. It is easily shown that the "proper

\

vibrations" will be decaying ones in the case in hand. Indeed, it

is known that the Hankel functions, tin contradistinction to the Bessel

function, always have complex values, and therefore the transcendental

equations

m VnC V> «£

2 ) <V> ' Vn<V> H

n

2

< k

af> (6)

may have complex roots; since f

is real, k must be complex; hence

follows too our statement concerning the decaying of the proper

vibrations'.

§!?. The diffraction of electromagnetic waves near a sphere

^ .

In spherical coordinates equations (l) (Chap. I, §1) have the following

form:

r sin

r sin

+

j e

c c I r — + 21)

c C / ©

iooS

+ Er

i u> r smin ©H

r

r sin 9Hn c ©

3 (H^ sin 9) d (Hq) “71 " 3 MH

r

) a (rH^ sin ©)

5 r

9

a (rH

@

) a(H

r

) 6 r d 9 J

d (E^ sin 9) d (E0

) 6 9 d

d(E

r

) a(rE^ sin ©)

60 dr 9

a (rEg) a(E

r

) dr o>

( 7 )

( 8 )

( 9 )

( 10)

( 11 )

( 12 )

^G. Mie, Ann. d. Phys. 2£,

Handbuch d . Phys . Bd . XX .

1908; P. Debye, Ann. d. Phys. 38 ,
. . . ' ' • .. ' ' j • .
18

In analogy with a plane vibrational process (Chap. I, %l) , we

shall here consider separately two cases?

and

1) \

= 0 , H

r /

0

2) H

r

= 0 , E

r f 0 .

The general case is the superposition of these two component processes. In case 2) equation (10) acquires the form

a (E^ sin 9) a E

e

a © “ ~ " ~ST~

= ° ?

which may be satisfied by putting

F

1 d

2

(ru) . = 1 d

2

(ru) n0 r sin © d 0 a r

5 9 r d r 9 9

( 13)

now from the system ( 7 ), (8), (9) we find all the components of the

magnetic vector and E^.

Substituting in (ll) or (12) , moreover, we are in both cases led,

for u, to the vibrational equation in spherical coordinates:

1

a

2

(ru) + 1 9 sin 9

»u

e

+ r dr2 ?sin 9

d 9

In case 1) , in a completely analogous

TT = 1 a

2

(rv) . TT 0 r sin © 3 rd 0

9 ^9

1 a

2

u .2 __—__ -—. + k u

r sin 9 d 0

manner, putting

i a

2

(rv) r 3 r 3 0 }

. (1U)

(ISO

we determine the other components of the field and obtain for v the

vibrational equation (lU) .
.

:

'

'
19

The components of the field are obtained in the general case by

superimposing the two processes just considered by means of the following formulas?

E. + k

2TM ,

E = 1 d iru) +

66 1 d (rv)

0 r sin 9 9 r 3 0 c r d 9

.CO E -

d

9

~ r arj c r sin 9 2> 9

3 (rv)

H - r

2,

d r

a

‘‘(rv) +

’ 2

k rv

(16)

Er/ = V

_ 1 f 1 ^ (ru) + 1 ^(rv)

59 r sin 9 <?r s 0~ ,

%

- ±cO 6 4. \ 1 d (ru) + 1 3 (rv)

c / r sin 9 5 r d r d 9

The field exterior to the sphere will consist of an incident disturbance, of a given potential u

e and of a diffracted potential u*. In

addition, a refracted disturbance arises within the sphere; it will

be described by the potential u’

.

In accordance with the general Maxwell theory, the boundary conditions have the following form?

( 17)

Here the indices a and i indicate the external and internal nature

of the components of the field.

20

The incident disturbance we shall assume to be plane, from the

region of positive z perpendicular to the x-axis, moving in a linearly

polarized wave:

ik z „e a S

x

= e j *

® - E® = 0 5 H®

3

3 Y

ik za

I if = H® = 0 . (18)

Equation (lh) will be integrated by separating the variables:

u - f(r) Y(0, Q) ,

where, as is known, Y(0, 0) turn out to be spherical Laplace functions,

Expanding them in Fourier series, we obtain

T W, 8 ) = £

P„

B

(a cos s0 + b sins-jO , S'—o

5 5 ^

where P are the associated Legendre spherical functions of the ff-th

n,s

order with argument cos 9:

V (x) = - x2

)

s d’ Pn

(x)

‘ dxs

'

It is obvious that P "0 for s > n. n.s —

1

As regards f(r), making the substitution f(r) = (kr) 2 C(kr) in

one of the equations obtained from (lU) after separating the variables,

we obtain for the determination of C(kr) a Bessel equation with

characteristic numbers (n + |-) , and consequently

C

n*J

An

Jn+| + BnY

n+J

Thus we have as one of the solutions of equation (lU)

21

u

n

(kr)

v4c r

and accordingly the following general expression will also be a

solutions •

oo n u = E E ( kr) 2 C

n+

i(kr) Pn(cos 9)(a

n cos s0 + b sin s0) .(19)

n=l^=o

Putting this in and taking the Bessel equation for C„. i(kr)

into account,, we obtain for and H

r the formally identical expressions

E E ( *&) C

nT i(kr) Pw Jcos 9)(ayi m cos s$ + b„ „sin S 0).(2O)

n=l s -o \ / r

n,s n,s n? s

For the determination of a and b as well as of A and B , n5 s n n*

we shall express the components of the incident disturbance in

spherical coordinates and afterwards expand E^ and in series of

spherical functions^ upon which the following results are obtained;

CD .. 1 E

® * ^

1

1

(2n + 1) /f(kar)

-2 J^lOy) P

n>1 (cos 8) cos 0 ,

00 IF*** 1 ^

3 E

±n+1

(2n + 1) /t (kar)”2 J

n+

i(k r) Pn 1

(cos 0) sin 0 . n«l v 2 9

Gomparing these expansions with (20) and (19) 9 it is natural to

6 require that in the expression for u we have

if .

s
/

'

'
22

8 / 1 I

a - s 1 , b ,

33 0 | n,l 3 n5 l n

0 , Bn k

1 rz, on-! 2n+X

ff i a

and that in the expression for ? we have

/

b

n? B n, 0

if

s f 1

a_ n ~ 0 ns ! n a l

B =0 A - i /|f in^

^n+l

n 5 Rn IT \/2 x ntn+TJ 5

cL

thus it becomes evident that the magnetic and electric potentials of

the incident disturbance have the form

e ru

rv

e

Jn4(kar) P

n,l (oos 9) 005 **

tI^TT Jn+|( V5 P

n,l (cos e) sln * ©

# -ft- Proceeding from the form of these expansions^ we seek u , v and

u^ and v^ in the form of analogous series,, where for the first two s in accordance with the Emission principle,, the Bessel functions are

to be replaced by Hankel functions?
v
23

oo „ Afk r l (9 x nT -~n Y_ »C

y -5- H

n+j(ka

r) P (oos S) 00s k n=l. a

rv* 1

k n=l

a

00

00 /

/^f k f \ X K / “T" Hn4(ka

r) P

n.l(oos 9) sin 0

, . fifk . r ^ “ “2 E “n J ~T~ kf n-1 v

1

X l( COS ©) cos 0 9 lioX 9

, 00 . /tfkVr ' 75 S, Pn / -rJmi(ki

r) P

n 3 l

(cos 6) sin 0 • k^ nsl

d s

Making use of (13), (150 and ( 17) now, we determine the numerical

coefficients of these expansions^ so, for example, for the first two

series we have:

,

-5'r „ n* CK S2 = X n

-1 2n+l %(p)Y; (P) % W

?n(p) %'w-nL (p) y„(np)

2n+l ” t(p) T>) fnW

n n(n+1) N|n

(p) tn '(^>-Jn(p) YnO^)

p

s k

Q p s ( p* is the radius of the sphere) j

N

k

i i cr 1

i US~ “gO

As regards the convergence of the. series here obtained, the same remarks must be repeated as were made at the end of the preceding

214-

section o Numerical calculations and detailed bibliographic references

on this topic can be found in the Handbuch d« Physik Bd, XX, pp„ 307“l6 o

The results found in the Handbuch bearing on the phenomenon of electromagnetic diffraction near a sphere have been basically reproduced here,

S6 8 Diffraction of elastic waves near a clamped circular cylinder »

In an infinite elastic space with Lame constants X and

p

and density

era cylindrical excision of diameter is givenj an incident plane

wave of the longitudinal (or transverse) type is considered$ it is

periodic with the time, propagated in the plane perpendicular to the

axis of the cylinder, which we shall adopt as the z-axis„ Thus the

problem consists in the study of diffraction near a circular excision

in the plane xy-planeo

The equations for the basic potentials. In accordance with § 2

(Chap„ I) will be

the components of the displacement and the stress are given by formulas

(9) and (ll), (§2, Chap e I) „

Putting

- 9Keimt

0) 3 \ - ^(eia>t

f ) (21)

for the lagging potentials 0 and , we obtain

A0 + k^0 - 0 j A tjf + kg ^ - 0 5 (22)

where
.

'

.
2*

2 &id co \

s “ ~r 5 k

2

2 2

to to (23)

Let the incident disturbance be given by the potential 0f*

s

ik^r cos ©

We seek the supplementary potentials in the form

0# * 0* + £ (k^r) ( L

n

cos n© + M

n

sin n©) , n=o

00 / Y 85 £ H

n

^ k

2r^Pn

cos n@ + sin n@) • n=o

(2U)

In accordance with (9) , on the circumference r - there must be

fulfilled the conditions

^ 0a a0*

d X d J

or, in polar coordinates.

- dp Fa o

ay ax

i ^0*

Bf P

- 0 .

Using the expansion

CD “ J

Q

(k, r) + 2 £ in

J

n

(k 5 r) cos n© , n=o

the boundary conditions just written, using the series for 0^ and Y

in than, will give a system that is linear in L

n, Mn, P

n, from

this we find
.

.
26

p - M ~ Q - 0

n n n

b n - 0 5 1, 2 s

L

n

” + ^ ^ n

2i

“»J

pd n

n > Hr

/ (V)'V2

J

n(

k

lP) H

n

(2)

( k2f)/

^ - %

+ M

21n % J

n(

k10H

i

2)

( kl^-J

n(

k

ll,)Hn

(2)(klf,) f

. n

A “ «i

2)

<V> Kn' <V> kl

k

2Hn

V ‘' <M H

n

(2)

<V>

Inking into consideration the known formulas for the differentiation of cylindrical functions s it is easily shown that with real k^

and kg

A ^ 0 e

§7® Diffraction near a circular cylinder with a given stress at

the boundary „ In this and in succeeding paragraphs we shall use for

stress several symbols differing from those of §2 5 Chap„ I 0

The vector of the stress for an element on the circle with normal

n = r has the following form

s p( x) cog q + sj_n q o

The conditions of absence of stress (free borders) at the boundary are given notationally thus;

p^r) _ ^ © + % sin 0 s 0 | xx “ v cos © + sin © = 0.(2!?)

xy j 2^r y

’

where (<v- , Z ) are the components of P " and (V cr ) are the

x* xy

components of P

(y)

xy* y

Formulas (11) , £2, Chap c I, now take the form

2 ?

X

03

a % d ’ t

JyZ

*

9 x» y

5 V = 2

P TVTy “ - 2

f

cr.

2,

2i

(26)

7

' • r * ~r ^2 t axay

Having expressed the boundary conditions (2^) in polar coordinates,

and introducing the values of o~^s "C

a °"'j from (26) and of 0v, and ~'{J

from (2U), we obtain, for the coefficients P

n

, Q

R , the formulas

link. VM~~r

lin .n. “W 1 v n n

9 O ft - Aj^k^kg)

D

H

(k1

)DH

(k

2 )”AH

(kl5 k

2 )AH

(k2s,k1

)

Ilk 2k2 " JT j^n'-TL*' p

1

V>

DH(ki)

Q.

2 ii2i^\ _.n T /, - c2“ ~pT] x J

n'

kl^ f

1 u n^l

1

-X(^Lf)

n

AH

(k

2 ,k1

)

D^(k1

)DH(k

2 )»AH

(k1

,k

2 )AH

(k25k

1

)

link.

/> 1 *XV> - ^

M P n n

DH(ki)

AH

(ki,k2

)

-w H

-,2 n

(n - 0, 1, 2, ••• ) J

f ~n ~i<

2k. n , 0

\ ~ h' ( 2)

(k, f>) I

r 2^_ k

21 u(2) !)

Hf

)

( k2f) -7 H

n

(2)(k

2 P> *

aH(k2’k

l

) “ (y?

" k

2y

H

n

2k~1 „*(2) H

v ^(k_/)

y? n 1'
-
28

§8„ Diffraction near a eylindric al. occlusion B In a region of

” falling" potential Y

" there is a circular occlusion of elastic

material with constants and density Z

7

« In this case the

boundary conditions express the continuity of the components of the

displacement? and the stresses %

u

(i) s U. X x

3 (r) s p

(r)

' 2S-, X

jL a

.(i) (a)

u |

y

[r)

( 2 ?)

V. '

7

.

In consequence of the evident regularity of the components of the

refracted fields only cylindrical functions of the first type must be

entered in the expression for the inner potentials „ The external potentials 9 however5 are in accordance with the Emission principle expressible by means of Hankel functions?

s 0'- a '® + E (k^r)(L^ cos n© + sin n©) , n-o n n

* E H. nso n

/ co 0

1 - £ j

Y

^ = E

Jn C^r)(Pn

cos n© + Qn sin n©) |

n=o

oo

n

(kgr) (Pn

cos n© + Qn

sin n©) |

cos n© + M

n

sin n©) j

n-o

( 28 )

both kn and k« retain their former values, but k^ and k. are defined

.the formulas;

, 2

h

P Ui‘

h

CO

b

2 2

( 29 )

29

Expressing the boundary conditions in the sought potentials by

means of the formulas indicated above } and rewriting everything in

polar coordinates 9 we arrive at the following system of equations for

the determination of the unknown coefficients of the several series (28)

L

oH

o2(klf> - VoM - -

9 9

l' IJ k

2

H<

2 > 2kl

' + H

p o

3^J0

(V) - ~r J

o (kid

L

o [

k

u

JoM + ^ J

0

6k^

o (k

3

?)]

=

kA(2)(ki/) L

n

+ p

H

i

2)|

:2?> «n

“ k

3

J

n

(k

3 h L

n “ j

Jn<V> Qn

n

j + k£,H n 2 n <V> Qn

- 2 Jn(k

3

f) L

n

- k

u

J

n

(k

u

p) Qn

2in

|

J

P n

AH

(k

2

5 k

x

) L

n

- D

H

(k

2

) Qn

- Aj(k^ k^) L

n

+ Dj(k^) Qn

Ilk, 2k

2 iin^ ] .n. i

n

J

n

(klf) + -y i

n

J^(kiP i

Dj^kl) L

n - k

2

) - DjCkj) L

n

* ij(k

3

, kj3 Q.n

-

link. T

1

inJ^ (ky) - if in

J

n

(k1

p)

P

M S P' s m' =p' =0

n n n n

(n s 0, 1, 2 5

• ° °)

The symbols adopted here will be clear from the explanations

given at the end of the preceding section 0 The subscript J

30

accomparming D and A indicates the presence of a Bessel function, and

H to that of a Hankel function 0

The problem of investigating the determinant of this system we

shall here leave open.

The rapidity of convergence of the series (28) depends on the

^ 2

<Tf smallness of 2 f in comparison with the length of A &

$•9® The diffraction of an immovable clamped elliptical cylinder .

The problem statedg the differential equations „ In the preceding

examples circular and spherical functions turned out to be the fundamental. functions of the problems there considered. It is clear that

this circumstance is the consequence of an appropriate choice of the

coordinates in which these problems were studiedo

In the case of an ellipse, the most natural functions in which the

solution of the problem can be expressed are the functions of an

elliptical cylinder-”the Mathieu and Mathieu-Hankel functions.

The theory of spherical functions has been sufficiently well

developed for the solution of diffraction problems j the theory of

functions of an elliptical cylinder, however, requires additional investigation in many of its parts. In the next several sections we

shall be preoccupied with this matter.

As regards this problem of diffraction, near a clamped elliptical

cylinder, of the wave (we assume the field to be independent of one of

the coordinates) s

( 1 )
"1
31

( 1 ) we note that K„ Sezawa was concerned with this problem in 1927 0 He

did nots howevers obtain a solutions in consequence of an underestimation of some of the principal circumstances connected with the Emission

principle 0 Let us introduce elliptical coordinates!

x e ch cos 7 “ sh sin ©3

( 2 )

The entire plane will thereby have been covered with conf ocal ellipses

and hyperbolas of the families

COS T| sm

= 1 (3)

The curves f = constant^ 7| = constant are obviously mutually

orthogonal e Let us remark that from, the equations r s

0

- arctg

7J th f g it follows that

©$

lim (r - ch | ) s 0 j lira, (r = sh |

) s 0 j

f "^00 | =“^Q0

lira 0

s Tj

j,

(10

i.e«, that at great distances formulas (2) reduce to the usual expression for polar’ coordinates , x - r cos '0$ y

s r sin 0„ Thus our

v Scattering of elastic waves and some allied problems” Bull,

I Earthquaque Res« Inst OJ vol e IVs ±927, Tokyo,, See also V c Kupradze 3 "Diffrazione delle onde elastiche sopra un contorno ellastieo Atti

d. Accad, dei Lineei 5 Ser e XI a vol« XVTII, f« 3“U»

32

isothermal net asymptotically approaches the polar net of concentric

circles and rays. This circumstance permits an a priori expectation

that the asymptotic behavior of the special functions that will appear

here will be analogous to the behavior at great distances of the usual

cylindrical functions.

= <3? J

(x, y) ,

We shall utilize this circumstance further on during the solution of the problem. If we suppose

the solutions of the problem to be

periodic with the time, i.e.,

reckon that

Y1

=Sleiu,t y (x, y) , (?)

then in the new coordinates the fundamental wave equations will acquire

the form

where

( 6)

cr is the density.

If, moreover,

and ju the Lame constants.

(]) and ~^*be sought in the form

F(f)G(I)) , (7)

33

equations (6) will be transformed accordingly;

1 d

2

F A ,2 u 2 ? 1 d

2

G , 2 2 ^ KJ1 7p

k

i

ch 5 " gHTT J^

+ h cos ^ •

Hence, equating to an arbitrary constant A, we obtain

+ (k

2 oh2

f + A) F( |) = 0 , (8)

d !

2 " - (k? COS

2

7) + A) GO]) = 0 . (9)

d

1

Comparing these equations with each other, we see that if G(7J ) is

the general solution of equation (9), then

G(i f ) * F( O ( 10 )

will be a solution of (8), and, conversely, if F(f ) is a solution of

( 8)

,

then

G( 7) ) - F

[

( 11)

will be a solution of (9)

.

There is therefore no need of investigating

these two equations separately.

We shall be occupied in detail with the equation — - (k

2 cos

2

7] + A) G(7| ) = 0 , d7f

1

which we shall put in the form

2

£ -111 + (a + l6q, cos 2 7) ) G( 7? ) * 0 , &Y

where

Thus if the general solution of equation (9) , G(7/), be constructed, the general solution of (8), i,e., F( | ), will be found in

conformity with (10), and accordingly the general solution of (6) will

be found, of the form

F(

l

) G(l) ) .

Physically, however, it is obvious that of all such products only

those that are single-valued functions of the point are suitable . For

this the function G( ^ ) must be periodic, of period 2 it. By this condition, in turn, a discrete sequence of values of the parameter A is

determined, for which equation (9) admits of such periodic solutions.

As regards the function F( |), only such particular solutions are

suitable as are amenable to the conditions of the Emission principle

for infinitely large f „

Thus the mathematical part of the problem before us consists in

seeking particular solutions of equations (8) and (9) satisfying the

stated conditions.

v §10, Mathieu functions . It is known from the general theoiy of

differential equations with periodic coefficients that if periodic

solutions of equation (9) are to exist, it is necessary that a definite

dependence between a and q^ be established.

This relation of dependence will be given below| it is, for given

q^, the characteristic Mathieu equation.

In the special case q^ = 0, and a

53 n , equation (9) reduces to

the followings
. ' . ' M. " . . : • 1 ' '

:• • - . '
y?

ai\

with the fundamental system of solutions

(13)

sin n )j 9 cos n tj

¥e shall adopt a terminology calling such periodic solutions of

equation (9) as for q^ ~ 0 reduce to cos n

7| and sin nrj Mathieu funcTM

tions of the n th order.

and the second (the odd) 3 by the symbol sen(7^ 9 q^) „

In diffraction problems we shall have occasion to meet with all

types of Mathieu functions } which are comprehended in four classes

represented by Fourier series

„

We shall give them a scmewhat special notation convenient to our

purposes?

The first of these (the even) are denoted by the symbol

( 111 )

s<L

+ cos (2n + l) Tj + ][) cos (2n + 2s- + l) ) (l£)

s sl

36

se2n+l (?

? » ^ " E ts

2n+l) sin (2n ~ 2s + 1)7

/ s=l

a?*)

+ sin (2n + l) 7j + £

^(2n+l)

s,_n (2n + 2s + 1) 7] . s-X

Let us recall at this point that for fixed a equation (9) does

not simultaneously admit of two differing periodic solutions, wherein

it differs from equation (13), for which, on the contrary, this circumstance obtains.

This property of equation (9), important in principle,, long remained unnoticed until it was proved in 1922 by Ince^ . (Judging

by his works, it is apparently even yet unknown to the Japanese

seismologist Sezawa.)

Series (llq) ra (l^ ) are known to converge over the entire plane,

anci consequently the coefficients <* , (3 3 S R

diminish more

rapidly than the n-th power of any number.

However it is of essential importance to us in diffraction theory

to obtain more exact estimates of these coefficients. In the following

section we shall give the form of these coefficients and the estimates

we need.

§ 11 . Construction of the Mathieu functions , Periodic solutions

* ( 2) of equations ( 9 ) mgy be constructed by the method of Frobenius v

, as

(3) has been done by Whittaker and Watson

L. Ince. A proof of the impossibility of the coexistence of

two Mathieu functions. Proc. Cambr. Philos. Soc aj 21, 1922.

( 2

2rourn. f. Math. LXXVT, 1873, pp. 21U-22U.

Course of Modern Analysis. N.Y. Macmillan, 19i|6. p. l\21 ff

.
' vs

...... •

.

;
37

However^for our purposes it -will be more convenient to proceed

otherwise 5 thereby cohsiderably simplifying the derivation of the

functions we require,, as well as of several of their properties 0

Let

a - (2^^ + 8p e (l6)

Equation ( 9 ) t^en acquires the form

2

£ ^2 + Itn2

G( If) - 8 (p + 2 q1 cos 2 Tj) G( Tj ) d1

Let us employ expansion (lU) in this equations

( 17 )

£ U(n - s)

2

o<g

2n^ cos(2n - 2s) - Ltn 2 cos 2n 77 n

k

- £ U(n + s)

2

f3^

2n '

1 cos(2n + 2s) nj + £

<Y^n

'cos(2n - 2s) ^

+ S—1 S=1

,(2n).

00

,(2n)

n

(2n)

+ Umcos 2n7j + Ijn £ /ir n; cos(2n + 2s)7J + 8p £°<g 'cos(2n - 2s)7| + s-1 s=l.

co /« \ + 8p cos 2mj + 8p /#' y cos(2n + 2s) Tj + l6q-^ cos 2m] cos 2 t] +

s=l

+ l6q-^ cos 2

7) ^ c* ^ 2n ^ eos(2n - 2s) 7| + s-1

+ l6q, cos 2 Tj £ /3g

2n^ cos(2n + 2s) Tj - 0

s = L

Cpmparing the coefficients of cos 2ni|, cos(2n - 2)tj ,

• •• 9

cos(2n - 2k)T] , cos(2n + 2 ) s etc., first having resolved the product of the cosines:
: ' . ' - . > - ' : * , .
38

l6q^ |^«tg

2n

^cos(2n - 2s)7] cos 27] = Sq^ j"<*

g

2n^ [cos 2 ^ n ~ s + l) 7

/ + s=X s-1

+ cos 2(n - s - l) ?{ |

°° (2n) , 0_ , 0 - TM

o«- ? f*(2n) 16q^ Y, /3g n cos(2n + 2s)yj cos 2 7]= 8q^ ]T n

[cos 2(n + s + 1)t] +

S=1 S=1 l

+ cos 2(n + s - l) TJ ] j- |

l6q^ cos 2m] cos 2*7] * 8q^[cos 2(n + l)7| + cos 2(n - l) 7] ] ,

we shall obtain the following system of equations!

p

+ +

x?i

2n) - 0 ;

s(s - 2n)<r< 2n> =

2 {p«<2n> «(2n) +0((2n): S-1 j S +1 J

5 (s =

(18)

1, 2, ••• ,n(1)

) (19)

= 1, 2, S ( 5 + 2n) (3< ••• oo ) (20)

2n> - 2|p ^2n)

(9^]} , <i

( 2n) ( 2n) where 1 = /S^ ' = 1 and with respect to «( it is moreover necessary to keep in mind that £

varies from 1 to n.

Let us introduce the notations

°s

= Uq^sCs + 2n) - 2p|

1

$ ja

g = - Uq1

«js(s - 2n) - 2p|

1

.

Eliminating

0 ( 211) (2n) ... o(2n) o(2n)

» f J 2

5 5 ' s-1 5 ' s+19

from system (20) , we obtain

(«„ ... Will n4* T.TV>1 4* /\ . rn,m .VI ^(2n) For s = n, one must write —

n

2p ^i

2n)

+ 2^^

See editor's Note 1,

39

(-) TT

r=l

( 20*)

where A_ is an infinite determinant of the von Koch type:

s

1, ^s+l» 0 o,

°s+2> *s+2» 0,

^"s+3* 1, *“s+3>

This special determinant is absolutely convergent, since the

necessary and sufficient condition of this, consisting in the absolute

convergence of the series

^* +0* +©* +•••

s+1 s+2 s+3 9

is fulfilled.

Moreover, Whittaker has shown^ that

lim A

g = 1 . s~>oo

It can be similarly shown that

^

2nV(-)s friv>s , r=l

where

1, ps+i> °» •••

fW> X > Ps+2’ °»

D

s =

0, Ps+3> 1 > Ps+3’

"•

^Ibid., p. h23.

(21)

(19*)

Uo

is a finite determinant,* since here s varies only up to n. Equation

(l8) plays the role of the characteristic equation* Let us transcribe it in the following form;

PA0D

0 - q1(^1

A1D

0 + p^Dj) = 0 on

Multiplying this expression by

ooW r n i r®n

TT j r=l L ^ “ r(r+2n) }

p~n

• U

r=l

it is easily turned into an integral function in p and ,

Having fixed n, it can be shown that the root p of equation (18“)

can be represented as a power series in commencing with the

term q^.

If

jq-jJ

is small enough, we have, for the value of p already found,

*o^° . Do^° »

since for q1

= 0, A

Q

* 1, D

Q = 1.

In addition, it will be shown below that for small and with

any h, the following inequalities hold:

2p 4 r(r + 2n) , 2p 4 r(r - 2n) , (21*)

in the latter of which inequalities r runs over integral values up to

% n only,

'[he number p thus determined in the function of n. and q^ plays

the role of the characteristic number for equation (9). In future it

will be essential to know the asymptotic behavior of p for n tending

to 00

ia

§ 12 o The asymptotic behavior of the proper values of the Mathieu

equations e The assumption was made above that 2p £ r(r + n) for any n,

and r = 1, 2, „ We will now elucidate the conditions under which

these inequalities actually hold, and in addition, will study the

asymptotic character of the distribution of the spectrum of proper

numb ers

Setting A - (2n)^ + 8p together with a = (2n)^ + 8p, the Mathieu

d

2

u

equation —^

+ (X n

+ l6q-^ cos 2x) u = 0 is an equation of the Sturmdx

Liouville type, and to the theory of its proper values (characteristic

numbers) one may apply the theory of that type of equations, based, for

example, on the familiar variation principles

It is possible to show in just the same way as is done for nonperiodic boundary conditions, that the characteristic numbers corresponding to periodic solutions form an increasing sequence

< P

1 ) < ••• (i - 1, 2, )

The two superscripts are to indicate the evenness (l) or oddness (2)

of the corresponding fundamental functions » On the other hand, it

was shown above (with an insignificant variation in notation) that

to the numbers

“ (2n)? + <4n ( V > *2n ' ^ + 4n

^ >

*

(2nh (2n + 1)2 +

> 4nh m (2n + 1)2 +

there correspond the solutions

9

b2

GO /p \ ce2n^ > ^

= E *2S cos 2sT

l s=o

ce12n+l(V 9 \) 52 E b2svl

1 ^ cos ( 2s + !) *) i s=o

00 - ( On \ se2n^ s = E aps

^ sin 2s 7| , a

S-0

se2n*l (ll>

^

= 2 S2s+1

+1) sin (2s + 1)7I ’ s=o

$2

^ (q^)j ^2n+l ^l^ > = 2) being here special power series in

cjp reducing to zero for q^

= 0 ; and a, b, a, b being coefficients

found in the manner indicated above.

We shall show that

i = 1, 2

'1 P

l“

1

5 \j * 1, 2

y i„e., we shall prove that the classification of the proper numbers in

accordance with the principle adopted by us coincides with the classification that emerges from the variation principle.

It is obvious that

A^1 ) (o) > Ap, (o) s

(**)

Pi p r

now for =

q£ / 0 9 let

Two cases are then conceivable:
.

.
k3

#1. To the characteristic number ji there correspond two different

periodic solutions: ce^ and ce^ se^ and se^ ce^ and se^

or lastly se and ce , which is impossible according to Ince’s

Pi P]_-J- theorem mentioned previously,,

#2. The characteristic number

p

is multiple, and moreover some

of the pairs of periodic functions indicated above become equal, which

is also impossible, for obvious causes.

Thus supposition (2) is untrue, aid inequality is consequently valid.

This examination shows that the classification of the proper values

set forth in §8 is also an increasing one, and therefore the asymptotic

character obtained for the characteristic numbers on the basis of the

variation principle is valid for th,ai too.

In accordance with this principle, the characteristic numbers of

the equation

which give periodic solutions of period 9 are given by the greatest

of the least values of the integral

u + l6q-j_ cos 2x • u + A n

u = 0 ,

on condition that

l

2 if

o
'

-

-

.

•

:?

: ;v
uu

It is clear that the sought raaximinimuiri is less than

.2TT

maximin

Pi2 - / u

“ dz + max (l6q^ cos 2z)

but

»2iT

maximin

f 2

u dz

are the characteristic numbers of the equation

u + ku = 0

for the interval (0, 2 TT)

.

Accordingly A

(i)

(p-i V

cannot exceed |l6

q-jJ,

whence

+ a finite quantity, where this latter

p < 2q

x ,

and if |q^| < 1 under the condition of the problem, we have

2p / r(r + 2n) 5 (r - 1,2,3 000

)

Q „E «D

.

Comparing this result with (16) and (L8“) , we conclude that the

values of p that are the roots of the equation

PAoD

o

<*L

( ''lA

lDo

+ PlAo^*f = 0 »

satisfy conditions (21'') and remain bounded as n increases without

limit
.

*

"•

'

;
Utilizing this property, we shall give several estimates for the

coefficients of (20“) ;

/3 C2n)

s

(3 (2n)

s

r(r+2nj~2p 5

*L

(-)% sr1

7“rr~ TT

(si) A

o r=l

>

where s^ is so chosen that 2n > which is always possible in consequence of the boundedness of p for any n.

Under the last TT sign it is possible to neglect the positive

2 n 2*0 summand in the denominator, and the entire product is less

r in modulus than q£

S1

As regards the first sign, we have there a finite number of

finite factors, and the entire product is consequently bounded.

It is clear that for any n this product will be less than some

sl"l

number Mq^ $ finally

<

s (sl >\

( 22 )

In addition to this formula. one can obtain another inequality

U6

f3(2n) . ( _)« ^

fjo r=l 2n°r fe+1' ml

(-)V S 2"1

S tV

(2n)

s »slA

o r=l

sIT

r=s.

*L'

^2 p where S

2

is so chosen that ~2

~ > ^

3 which is always possible in consequence of the boundedness of p for any n. Furthermore, reiterating the

foregoing reasoning, we obtain

p

( 2n)

<

1*

' s (2n)

s -slA

0

Finally

(3 (2n) (-)%, A

A ~ TT

r r(r+2n)

(2n) l A

q

sl(s+2n)

x2p

r(r+2n)_

1

sTT —2

^-

and since 2p ^ r(r + 2n) for any n, and r - 1, 2, 3, • •

° , we have

A (2n) Hq1

(2n)iA

s s l(s+2n) IA0

( 22*)

A

Q

, for small q^, is different from aero, since A

Q (qy =0) = 1,

Inequality (22) is stronger than the two preceeding ones.

Let us pass on to deriving estimates for <=< (2n) From (l9~' ) we

have

U7

<*(2n) . (~)

s D s ir TT

O 37=1

Q1

r(r-2n) 2p

r(r”2n)j

D

s ^(2n-s-l) 1 s 'B'Bt&ixn Ji

1

1- 2p

r(r-2n)

9

Having noted that

it —-—- fr fi + —

,

r=l -i 2p r-1 I r(r-2n)*»2p

rC?-2n)

(23)

s s and utilizing the fact that Jf (l + a^,) < e * where u = ^ a * we

r=l r=l

shall have

1

1- 2p

r(r-2n7

<

® 2p

rssl r(r-in)-2p

e

Since 2p ^ r(r - 2n) * this product remains bounded for any s - n. Let

us recall that in the expression for s is always less than or

s equal to n.

' CD

We finally have v y

«(2n) Sqf (2n-s-l) ID

1 s

s D

d .sl(2n-l)J

(23*)

s s We shall demonstrate below that the ratios ~r~ and *e- are

A

o „ u

o bounded for any s* and therefore* from (220 and (23") we obtain*

finally?

( 1) In all of these arguments we have considered p

> Oj in the

contrary case our inequality is valid a fortiori .

3 ,(2n) i ai^(2n-s-D

s

* ^ s~irs=ivr (2f)

(22*)

1 ^ s~t(s~+2n) l

0) where a, a are finite positive quantities not dependent upon s and n.

Ar Let us prove the boundedness of the expression -r— for any n.

A

q

Utilizing the Hadamard theorem concerning the upper bound of the modulus

of the determinant j we have

|A I mod » r» m *

1 o'

r+l 5

crr+2 f ^ 5

C r+3 S

r+2 s s

1 v±y

o o 1

r+m5

< /(! + ^

0 )(1+ 2<? o) ,,8 (l + 202

i v ,. ) \ r+1 >• r+2 rim

M - lim I A

r m

l - UTM I /(l + 2^)1 “ m—»oo m—>oo

r+1' r+m

m

| 2 in(l+2<7- , .) lim e m—>oo

k=l r+k

oo but dince ]T |/n(l + 2 | has a common finite upper bound for

k=l

any n9 the boundedness of Ar

for any r is proved with this* Moreover,

since A

o for q^ = 0 is equal to 1, A

Q

is for small q^ different from

A^p

zeroj hence follows the uniform boundedness of the radio ~^-9 QJS.D...

o

( 1 ) See editors note 2.
. • , : i . i , •
§ 13 e Properties of the Mathieu functions. It can be shown

1( 1)

that the functions , q^) and se2n+l^ sOp) are solu'

i:' ions

equation

oCtj ) . X

P** J

LkislnT

l sin 9

G(e, ^ de . (2W

•fir

In addition to this,, the equation

2tr

/

ikn cos 71 cos © ^

s , A

• 1 ' Ce

m

(S ’ ql

)d® (25)

is valid for all even functions, and

,2 IT'

se, (TJ 9 q^) = ^ Jsin(k^sin rj sin ©) se2n+1 (©, ql^

d0 2n+l

( 2 ) E. Poole has, moreover, obtained an integral equation for the function

se

2n*

^Whittaker, On the partial differential equations of mathematical

physics , Math,, Ann. Vol. 5>7. See also the same author f s work in Proc

Int. Congr. of Math., 1912. Mhittaker ! s derivations must be somewhat

modified as to form in the following manner. The integral

•hir ik-, sin 7j sin ©

A / e G(©, q^ d© (2$*)

having been substituted in equation (9) following two integrations by

parts, leads to the equations

“I

4* If* ik-, sin T) sin© , r, n ik-, sin 77 sin©

+ J

e [G,, (©)+(a+l6q

L cos 2>/)G(©)]d©=0 . -if

If G(0) is an even function, G*(©) - 0 for 9 =+ Tf 3 if, however,

G(©) it an odd function, G

1 (lT

) = G’(-'n'); the integrated function

equals zero in both cases, therefore. The expression standing under

the integral equals zero in consequence of (9) . Thus integral (25*)

(Continued to page £0)

(Footnote (2) also appears on page £0.
.

l
2ir 50

se

2rS~^ 9 ^

~

^ /

cos7| cos © sh(k-^sin7) sin @ ) se

2 n^3 q^)d© .

Integral equations can serve for the effective construction of the

Mathieu functions j we shall show -this briefly in an example for

ce

o

(l

) » <J)>

00 ce

0 h > s 1 + I * r

cos 2rr) • r-1

Substitute this expansion in (25) and put Tj = -Xs

f

v 20^

=e

0 (f- » qj / oe0

(S, d9 « 2irX

a

J *f? ce

o (3"» %) •

Now (2?) may be written as follows:

/ \

2ir oe

o

(7l »<Jl> " 2V ce

o (f-> %) J

9

k^cosq cos 9

( 26)

ce

o (© ? qj_) d© o( 27)

But

k, cos t') cos © o° r~_ - J

Q

(ik-. cos ©) -+ 2 £

j/^ik^ cos 9) cos n>]

n=lJ~

=n

Introducing this expression in (2 7) and comparing Ihe coefficients of

cos 2n?] 3 we find that

~~n°r —

_

v y Footnote continued from page k9i

is a periodic solution of the Mathieu function. It is not possible by

the same reasoning, however, to obtain integral equations for ce^n-KL

and se2n+1 , since

+T1 ik, sin r] sin© +ir ik, sin rj sin©

J © se2n(©,q1 )d© = / e ce2n+1 (©,q1 )d© = 0. “TT —IT

(2)On certain classes of Mathieu functions. Rroc. London Math,

Soc., 20, 1921. (Footnote reference on page U9) .

ce - 1 lf9ql‘

21r ——

— J

J

2r (ik1

cos 0) ce

o (@ 3 q^) dG

If we now make use of the familiar equality

2if TC •(2m)

I

cos V cos 2n0d©

we will obtain, from (28)^

2r

2^m”

1

(mtn) t(m-n)

r(3r+lr)k^

r+^ <

o)

- ( -1)r

/ - ^

(r,1)v(r;1) ;

Si

r * 1 .(28)

The orthogonality properties of a Mathieu function follow from

the homogeneous integral equations with symmetric kernels that they

satisfy:

•=**» J

cem

( 7

7 3 s en(7} 9 <%) dy\ = 0 9 -ir+ ff J

Cem^ 3 cen

(^] 9 9.]^ d - 0 | m / n (29)

rf + fT J

n

s Qj_) sen

(T)

, q^) d"Y] ~ 0 . -IT

It has already been pointed out above that equation (9) belongs

to the Sturm-Liouville clhss of functions.

It is known that the proper numbers of this equation, under given

boundary conditions, form a countable non-diminishing sequence

~K A

2

? •’*

, ^

n

, ,and that the corresponding system of characteristic functions forms a complete, orthogonal system.

(l) / v 'P® Humbert, Fonctions de Lame et Fonctions de Mathieu* Mem.

Sci. Mat. 926o

*2

Any continuous function with piecewise-continuous derivatives

can be expanded in an absolutely and uniformly convergent series of

these characteristic functions.

By means of this proposition, it can be shown, in particular,

that

j 00 / r, \ oe2s

(l> « ^

? “2k se2k

(l

l *

<’2 ) »

oe2s+l^ > **L^ = k

2

o ^k+l

^ se2k+l ^ ’

| OO se2s^ 3 ^ E ?2k ce2k^ » *

(30)

^ OQ / p “f0"] ^ se2s+l^ » ^

= X q2k+l ce2k+l^? * q2^ 9 k-o

where, on the basis of (29), we have

+ir

. / ce

2s

(ll»*L)se2k(l

> » %

)d V

+y

- “

c/ se2k(i) J q2)d1l

n,(2s+l) J

Ce2s+1 ^ 5 ^^k+l^* (fe^

d7? 2k+l ,+r

2

J

se2k*l^l 5 q2^ d ^

tr

+ r

f, se

2s (l*^ee2k(

'1l>% )d/>l 4k =

+ir

</* ce

2k^

7

? 9 d ^

»tT

(31)

(2s+l) -rf se2s+l^ 5 qL^ ce 2k+l^ 5 q2^ d7l 4k+l

3 “ y

, °e2k+l^ 9 q2^ dT* — tr
ni;>

m:. .

-
53

§liio The Mathie'u-Hankel functions 0 To conform with the Emission

principle, it is necessary that non-periodic solutions of the Mathieu

equation (8), F( § ) be introduced such as to give a. phase outbound to

infinity and with an attenuation of 4s B VT

As a preliminary let us prove the following proposition.

If a function £\ ( | s ljj) 5 periodic with respect to 7} s is a solution of the vibrational equation

a A

2 il

(f

,l|) + k

2

(ch

2

f - cos

2

if ) Q- (f 9

lj) - 0 (6*)

and G(TJ ) is a periodic solution of the equation

G"( t| ) - 04 cos

2

7j + A)G(7) ) « 0 , (9*)

then the integral

2ir

l

TW

jO.

(f S, V ) <3(T? >d1) " F(f ) (32)

is a solution of the Mathieu equation

F" ( f

) + (k| etr l

* A)F( |

) » 0 ( 8*)

where we shall, of course, consider solutions that differ from zero,

Indeed, we shall introduce (32) in (8

>

') and prove the equality

2ir

4f i —il#! + eh2

f

* A) A (| , Ip

B

s

G(7j) d7) ® 0„(33)

¥e rewrite (33) with the aid of (6*). 8

2 if

1

^tT ^5 ~ (k? cos

:"

Tj + A) il

a r

1 ! G( T] )d Yj « o Oh)

2 r .2 Applying a double integration by parts to / G(7) )• dTl, and

o a

1 '

taking into consideration that G(2 'TT') = G(0), G ' (

'tT )=> g’( 0), we find

that

I

0

Now (3U) gives

2lf

1

TW

o

21f

? e V** o 0(7) ) d T) - / «<§ ’ r|)±J^Lld>

l

21 /

2r

dT)

d G

(k

2 cos

2

7) + A)G( T) )L XL

(| 9yp - 0

in accordance with (9' ) } and our statement is proved,,

This result gives us the possibility of easily constructing solutions that conform with the requirements of the Emission principle

„

It is known that vibrational equation (6) has, for example, such

particular solutions periodic with respect to 7] t

H^Ck^r) sin 2n0 , H2n^ k

lr^ cos 2n@ 3 H

2n+l^k

lr^ sin^ 2n +

9

H

2n+l(

k

l

r) cos ^ 2n + 3

where

ch l sin arctg

I

tan rj

jOn the basis of the lemma proved above, let us set up particular

solutions of (8")s
’

.

'
1_ rr o

2tt

o

o

2if

F/ 4l£l(,!i

r)cos(2n + 1)9ce2n+l

(l

I > <»L)d1

l " Ze2n+ltf’ 'W

o

If with a Hankel fmiction of even order we take under the integral

a Mathieu function of uneven order 5 and oppositely, the integral will

reduce to aero? this will be immediately apparent if formula (5>CT ) be

taken into consideration. In addition, we remark that an identity of

the indices of the Hankel function and the Mathieu function under the

integrals guarantees that they differ from aero, which will also be

evident from the asymptotic estimates that are subjoined.

Let us give asymptotic representations of the constructed functions

as ^

“=* 00 ! we shall utilize the asymptotic expression of the Hankel

functions

H,2n

( 2) + o<*)}

and rewrite (l) in the following form:

jl_ vr k^ TT

2 ^

i-kiii If e ~ -ik^r -ik^ch |

e e

£6

-ik-^ch

VoiTf

V^h l

sin 2n0se2

m

(T) , q1

) d 7] =

. hn+1 XT- * # /

2 if -ik^chl

vSTf

sin 2n7^se2m

(Tr

} , q3 ) d TJ +

2 ir f -ik^rik-, -ik^ch f

/ vVr \/ch ^

sin 2n9s e2m

(l1’ V d

’l +

2 TT -ik^ch \ + / (sin 2n9 - sin 2n T])se?

(TJ , q-, ) dTj

J VchJ

27f

°( J7T] sin 2ne se2m(7

? ’ ‘’l

5 d1

1 " Se2n( S

Since as f —

oo

th ? -» 1 ,

we have

lim 9 = lim arctg tg TJ • th J - , S->°o

and

' lim [sin 2n9 - sin 2n Yj] = 0 ;

t -* 00

here the order of this difference for large \ will not be less

1

than 0

cosh^ f

l indeed, since

£7

i 1

1 TT

lira 9 = lim arctg tg T? -

1+ 1

9 will approach 7] , and accordingly sin 2n9 will approach sin 2n 7| as

fast as “-is- or approach zero (as £ —£ oo) . “Tf e > Moreover,

ch‘"7

cos (k^ch \ ) »i sin(k^ch |)+0

6

£8

therefore, assuming that f —» oo in the preceding sum of integrals,

we may write that for sufficiently large \

:

Se2m^ = Tf Jk^rr ch |

1 / T i(k^ch § - -fei-rr) ^

J

sin2n ij s e2n

(i|

, q^) d rj

,cKTj 21

/l- Si ^ ch2

/ sin2n 1) se2nd TJ + 0 ^

x -llc^ch

£

ch.2/c

I

2ir

/ °(>) sini 2mjse2

7

J

(^ 5

and finally

( 3£)

. 2

if. for large ^ s r - ch

£ /l - be written for ch |

. V cn

|

Analogous asymptotic estimates may be obtained for the three other

Mathieu-Hahkel functions.

We may thus affirm that (i), (il), (III), (IV) represent those

solutions of equation (8) that satisfy the Emission principle

Separating (i) and the others by the formula

^ ' J2n

" iy2„ , ( 36)

where Y

2n is the Neumann function, we have, for Se2n(^ 3 q^) , the

complex representation

59

$ (Se

2 n

^ + l(Se2n)

* an<^ so ^or’^1 * (37)

#(Se2n) and l(Se2n) are two linearly independent solutions of equation (8“)j the coefficients of which have no singularities \ accordingly Se2n(v| ? q^) does not have real roots for j[ 9 since in the

contrary case 3R (Se2n) and I(Se2n) would have a common real root. It

follows from this reasoning that neither does Se 2n(^ 9 q^) have a

real root of J

.

§l5. Expansion of the Mathi eu-Hankel function . Having remarked

that

1 / 2 1 ^ -2 l

' 0 ~ r = ^ v e

3 + e

6 + 2 cos 2 7/ s

A sh£ sini) n ch f cos 7? sin 9 = -— ; cos 9 = ————

(38)

let us consider the following solution of (6T

V

)

:

H

^ + e”

2 ^ * 2 cos 2 7j J (cos n9 + i sin n9) -

» H

(2) e-2 ^ *2 cos 2T)j ^12201.i-MniUlV

= H^(^/e21 + e-2 ^ /T

= (~)

n

einl

?

2 cos 27|

+e ^ e”

2il?

e

5

' 2r

^k.

(39)

^2$ +e +2 cos

^"^ ]

v4

2 ^ +e~

2 ^ +2cos2 7]

*

Make the substitution

2 7] = TT — co

and for brevity designate

60

” R \ -I

-s-t” e

(39) then takes the

H

k

n

= (-) n

ein ^

1 -^ e2s * e

ito» _

/>e -R

2 cos 2 t

|

e'in@

in

(Uo)

p

2

-2Rp

cos

R + f

" 2 R p cos oo

Dropping the factor (-l) n

exp i nlj in the above solution of the

vibrational equation* let us study the expansion in a Fourier series

of the following;

,( 2 ) [Hn

R“ * p

d - 2 R f cos ua )lexn^

It is obvious from Fig 0 3 that

sin f sin

R‘~+/>

2

~2Rp cos

cos 0

im

P cos a> - R

R

c + /

j ‘““2R p cos a?

iw

-R

/R + p^»2Rf cos uJ

. “iw _-R

n

R + f -2R f cos u>

See editor’s Note 3

61

and it is obvious that the function

u - H^(y R

2 + f2 - 2R/> cosoj)

as a function of p

and satisfies the vibrational equation

iis + ifs 1 u -o ap ^ 3/ P^ ^ a?

as does the function

(2)/ / _2 0_ _ “7 -in0

n v

R + ” ” 2R p cos

y

e

H^“ / lv

r R

6

- + p” - 2Rp cos uj^ r(2)

n

T

-iu) „ Pe -R

~i n

- / R^+/-2Rp cos u)

Hence we see that the function

»inu) , u(2)( /t-,2 2

v - e • u " H^ l v R + p

2R^> cos u> je^n^ e ^nU>

H^2

^ / R

2 + p

2 - 2Rp cos 03

-iu? -i n

AV-W cos “*

as a function of R and w is a solution of the equation

<? 2 v1 <9 v 1 <9 2 + v , „ n

n Tr + T

+ v - 0

d R^ R 5 R

R^ $ u>

From these remarks it follows that within the circle J?| < |R|,u

may be represented in the form

s=+- oo u - E Vr) j

s

(p> eis lO (la)

S=-00

62

and that outside the circle f> <|r^ v may be represented by the expansion

s=+co

T - E C (R) K.(/>) eIS u> (U2)

s=“00

where C (R) is a certain cylindrical function whose character will be s classified below^R Hence

u - eln “’

• v - £ C (B) K (F)e

3-‘i,(ws)

. Ui3)

S=-00

Comparing (U3) and (Ijl), we obtain

VR) - c

s-n

( R) •

Consequently we have, finally,

H^H/b4 * P

1 - 2R? cos u>) » (W»)

00

H^2) (/r2 + p

2 - 2Rf cos to) e-1”** - £ C (R)J ((-) a

15 " . (1£)

Let us determine the coefficients of these expansions. Employing the

general theory of Fourier series, we obtain

C (R)J

s-rr '

i 2vr , = ~ /

H

n

2

V

r5_+ ^ " 2Rf> cos^J

iu) „ f e -R

-in

- v^+p^-2Rf COS UJ

isu) , e dcu

2fr r -t/ ‘t/5

-iV^2

+f

2

~2Rf cos u) + ^£1 it i

>(._l)

n

(e

lsai + fi)du>.

®At this point K ( ? ) is an unspecified function of p, and it

should not be identified with the Bessel function K (f). Editor.

63

where

£ R

3/2 4 M ,

M being a finite mmber.

Furthermore, expanding

r'22

V R + f

- ZRp cos u> s R — f> cos uo + € ^

we obtain

C

s_n

(R)J

s (P) - ^ e

2n+l_ . 2tt iR+ —t

tri / N n ^

l

f-r cosa3+isch

dij0 + c

2

5

where

£ 2 s 0(R”3/2

)

It is known that

27/

j ( X) . iziL /•* / ei(x cos <*) +g u)

)

s ^ir

o therefore

^s-n^J s^ z J~¥s.

-iR+ Sljti IT i

e ~

(i) s (-l)

n

J (/>) 5

hence

c

s-n

(R)J

s

( P ) = Jwn e

-1R+ (_ | +n)iri

2e (»l) n(i) sJ (f) . 5

Comparing the asymptotic expansions of both members, we see^ that

<WE> H

s-n < R)

Thus we finally arrive at — iL4 . '«"» wThe result can in fact be obtained directly from the addition

theorem for Bessel functions. See G. N. Watson's Treatise , Chapt. XI,

eq. (6). Editor.

H

n

2) + P

2 “ 2R f cos ^

ein^ - £ H^(H)J s(f) eis *

,

61*

(li6) s»- 00

and quite analogously, at

H®(/e * = ^

u(2) 2 - 2Rj? cos u,j e-^ - £ ^

H^(R) J

B

(f) e

Compare (1*6) with (I4.O) s

is a)

. 0*7)

S--00

H

n

2

^ (^T / e

2

^ + e“

2

f + 2 cos 2 7}^

in©

(-)” £ H

s-i(R)J

s

( ^ ei<sa,+n1

) ) = S——QO

- ('

)n

s 5co

( ' )SH^(^ e

j)

J

« (v e_l

)

e(n'2s)l

f i - ( 1*8)

00 = £.(-) [cos(n - 2s) TJ +

3==“ 00

+ 1 sln( n - 2s) V ^(t ei

)

J

S (t e~l

)

In case n = 2k we shall have

nw(\jjrT7rT + 2 cos 2 7j i e2k©i

k-1

S=-00

00

E

s=»k

_£_(-l5s [cos 2(k - s) T i sin 2(k - s) 7, ]H^fc ^ j) (2) (h V) , Al -5

+ £^(-l)~S [cos 2 (k - s)t[ + i sin 2(k - s ) 1/)H^2k("A e

^)

J s('Ae

Let us substitutes in the first sum, s

53 -r, in the second,

s = 2k + rj having made some simple transformations , we obtain:

6£

In case n - 2k + 1, we shall obviously have

n^k+l (t ^ + e"2

* * 2 cos 2 1

) ]

e^

2k4“ l)0i -

oo£ (-)

r

-i jfe-k

H

(2) \

r+2k+l l T e5 J _-S0

r iT e

* H

r

2) (v s!

)

J r+2k+l\T i cos 2(k + r + §) T] +

H

( 2 ) r+2k+l ft •>) 1

Hw Airj 4 i -i

l T eI V2k+ll T e i sin 2(k + r+|-)^

(50)

Replacing

7J by -Tj in formulas (U9) and (50), and combining the

two equations thus obtained, afterwards introducing new indices for

the summation

s = r + k - 1 and s = r + k 9

66

we are led finally to the following four^ ) expansione

/

2 ^ + e"

2 ^ * 2 cos 2^ sin 2k© = H,

• *’)

- »S.i(t •’)

s-k+1 l ~T

. A,-s'

J s+k+llT- e sin 2(s + 1 )Vj i

H, - J

2k \T"

+ e”^ ^ + 2 cos 2 7} J cos 2k© = H^w\jr e

-1

•) f)

^

i - r.

-j- e I +

. H.01 . i 5 .1 k+l\T e / J

k

i -S

s+k+1 V ”2~ cos 2(s + i )yj t

^kil (if v4

2

^ + e“

2

^ +2 cos 2 7)^ sin(2k + l) © » (SO*)

oo '£ <->

s=*k s=o H

s

2+llft e V

a k+l^lT e ^ ]

sin 2 (s + i)?| 5

H

2k+1 (t + e”2

^ + 2 cos 2T\j cos (2k + l) © =

00

E

(-)

s-k

S 30

H

s'k+lfe el

)

J s-kfr e" ]

)

+

+ H

s~k (v e

j)

J s+k+l (v e" J) ]

cos 2 (s + §)>] .

The uniform and absolute convergence of these series is proved

just as for the series Se

2m( f

> Q]_) c

TE

'See editor’s Note 3

67

Putting these expansions into formulas (i) - (IV), and using (*)

( %1h) and (§12), respectively, we obtain the fundamental expansions:

'!.<! . 1> £<->”" "41"’ *

!

)

s~n l T”

f

( 2

) (

k

: s-n \~T’

“W* > 9l> - |L (-)

S"n'1 [h£> (J el) J

s.n.1 (^

- Hs-n-i(v el

) '«(r e" S ) ]

*

Ze2n<! - *1> J1

<-> S-n

4s

n)

[

H^ (5= •*) Vn(v e_f

)

+

(v ef

) Vn(r e_l

) ]

+

USD

( 2 ) s-n

+ 2a^ H

o

00 ^ e2n+l^ ^ ^ ~ !C 6~)

?> (£ •;) »„($ •-*)

K2fe -0 s-n-1 . (2n+l)

d2s-1

(h -i ^

, „.„vr

B

y "s-n-iir s=l u \ \ )

+ H

( 2 ) s-n-1 <) v„(^

.- !

) ]

•'

These expansions merit attention not only because they are simpler

than any expansions yet known of functions analogous to the MathieuHankel functions constructed above, but also because they converge

uniformly (and absolutely) for all values of § , whereas Heine’s expansions converge only for a it larger than a certain finite number,

and Sieger ’^expansions, for § > 0.

^'As stated in Editor’s Preface, Sieger’s expansions are also valid over the entire complex £-plane *
•-

.

-

ss-a:

spm

mmm
68

For proof let us look, for example, at the series for Se^C^ , q^) .

The expression for the remainder of this series for s > N, N being a

large number (utilizing the well-known expression^ for Bessel functions

whose arguments are smhll in raodblus compared with the order), can be

obtained as follows;

lx',

x /, . -v i s* i . e -1

2

s

-s

J ( x) = -5— (1 + € ) 5 \t \ < 5

2

s -si x

h<

2 ) (x) - (i + e

2

) s | e

2 | < ,

(?2)

acquires the following form:

|RNl

oov ,

\ s-n-1 -(2m)

U. a2 s s=N

J - J

s+n s-n s-n s+n

y

g i^~n(2n)l

S =H (s-m)l(s+n):

2s+n(s+n-l) fe.-i)

"

s+n

2

s”n

(s-n) l

2

s~n

(s-n-l) l e~ s-n

ft 2 s+n(s+n)

For each Mathieu-Hankel function of number 2m, N is to be chosen

such that

N - m > 2n ,

(l).See editor r s note U,
- • ; •>

..... .... • • .
69

which is always possible,, since any finite number, for instance 0, 1,

2, 3 S etc«^\ may be taken for n, aid 2m is a given fixed number 0

Then (s - n)l > (2n)l, and simple transformations give

oo (2n) 5,(Cq®”

m

,Rnl < Cs-^Ks-e) '. ’

where c and n are constants independent of £ j choosing N suitably,

we see that for any |

< G 9

where £ is any positive number arbitrarily^^small

The uniform (and absolute) convergence of all series (5>l) is thus

proved.

§X6 0 Diffraction near an ellipse (Fig. U) ® Let us expand the

falling potential

f =&

iO^y+oJt) iu)t ik^hf sin 7) a e ° e

in a generalized Fourier series in terms of the functions of an ellip-

( 2 ) tical cylinder

ik_ sh I sin7| co e = £ f\cen

(i !* k

l

)cen

^ 7

] > V + Cn

sen

(i

§ » VSen

(7

7 > kl^ • n=o

^Cf« the Heine, Sieger, and other functions exhibited in M.T.O.

Strutt’s book Lame’sche-Mathieusche F^nktionen in Physik, etc., Berlin

1932, ’pp. l£-U8.

^

^Whittaker and Watson, Op. Cit..

^Provided all . exist. See editor’s note 3.
'

.

:

.

...
70

The coefficients B and C are to be determined on the basis of (29) n n Noting that

/

+lf ik-^sh f sin

e

-Tf

+Tf P ik^sh l

J e sinTj

-ir

ce2n+l^ > k

l)d7t s '° »

se2n^ V

~ 0 ,

iuo t

( 53)

the final form of the expansion will be the followings

. ikTshfsinri

UeluH . e 1 5

’ 0°

f Z B

2 soe2s(i tk

l

)ce2s

(l

?* k

l

) + C2s+l

se2s+l (i f<k

l

)se2s+l (ll> k

l

) S-0 L J

This wave , on meeting an obstacle, generates two types of new disturbances, which are characterized by the potentials J

and $ as has

been already pointed out above, they (the disturbances) must be solutions of the equations of elasticity, must satisfy the conditions on

the boundary, and must represent a wave with phase outbound to infinity

and dying out there „

In order to clarify the form in which the potentials (| and Y

should be sought, we will make use of some mechanical properties of

the problem in question e

The physical symmetry and homogeneity of all the conditions require that

Ti(x, y) s - u(- x, y) 3

v(x, y) = v (— x, y) e

We shall endeavor so to construct the potentials as to satisfy these

conditions e

(5U)

For the time being let
.

'

:.

: V ' ;

-

‘

' -
71

$

(x* y) 3 IxCx, y) * J 2(x, Y( x5 y) s *4x ( x* y) + Y2

( x> y)

and together with this 5 let

$ 1

(x J y) * - $ 1

(» x, y),

“

yP’ 1

(x, y) - - ^C- x5 y) ,

(55)

$ 2

( x* y) - l 2

(- x, y) a ^(x, y) = Y2

(~ x? y)

Then

u(x5 y) s -g- +

v(x# y) =

3x ay ax ax ay ay

Sf

_ 3|x + 3^2 9yx »lf2

ay

ra a x_ <3y 3y ” ax ” ax

(56)

whence

u(- xs y) =

v(- x s y)

JL

3 x il + ^} + Ty{fi + T2}] x=

Ty{ii^2}--^fr1

+ y2 ^

= -X

y = y

X--X

y =y

(56*)

satisfying the conditions (5U) from (56) and (56'

r ), we obtain:

3$i _ «Y2 _ 3 %

d x

“ "

a y

1 ay"1 ax 5

and accordingly

^(x, y) + if-jU, y)

is an analytic function of the complex variable z = x + iy and ^2

and are thus solutions of the Laplace equations
.

'

•

• •

•

• ' • •

.

'

•

-

; I
A

2

¥2 " 0 , A

2

f x

- 0 .

On the other hand the equality

$1 + k

l $1 = 0 5 a

2 % + % = 0

must hold,, whence

I^Cx, y> = Y2

( x^ y) 5 0 s

thus, desiring to satisfy (5>U), one must put

$ (x, y) = §2^ x? y) i.e,, an even function of x , ¥ ( x * y) s y) i.e., an odd function of x

Now noting that

72

se2s^

r)* k]_) ~ "" se2s^^ ” k

i^ 9 se2s+l (7l® ki^

~ se2s+l^ ” 9 ^1^ 9

ce2s (1

? ? = ce2g (^ “ If] , \) , ce2s+1 (7), =-ce2s+1

(ir - Jj

, k-^ ,

we conclude that (| and ^ should be sought in the form

I (? , T\ ) -

00

E [a2s

Z e2s (? 3 k

l^ ce 2s^ J k

l^ + b

2s-KL

Se2s+l^ » k

l^ se2g+l^

s so f (f , T)> -

00

B ^ c2s+lZe2s+l^f s k2^ ce2s+l^ 5 k2^ + d

2sSe2s^ 5 k2^ Se2s^ J S=0

(£7)

9 k

l) ] m

k

2

)]

f

(Here we have not written factors depending upon the time, for the sake

of brevity.) One is readily satisfied that the boundary conditions of

our problem have, in the new coordinates, the forms

73

r

1 h + 9j_ *£SL' = 0 : 3 If +u _ Jl at <3 t}

J f i o

3 rj J l m

i o

Rewriting these conditions by means of (S>3) , (5>7) and (£8)^s

co£ B

2s

ce2s^o* k

l^ ce2s^ J k

l^ +

^2s+lSe2s+l^o3 k

l^ se2s+l^ 3 k

l^

+ S“0

+ ^S20

2 s ^ o

3 k

l^ + b2s*lSe2s+l^o3 k

l^ se2s+l^ 9 k

l^ +

+ c2s+lZe2s-KL^ o3 k2^ ce2s+l^ 3 k2^ + d

2s

Se 2s^ o3 k2^ se2s^ k2^ = 0 3

JE®2sce2 S

(i ^ h^U^’h) + 3

2S+lSe2s+l

(i

*o> V +

o

+ a2

sz®2 s(f o’ kf

oe2s^’ + b

2s+lSe2s+l^o> k

l^ se2s+l^’ kf

" °2s+l

Ze

2s+l ( ^ o> k

2

)oe2s+l (7l> k2> * d

2s

Se2s^ o> Vse

2s

(l

? » V ’ 0

As will be shown below3 all the series met with here are absolutely convergent^ and a rearrangement of them is permissible 8

Using formulas (30) , the following formulas are readily oboo2

3-0

tained

oo

s=*o

00

^^s^s^o3 k

i^ se2s^ 5 k2^ = E ce2s^ 3 V 5 ‘W^s

^ Se2r^oJ V * s=o r—

x>f b

2s+l

Se2s+l^o> k

l

)se2s+l (11 ’ Kf

(2r),

00

£

s=o

E ee2s+l

^ 9 k2^ E b2r+l4s+l >>Se S"0 “ ' 2r+l^o3 V

00

£

r=o

(2r+l),

( 1) B

2s

B

2s

* 1 1 C2s+1 C2s+1

° 1
•

1

v.
7h

00 00 °° / 0 \ £

a

2s

Ze2s^o*

^ = se

2 s ^9 k

2^ J£q

Ze2r^o*V

00

£ c 2s+lZe2s+l^ o» k2^ ce2s+l^ 5 k2^ ~

ooE se2s+l^ 3 k

l^

00

I^O

(2r+l)„ /£ c

2r+-ln

2s+l Ze 2r+1 ^ < k2l

Moreover^

oo

? oo OD

^ 2 k

l^

ce

2s^

k

l5 ^ = 2 B

2sce2s^

i

^oJ ^ X “2^

s^o • s^o k*o

se2AV

“ E se2k^» k2^ 2 B

2s

m

2k ce2s^o* kl^ k^o s=o

oo oo , = £-2s

^

k2> 2 B2rm

2s

ce2r(i ?o^ V * s-o r-o x

and, analogously,

oo _ , 2 ^2s+lse2s+l^'

L ^

5 k

l^ se2s+l^ 5 k

l^ s s^o

00 CD / p +1 V 2 ce2s+l^ 5 k2^ 2 C

2r+lq2s+l ce2r+l^o* k

l^ ° s”o r=o

Employing these equations in the preceding boundary conditions,

collecting the coefficiaits of identical Mathieu functions and equating them (the sum) to zero, we obtain;

?5

^2sce2s^o> k

l^ + a

2s

Ze2s^o>

^ +

00 (9^

+ E d

2rp2s

Se2r^?o> k2^ = 0 3 r=o

^s+l^s+l^o* kl^

+ b2s+lSe2s+l^ o> k

l^

+

r?0

C 2r+lm

2s+l"

^ Ze2r+1^ 0

* k2^ ~ 0 1

0° /p X E ®2rm

2sr ' oe2r(i £o»

k

l

) +

CDE

r-o

(s = 0 , 1 , 2 3 *•• ,

)

oo , X + E*2A Ze2r ( ?o> V. - ^s^Wfo. V - 0 5 r=o

ooS^2r+l q2s+l

^ se2r+l^ o> k

l^ + r=o

E b

2s+l q2s+l

^ Se2r+l^o5 ^ ” c 2s+lZe2s+l^o> k2^ “ 0

oo + E b

r-o

•( 59)

Sly. Proof of the uniqueness

Let us snploy the notations:

B

2sce

2s

l' i

^o5 kl^ 5 E

o S

2rce2r(i ^o>

a2s

Ze2s^fo5 k

l^ = *2s 5 d2s

Se2s^o* k2^ s J2s 3

( 60 )

and in the first and third of equations (£9) let us withdraw from

under the 2 sign the summands for r = s (the main term) , noting this

with a * on the 2s
/
76

o’

k

l^ * p2s^ d2sSe2s^ o

s k2^ "

" h

2s “ E

q

p2s o*

k2^ 5

" ^ Ze2s^ o s k

l- + d

2sSe2s^f o 9 k2^ =

(61)

Hence we have

Ze2s^ o*

k

l^ (2s) X

2s Ze

2 s

( f o’

k

]7

P2s 72s ”

" • l' P2s

r)d

2rSe2r( f o’

k

2

) 5 2?“0

„(2s) A

Se2s^ o ,k2^ " W2s S^Tf^T J2s

=

®2rZe2r o>

( 62 )

77

Let

2 s Ze

Ze2s^ o

jkl^

2s^S o

jkl

}

Se2s^ o*

k2^ (2s) (2s)

Se2s^V + ^

P2 s (63)

Then from (62 ) 3 with the aid of the notations (60 ) 5 we obtain

y2 s A,2s

X + ff2s rn (2r)

2s Ze

2g

Ze

2g rQ

h ^r

, (2s) (2r) Si _(2r)

Tr bas^s ~ ^s ^ p2s y2r

rso

Let ns moreover designate z2s> y2s

= z

2 s+1 * then we

have the system

78

Z

2s * A

2s

(2s)S'f2r) , Se

2s S' (2s)

P2s ^“28 2r Se

2s £o P2s Z

2r*l

2s

h,

Se

2s v n(2s)

^s Se

2s

,t2sp2s

'2s-KL A

2s

Ze

2s (2r)* (2sj

ZeT" 1 “2s

S2r * ^s

GO

2s r=o r=o

2s 2r+l

>(sO,i 5 2, - • •) (6U)

a2s 2s 3S£ +

¥e shall prove that

oo. 0 <s

i<' 1 - .Elhkl

1-0

<e

2

5 €

1»

€

2 > °> ( k = ° ? 1, 2 5 ••• i - 1, i + 1, •••)

A

ik being the coefficients in the k-th equation, 6 ^

being finite

numbers independent of k.

Copying in somewhat more detail, it must be shown that

0 < Si < 4*

(tf)
'

. 4*. : •• t

'

• f

s

i -

;

. •' "i
79

C 2r) ( 2r) In these inequalities > p^ s have values determimable from

formulas (3l)

Comparing the Mathieu coeffieients from formula (“) with their

expressions from formulas (li|)-(]5'

r

) 9 we find

a (2r) = 2k

,(2r)

r-k

P^

2r)

' k-r

r > k.

r = k« a

r < k;

Therefore by formulas (22") and (23") we obtain the inequalities:

(2r)

2k

;(2r)

'r-k r > k.

r - k 3 (66)

s(2r)

-r (3^;:/ r < k

for r > ki |a^r)

| - |a( ^(ql> | < ~

l^q”^ (

+k“l ) 1 (r-k) l(2r-l) l

for r < k: |a (2r)

] - |(3

(2l

tq ) lor r <k. I < 3k?L^!liL_ .

I a2k I I ( k-r^V I < (^r) l(2rHc-r)

39 l

ql"

r(2r)!

T 3

+<r /

ce2r(ql , » q2

)d1I " - S 2k a2k

r)t

^-)i2k

S)(<}

2

) -IT

00

l

k«o

(67)

( 68 )

/

se

2

s

(l) , q2 )dT| = S^a2k

S)(q

2

)]2 ’ U k=0

?? 0, (2r)/ \-(2s)/ \ ^ a^2r)/ > 3 ^2s)/ \ 2 2k ^

,E_ 2k ®2k ^l'^k ^ q2'

kr)_ ** ^ **

. (2r) .A

^s oo \ «

J q 2s % r— (2s) ( X-,2

E [a2k (q2

)]

k=o

V r (2s), u2

E ^ a2k q2

k=o

80

Let r < sj then

l4

2r)

l < x 4k

s)

k=o <v

(69)

h*4lr)

4l

s) + 1

2k4k

r)

4k

s) + \

2k4k

r)

4k

s)

Let r > s', then

(2r)

£ 2ka^

2r) I(2s)

k=o 2k

r

L

k=s+l

2k

(2r) (2s)

^k 2k

oo

" E

2k

k=r*l

(2r) (2s)

^k a2k . (70)

Let us by turns employ formulas (6?) and (68) for the estimation

of sums (69) and ( 70 ).

1. r < s:

f 2k |(2r)-(2S )|< f ^l^"k(r+k-1) '-^2qr

k(sH-k- 1 ^

k»o

' <

.^Q

Zk

( r-k) I { 2r-l ) l { s-k) I (2s-l) 1 <

r q^

S 2k

(r+k-l) l(s+k-l)

<E 2k

Tr-k'j iis-krrr^^i

)' kss-ttt 5 K"”0

on the assumption that ^J> *il ^

*^2 ^1 > ^

~ ,(2r)-(2s),

2k3Sq

1

(2r) l(s+k l)

p* l^k a

2k 1 < ^ir-rjuwius-kyi^s-iyi 5 (72)

where fB > 7J ^

oo / ?

n /« % Aq

2k"r“ S (2r)'.(2s)l

E 2k l^lc a2k I < E

2k

( k_r; Hk- s

)

» ( k+r) »Tk+sH

k^s+l

A >*$^2 • (73)

81

=#2. r > ss

« , (2r) (2s)

s

- ‘aqr

s"2k

(r+k-l)l(s+k-D*.

2k l*2k a

'2k I < k§0

2k

U^k)U2r-l)U&-k)H2s-l)'. 5

§ a i4k

r)

4k

s)

k=s+l K

r 2k‘B r"

s (2s) l(r+k~l) l c J

1

k=s+l (r-k) *(2r-l) l(k-s) i(2s~l)

(7U)

(7*)

00 , (2r) (2s), S AqJ

K"I

"‘ S (2r)U2s) l

k=5+i

2k a2k a‘2k <

k^i

2k T^::5TITk::^^ (76)

In inequalities (65) 5 which are to be proveds it is necessary to

estimate expressions of the form

oo 0 m, r=o

2s

It) (r - 0 5 1, 2 9 «»• s ~ 13 s + 1, •••)

Let us distinguish two cases,, r < s and r > s:

S"1

(2r)

00

£ ”4s

+ r=o r-s+1

£ . ”2

(2r)

In estimating the sum

(77)

s-X

m, r~o

2 s

we shall use inequalities (?l)""(73) 5 and in estimating

® (2r)

L »2 S r®s+l

we shall use the inequalities (7U)-(73)

The following sum must thus be estimated

00

, (2r)

S"1 r 2kOJ<q^"

s l(s+k-l)l

Ks

I <

JE

q

Ti^k )

* ( s-F)T(2r-l

)

'» (2 s-1 ) 1 +

I**0

82

s-1 2k#q| r(2r) l(s+k-l) l

r?0 k=£-l TF“fTTTk+r) l ( s-k JTT 2 s-I .) I +

s-1 oo 2kAc^

k“ r”s (2r) l(2s)

+ r^o k=?+l Tk-rJlTk-s)'! (k+r ) i (k+s ) 1 +

oo s 2k Vi qj

*s~2k (r+k-l) 1 (s+k-1) 1 + k?0

(78)

oo r 2k$q-ps (2s) i(r+k+l) l + r^+l k=?+X (^k)H5r-l)lCk-S )'.(2s-l)i

oo oo 2kA.q^

r”s (2r) l (2s) l + r=?+l k=r+l (k-r) l (k-s) i(k+rTl'(k+s) l

The double sums in th$ right part of ( 78 ), for |q^j less than

unity, are easily estimated^ without exhibiting all the intermediate

transformations, we shall show that

®. ,j2r), ql (M+2S« 1

+l4Ae+2?J'D

3

+2a©

u

) i

o Ks I <

^

- h

> (79)

the constants being

M * V

j £0 U=k)i(s-kJT >

* lira 1

i

r+X 2X-1

TT '2r+T * 00

s ^3

k?Q

(r-k) sl s-kTT 5

9. lim (2X - 1) . »oo
'

’ ‘
83

In like manner it is shown that

f <• l P2s

r=o 1-

*L

where g is a constant bounded number independent of r and s

Using (6£), (79) and (80), we write

(2s)

P2s

A2s

00. £ 14

r=o

(2r) 1^1

(1~^)

Ze

2 sSe

2s m

(2s) + "T2sT Ze0 Se

2s

'2£> Os 2s<-* O

(i-qp

Ze

2 s . 4s

S>

P2

2

s S)se

2 s

w

CM <8

<

(80)

( 81)

It will be proved below that the denominators of the right members

of the latter inequalities are greater than 1 for any s e

Therefore (6j?) will be valid if q-^ be chosen so that the inequality

holds, i.e. s

(h+g) qi

‘<*1 < € £

2 > 0

( 82 )

Let us pass to the investigation of ^2s a

We apply the formulas for the differentiation of cylindrical functions?

4 cs (x) = - 1

C

s

(x) + Cs-l(x) 5 4 °s(x) = I

C

s(x) "

°s +l

(x)

8U

From (£1) we find

s=l

2nH^J a. s-*n s+n

+

(83)

s+n

^

s-n-1 s-n s+n-1 J _ - J 7

( 2 ) (For the sake of brevity, here and henceforth the arguments of Hr ' and

J are not cited explicitly, but it is throughout understood that |fk^e^

( 2 ) - £ stands with H' , and ^k^e s stands with J .)

The uniform (and absolute) convergence of the series (83) is

proved just as for series (£1), from which it nowise differs in

essence.

Series similar to ( 83 ) msy be written for the three other MathieuHankel functions too.

In future it will be necessary to estimate a lower bound for the

modulus of the quantity

J ql^ #

We have

^ e2m^ * Q}) = (")

m-n-l/ H

(2)

ra+n m~n

n+m m-n m-n m+n J
'• '

• ’

:
85

For § = where | is a real positive number or zero, the

expression

i -!

m-n\T e Vnl'T e

)

Jm+n( V e

I -fV(2)/1 m-n\

v T”

y

m-n

y

; €1 jm+n

L

^m-n u m-n

cannot be equal to zero for n / 0; indeed, since f 0 ~ 0 and < 1.

we have

k

l ~^o

0 < < 2, U3

and since in the interval (0, 2.1j5) none of the Bessel functions has

a root,

Jm-n(r e

f

°) 5

moreover

H h !o

m-i-n \T~ + 0 ,

since the Hankel functions do not have real roots, Therefore even the

difference

„( 2) j m+n m+n

-m-n m=n

for real f ,

The convergence of the series

00

1

4 0

_(2m)/ (2), _ „(2) , ^ 2s ( s+n

J

s-n s-n s+n
c

*

.

'

.

. •-

:

• V " »
86

was shown above. Accordingly the series

Q

2 ? (2m)

H^Js-n-HL

2

nVn

A 2-

' m+n m-n m-n m+n

also converges absolutely together with it. Let us denote try the

ratio of the largest in modulus of its denominators to q^

S then.,

using the inequalities for j ^ 2

3 we sha-*--*- have^

00 . 1 2M, q-, K\<*l £%

|s"m| < •

Let

< T+2IT ( 85 )

Then, from (8I4.)

jse9 (f , q,) I > [H V “ ; J - H J .

1 2m') 9 ^1' 1 1 m+n m-n m-n m+n

2M

l

ql

1"^ ( 86 )

Because they are completely analogous, we shall not write out

here estimates like (86) for the three remaining functions M - H.

^Editor’s note: The above is a literal translation of the

Russian, The author’s meaning is perhaps better carried as follows:

Let1 T denote any of the denominators in ( 81jg)| i.e„,

s

h

(2) j m+n m-n &*m+n

Form the sequence

|s-m| = P

and let be the largest in modulus of all jP s

|«
,

• •

T

' ; -

' " r

' U/-

:

' Vs

- V
8 ?

In consequence of the absolute convergence of series (83), the

series

r(2) Vn- H

m-n Vnl £

00

£

(2m)

2s

j

)' iwn m-n m-n nw-n

2n /H^ 1 J + J ^ - i s-n s+n s+n s-n r

eS Js-n - HE-1 J s+n,

e“^

( J - J 2 V s+n s-n-1 s-n s+n-1

(87)

converges absolutely.

On the other hand, we remark that this series depends on the num-

( 2m} ber m only through the coefficients of the Mathieu function, a^g , and it is evident from formulas (66), ( 23*) and (22^) that beginning

with some finite value, thqy cannot increase with increasing mj accordingly a positive number M2 , dependent on m, can be found, such that

I Tf Ss2m( i , V I < q* |H^ J

m_n

- JB+n | . (87b

Analogous inequalities also hold for the three other Mathieu-Hankel

functions.

Let us choose kl so that

Mglq-J < 1 . (88)

Finally, it is evident from (U?) that

2s+A . (2s) _ 2s+A W l . > P2s 5 ( 89 )
r

*

**••• « • • »•* *• » • 4

'

; v

; V
88

A* B* A1? B

1

are quantities which* given a choice of q^ greater in

modulus than 1, are dependent on s through the Mathieu coefficients*

and accordingly they do not increase together with s, beginning with

a certain finite value of s, Taking into consideration now (86) * (87*) and (89) a let us consider* beginning with a certain s sufficiently large* the expression

cd . J2s) + _1 Ze

2s

Se

2s

1 ‘ S Ze

2sSe

2s

If

Ze

2s

Se

2s is a negative quantity* then | u>

| >1; if* howP2 S

2s 2s

ever* it is a positive quantity* then

"*il > h

+ 2s+a

i p

2Mi

qll

TPBf L

1"

l-<3l J

| > 1

One may obviously show* analogously* that

Ze

2s .

14a

S)p

2

2

3

S)s8

2 s Z _ J 2s Se0

> 1

These inequalities prove the validity of inequalities (81) 0

Inequalities (65>) provide proof of the uniqueness of the solutions of system (61*) * given all the assumptions made regarding Jq^j

[see formulas (82)* (85>)* (88)], (it is obvious that these assumptions reduce to one inequality that bounds jq^| from above,)

Indeed* let system (61*) have two different systems of finite

solutions | then the corresponding homogeneous system will have solutions different from zeroj we shall show the impossibility of the
v
89

last assumption if inequality (6£) is to be observed* "Whereupon the

uniqueness of the solutions of the nonhomogeneous system ( 6U) will be

proved „

Assuming the existence of non-zero bounded solutions of the

homogeneous system corresponding to (6U) S the cases must be considered

where the solutions have a limiting point (exact upper bound)

Let x^ be the exact upper bound of the system of solutions! furthermore, let

be the n=th root, in number, of the homogeneous system! let us consider

the n-th equation

0

and let us demonstrate that such an equality is impossible. Indeed,

QD

1-0 X

But since

OD oo, „ uu X’>inx

i

4 X UinlKK*" X' lh„l

1=0

00

T

1=0

00

1**1 > 1**1 X' IhJ

l-o

to a finite number, N may always be chosen so that

•* N

i oo, r

>x*X IhJ > 1=0

and the more so will
• . •
90

£ , oo X

and consequently the equality

1=0

is impossible and our assumption falls.

We are thus led to the conclusion that the homogeneous system

cannot have solutions different from zero,, i.e., system (3>9) has a

unique system of finite solutions.

§18 . Proof of the existence and solution of system (61+)

Although system (61+) may be solved by the method of Schmidt for

numerical computation it is more convenient to employ the method of

successive approximations.

To show the possibility of its employment, let us subject systsn

(61+) to a preliminary transformation, writing it in the following form:

ooE * Ck = 0 , 1, 2, oo) (61+bis) r=o

Together with this, we employ the symbols:

l 2i c

(2o) .3

( 90*)

A

1

2i+l,2j ~ A2i

furthermore

(1) Riesz. Equations lineares, etc., Paris, 1926.
1

>

' ...

'

•

. " : •

' • - - . • .

:

*
91

A

2i,2i

“ 1 * A

2i 5 2i+1 0 ^

2i A2±

Se;

"21 sS“ ' xz± Pa

(2i)

i

°“2i+l " hi

y

Ze2i . , (2i)

*2i Za^ * ”2i

Let us estimate the order of diminution of the coefficients ( 1)

°~2i s I2i

**21+1

= t2i

, Se2i y

2i

ri2i Se^T ~

^21 p2i

y

Ze2i . . 2i

^2i Ze^T **21 “21

h

2i

“ ” B

2i ce2i^

1

^ o> k

l^ 1

oo *2i “ E B

2r oe2r (i

i o> k

l> *fe r=o

2r

i

(90)

Apart from this, from the expansion of eik^hl sin1l in 58 we

find

2^ ik-,shl sin*)

/ e ce2i (7] ^1

)d7)

•D — O

B2i TflT —— , ce2i(i$

o3

k

i) / [ce2i (7|, 1^)] dTj

whence

l%l <

2tr /

ik^sh^ sin 7] ce2i(7) s k

1

)d7/

( 1 ) Superscripts have been dropped for the sake of brevity.

92

integrating by parts , we put

ik^shf sinT|

u | ce2i(~*7 , k

1

)d rj = dy ,

ik, sh £ sin -q dti = ik^sh | cos Tje d7] | v " /

ee2i^ » k

i) d7) =

/• ® 2i

= J

£ ag cos 2k7j dl} - ^ 2i

°o oo ap,

kS0

d

2kJ

cos 2krl d ^

= + 7F sin 2k77 5 k^o

whence

V =4(Ji

where A(Tf ) = +

oo 2ie* 2i

2k

k-1

sin 2k T] e We shall prove that A (Tj) is a

number bounded above for any 7] of the interval (0, 2Tf)„ Indeed,

oo

|A(J1 )| < E £| 4 sin 2k n I < E 1|4| < 1

k=o 1 1 k-1 1 ’ % 2i

i-1 Uqi ^(i+k-"l)l • 00 2L ’L (2i)

5, ci-k)t(2i-m' k

+ .1

k-i.

k=l k=i+l

i-1 00 ^I^i”

1 00 EqJ

1

^

k=l KA1 KJ ' M+l KVK 3

f-j_ k ji-k [ J

, (E = const)

which proves our statement,

B

2i

Returning to the partial integration, we have

2tr 1

" 21

o o

ik18hfsLnii(l))

5 2i

o

Let us integrate once again by parts, putting

ik-^sh | sinp

cosTje doq

ik, sh | sin T) ik

x sh^cos^e = u ,

- A( >} )d>] = dv ,

93

du 2 .2 f 2

ikl

sh ^ sin1l sh $ cos 7) e ik, sh l sinrje

ik^sh^ sinr]

d7/ 5

•/

oo v = / A(7) )d7)

k§L ^ ^

/

Sin 2kT' d7>

00 - Z 41 oos 2k1i > k-1 2k^ K

whence

v = H( 7? ) “23“” ?

where

oo . 2 ?

. H(7? ) = - £ a

p

?; cos 2k r) k=l k

d

We shall prove that H( T] ) is a number bounded above for any T) of the

interval (0, 2 If). Indeed,

"mi<U|4

i-k

< 1 + i-1 %|q-7 (i+k-l) 1 /

. \ 2

k=l Ti-k7iT?I-Y)'r VI

® 45 1q^

1

(2i)

1 ( t \Z

„ |i-k| + l ———r i

< 1 + II Ec

5i

1 ^ k=i+l (k-i) l(k+i) l \ 1 k=L

Q „E .D .

Returning to the partial integration, we have

1 t ®2i

= “ 2T dki

cos r\ eik^sh j sirnrj ^ ^

+ ik^ sh |

sin K])e

21

ik-^sh | sinij

2lf 2-rf f ^ k

i

sh^I cos

2

7] (2i)

6 *

H( t

,| ) d T)

9k

Integrate once again by parts, putting

2 2 . i^shf sin T) (k^ sh £ cos 7} + fk-^ sh| sin 7} )e ~ u , H(tj )dTj = dv ,

p 2 du = [(- 2k£ sh

^ cos 77 sin tj + ik^ sh^cosT]) +

+ ik^ sh“^ | cos~ yj - sh~| sin 7] cos 77 ] =

3 3 ^

- = [ik£ sh^j cos" 7] - 3k^ sh^ £ cos T] sin 7) + ik^ sh ^ cos 7] ]e

2 , 2 ik£sh| sin 7}

_ y

>H ^ 7l ^ d7

J ~ “ (k) a2k J

C0S 2k1f

l d7) " " JC_

oo .2 0 . i 2i . 01 „ a9,sm 2k 77 ,

whence

v =

fitt ) ~S±~

where

oo i (7 ) * - (|J 4i oos 2k i •

We shall prove that A>( 7] ) is a number bounded above for any 7] of the

interval (0, 2 IT). Indeed,

12 « ' st(C

i-k

q1 (i^-1) ' 00 /

1 U-k)t(2i-Di

+ k £+1 \ e I i^ntiknrr +

v -1

0 (2iH

1<1+ E Eq1

ti'k l , k-i

Q .E J) .
.

'

.

'
95

B

Again returning to the integration by parts , we hare

1

21

(2i)‘

(2i)‘

,„2 .2 f 2 , . . ik^shf simj 2(T))

sh 4 cos T| + ik^ sh £ sin yj)e - -^1 -

2-Tr

2tf

^ J

[ik^ sh^ | cos’

J y)~ 3k^‘ sh^f sinrj cos tj o

+ ikl sh foos 17 ]e

ik

l Sh5Sin,, 2(^ .

Let us apply a final integration by parts, putting

3 3 3 2 2 [ik^ sir l cos^7| - 3k^ sn | sinTj cos i) + ik^ sh | cos 7)]e

ik^shf sin 77 = u

2(T))dr| = dv f

du ~ [”6ik^ sh^ ^ cos^Tj sin T) - ]jk^ sh^£ cos^ Y| +

P

o 9

ik.. sh \ sin T) + 3k-^ sh f sin 7) -» ik^ sh \ sinr) - k£ sh4

f cos 4

"Hie ,

v f e(y)dT) - - 4k f

sin 2k7> Jl ~E a2k

003 2k 1 5

whence

3(i))

where

00 A \ ^

( 3) = £ ( g) 4k

cos 2k T) . k

We shall prove that S ( 7|) is a number bounded above for any 7]

indeed

96 „ i-1 ^qJ-"

k

(i+k-l)i r \

<

k5T Ci-k>U2i-l)r ( £

s ^a°1(21) 1

k4i Tk^iTUk-i) 1 1

+ 1 < 1 + J Eq00 Ml

k=l

which proves our statement, Consequently

B

- 1

21

(21)3

2rf

(2i)

o o

2iT

—•jj /

[6ik^ sh^f cos

2 7)sin 7] - IjlcTsh

2

f cos

2

r/

? 2 ? , ) )

« i^shfsinh

* sin^7? - ik^ sh | sin 7} - k^ sir ^ cos

4

7}]e SC^dr) 3 + 31^ sh f

whence we obtain the following inequality

x

(91)

where is a constant dependent on f and k^ 3 and is independent of i<

On the basis of (88 ) } we have from ( 87)*

)u 1 *1 l^i^o^lM „ 1

1

(2i ) l(2i

)

2

(2i)

2

, v , *1 £ l“2i0e2r(1?

o> k

l

) l 1

On the basis of (89), from (86) we obtain the estimate we have

been seeking for the free coefficients of system (6I|bis)

S 1 L u

Se21 2i

1 si 1 < TT7 l A

2i'

h

2i Se2i

+ e2ip2i

(21)

h

?

^i+i 1 <

I

2i

[ Ze2i . 2i

_

1 621 Ze2i

+ V2i

(92)

. 1

. (2i) C

97

An analysis that is quite like this one shows that |cr-jJ and

| o-q J < —jjS in the course of this we use the properties established * o'

2

h- above that

= 0(2i) 5 = 0(2i) %

Se,

'2i

Ze,2i

Se 2i Ze,2i

are finite numbers for all i»

Let us now pass on to the construction of the solution of (6i|bis)

by the method of successive approximations.

Let us adopt, for approximate values of the roots, the following

quantities %

zk

= ^k 5 4

^ = ^k " £ *ksZ s

^

s ^ = 1? 2j s=o

the prime on Z indicating the omission of the value s = k„

Then

and consequently

or

o o «
•: ; ; ,0- f ...
98

Thus

,(D

ck

1

.(2) (1) (1) - [A k

o

z

ox/ + + ••• - *kk-i4-i +

+ A , , + . . . 1 kk+1 k+1 J 5

,(3) „ z

(2) . _ fx (z \\ (2) _ (1) A , (2) _ (1)

k V 0 0 V i i

A

k "k L/'k

o

Vi> o ” "o '

' ''kj^l ~ "1

.(3) ’’’ - 4-1 * (4-1 - 4-1) ••• ]

whence 5 since

zp^ < -i-1 5 and zP^ < —

—r for k & 1 . 0 ? k

(

k

) U

we have

kk-i (&i ~ 4-iJ

* A

kk+i ^4+i " 4iij

'+ )

i

h) - 4

3) - - [A. (43) - 4

2)

) - A,. (J3) _ .(2), +

> ( 93bis)

[s4

2) -

«t

1)

i < r I \s4

1)

1 < Jr £ l ^ks i <

^

j p < 1 : p - const ' s=o

P is a constant and does not depend on k, therefore

oo r,(3) ,(l)l [z < 1 y’ I X 1,1

k - zk J < -TT L I A

ks I C , S“0

r,(W _( 3

1

) i .1 v* I X I < -i

k

3k

J ^2 A 1 Aks^Ut

s*o 2V

and so forth j

consequently

99

i p

i* p-1 ' ( 91*)

We shall now show that z^, determined from the absolutely convergent double series (93) , is actually the solution of (6Ijbis) .

To accomplish this let us add the equations of (93bis), effecting

the summation by columns in the right member,, this being possible in

consequence of the absolute convergence of the double series (9U) ob~

taineds

cr.k e'

4“ * <4a - 4“> • <43> - 4^ s=o l + o

00 (95)

o". ^ ^kszs

9 s=o

or, since - 1, from (95)

oo

s-o

^ ksZ

s

" °~k 9 (k « 0 5 1, 2, ••• )

i.e., we obtain system (6i|bis)f this proves,, then, that series (93)

actually gives the solution of system (6!ibis) . Now if we rewrite

system (93) and (93bis) in the coefficients and we shall

finally have

oo, oo, oo, r

s==o k =\- Z ^ks^s * Z Z

I TM t

s— o r®o

(k = 0, 1, 2, •••) > (96)

OO. 00^ 00 |

S-o

Z

T^O

l

P“0

Z Vp + © • •

100

This series does give the solution of system (61|bis)| the coefficients

A. are given by formula (90“).

Reverting now to formula (I{5), we find a

2 s and dg

s which figure

in the Fourier series of the reflected potentials:

z.J 2s 2s+l

a2s

“ 5 d

2.. ' Se

2s( k

2 , f 0

) (97)

In obtaining formulas (97) s we have proceeded from the first and third

equations of system (5>9)j it is scarcely necessary to emphasize that

a quite analogous consideration of the second and fourth equations of

system (5>9) gives

J 2s+1

J 2s

J 2s+1 “ Ze, W ’

; 2s+l Se, (98)

where zf is determined by a series analogous to (96 )

,

and the modulus

of is estimated by an inequality analogous to (9U)

Entering ( 97 ) and (98) in (£ 7 ) and (£8), we obtain

(M)

- Ze2s^»V ®e2g+3_d jc. . . 2s

oe

2s

(l

? ’V + Se2s+1( S0 ,V

Z2sse2s+1 V • »

(99)

Yd

(100)

£

Ze2a,l(^2 ) V „ .. Jo «£&&& 2s+1° e2s+1

^ ’ 2> Se2s (f^)

22s+lse2s( ^ > k

2

)

101

The exact evaluation of series (96) will probably indicate a z. of

such an order as regards k as will guarantee the double differentiability of series (99) and (100) with respect to J and *] „ We shall leave this question open for the time being „ However

,

if the application of Fourier 5 s method to the diffraction problem be

considered as possible a priori , the question solves itself , since the

existence and uniqueness of the solution have been established by us

given the assumptions made regarding the smallness of q^ [see inequalities (82) , (8$) 9 (88)] „ On the other hand, since by (12)

where <*> is the frequency, the speed of propagation of the incident

disturbance! the smallness requirement on Jq^J is equivalent to the

smallness of cu or to large values of these two tokens are, as we

know, properties of long waves a Thus the picture of the phenomenon of

diffraction obtained by us above holds for long waves where they bend

around an elliptical contour.

§ 19 » The diffraction of elastic waves near a segment of finite

length e The fundamental functions of the problem that were constructed

in the preceding section, and all their expansions, as has been shown

above, preserve their definite meaning for f

- 0„

On the other hand, setting f

38 0 in our formulas, we shall by

that fact be considering the phenomenon of diffraction near a focal
.

-

•-

.

'

* X

.

. H

'

’
102

segment of the family of ellipses and hyperbolas considered above*

This is also the problem of diffraction near a segment of finite

length*

It is wholly obvious that a complete solution of this problem

is given by the formulas

oo Ze

2g (f 9 ce2s^ k

i^ +

9

® 2e2an(C . V

lo Ze2sfb0

> V

') S

f \ to, VXT

a2s+l ce2s*V7l> k2' +

Se2s^f > kz\ „ ss2bto; k

2t *2sh se2 8

(1

) » k2> »

which are obtained from (99) and (100) by replacing | 0 by 0 in them*

CHAPTER III

METHODS OF OONSTRUCTION

§20. General observations . While the method of curvilinear

coordinates may theoretically be applied to a wide class of contours and surfaces , the methods expounded in this chapter are distinguished in this respect by a more particular character; these

methods, however, possess a far-reaching effectiveness and offer

the possibility of an exact calculation not only of the qualitative but also of the quantitative side of phenomena.

On the other hand, the ideas lying at the root of the methods of construction expounded here are still far

from having been exhausted, and will

doubtless serve as the source of numerous generalizations. Some of these

will be indicated below.

We shall begin with an exposition of the idea of Sommerfeld,

who, using the Cauchy integral over multi-sheeted surfaces, constructs multiple-valued solutions of the vibrational equation

that have the necessary properties at infinity and acquire the

required boundary values.

Imagine in space a point source of electromagnetic disturbances that are distorted by the presence of a screen S with contour C that occupies part of the plane xz (Fig. 6). It is moreover assumed that the screen has an infinitely great conductivity.

z

Fig. 6

101*

This implies that the components Ev

and E are equal to zero on

the screen. At the point P the spught E and H will have a simple pole; at infinity the Emission Principle must be observed.

Using Thomson’s well-known ’’Reflection Principle," Sommerfeld

constructs a point P’ representing the mirror image in the screen

of the point P.

Then if u(x, y, z, P) is a solution of the equation

Au + k

2

u = 0 , (l)

having a simple pole at the point P, the function

v(xs y, z, P, F') = u(x, y, z, P) - u(x, y, z, P’)

will satisfy the required "boundary” condition on the screen. However, this function has, apart from the point P, still another polarity at the point P’, which does not correspond to any real equivalent in the physical picture of the phenomenon.

Let us therefore conceive of our space as "double- sheeted ,

’’

and for which the screen S represents a "branching surface"; let

us make an incision along the line C and so connect the two "regions" of our "double- sheeted" space that the different semispaces

are mutually connected along the screen.

The function v(x, y, z, P, P’), single-valued in such a space,

will represent the sought solution, since by now the pole P’ is located not in the physical but in the mathematical space.

Nevertheless in order to have the opportunity of applying effective methods of the theory of the functions of a complex variable, it is necessary to shift the investigation to the plane case.
"

: ....

v

' \

:

: •

"-J;
We shall replace the "sheeted space" introduced above by an ordiFig. 7 Fig. 8

nary double- sheeted Riemann surface; moreover, we shall for generality begin with a consideration of an n- sheeted Riemann surface

and assume that the sought n-valued solution has an infinite number of branch points.

Making an incision from the origin of coordinates to oo and

"stitching" the layers as is shown in Fig. 7, we introduce polar

coordinates (r, 0) in the Riemann surface constructed, 0 running

through the values 0 - 0

- 2n7T; thus in the k-th layer we have

2(k - l)ir - 0

- 2k ir .

In the plane of the complex variable z = r + i0 let us consider the function

fn

(z) = log

where 0 is a point on the real axis. Then we have by the Cauchy

o formula

eikr cos (0 - 0O ) , _i_

J

eUcr (» ' 0>f’( z ) dz ,

where

y

is a circle with center 0

q

, since the left member is

1Zn

*4r

- e n

106

exactly equal to the residue of the function under the integral

sign at its sole singular point z = 0 . It is quite obvious that

formula (2) remains valid with a continuous deformation of (y) into a curve consisting, in the drawing adduced, of two branches

C(0) going off to infinity and of the two vertical dotted lines

adjoining these branches. Clearly a deformation like this must

be realized in such a way that the integral does not lose its sense

at infinitely distant parts of the contour. But as is readily seen,

in the cross-hatched portions of the drawing,

R[e

ikr cos ( 0

- 00 )] < 0

and at infinity the contour C(0) lies wholly within the crosshatched region. In consequence of the periodicity of the integrand

function (the width of each cross-hatched strip equals tt), and of

the contrary direction to be followed in integrating along the dotted vertical lines, they are mutually cancelled, and we have

ikr cos (0-0o ) _ 1

2iTi JG(0)

ikr cos (r-0)

f

1 (z)dz = u^r, 0, 0o

)

Now let us consider the function

u

n

(r - t-to ) ‘ 2iFH Jg( 0)

eikr cos <z-0) •( )d n

v ' (3)

is a solution of the vibrational equation, as one may easily satisfy oneself by executing the differentiation under the integral

sign, which differentiation is legitimate in this case. The period

of this function is equal to 2nn' and to make this function uniform

(i.e., single-valued) we must utilize the n-sheeted Riemann surface.
; ' ' 9 '
107

We shall in addition show that u

n

(r, 0, 0 ) has the following two

properties

1) u

n

( r > 0* 0Q

) K u

]_( r , 0, 0Q

) as r oo and |0-0 | - <x < IT

j

2) u n

(r s 0 J 0 o)-»KOasr->oo and Tr ^ (3 — 1 0 - 0o |

For proof of these statements we shall investigate the first and

k-th sheets, and by means of a continuous deformation we shall modify

C(0) and C[0+ 2(k-l)ir] in such a way

that they lie wholly in cross-hatched regions (Fig* 9). In the first

sheet we shall of course avoid the singular point

z =

i*o- Having accomplished the aforementioned deformation, and passing to the limit as r •* oo , we see that since

R |e ikr cos (0-0o

)J < Q

in the cross-hatched region u

n

(r, 0, 0Q

) in case 1 (i.e., in the

first leaf) gives u^(r, 0, 0Q

)

and in case 2 (i.e., in all the

other sheets) gives zero. Both our statements are thus proved.

We shall show that generally

u = Z u<

S

>

, (U) 1 S=1 n

where is the value of u at the point (r, 0) in the s-th n n

c ' r

sheet.

Designating by

/(s+1)

the integral along the contour C(0+2sT),

we shall have

108

where f is composed of the curves 0(0), C(0+2Tr) ,

• • • , C[ 0+2(n-l)lT] .

Closing this curve with two dotted vertical lines (Fig. 10) arrayed

at a distance of 2nir from each other, and taking into account the

periodicity of the integrand function, of period 2nir, we satisfy

ourselves of the validity of equation ( ij* ) . Summarizing the inquiry, we are led to the conclusion that the function defined by

(3) is an n-valued solution of the vibrational equation having at

infinity the character of a plane wave incident along the direction 0

=: 0

q

; it describes the condition of an undisturbed electromagnetic field in an n- sheeted Riemann surface.

§ 21 . Diffraction near a semi- infinite segment y

= 0 , x > 0

As an example of the Sommerfeld method, let us consider the cases

of diffraction near a segment y = 0, x > 0.

109

Here in accordance with §1, Chap. 1, we distinguish two cases,

which in polar coordinates give the following two conditions on the

screen y

= 0, x > 0.

1) u = 0 , 2) r|| - 0 .

For the solution of both of these problems, it is enough to

construct a double-valued solution (n = 2) of the vibrational equation and get its "mirror image" in the screen.

Let us consider a plane wave incident at an angle 0 = 0

q

. The

double-valued solution that we seek will then, in conformity with

what has been said in the preceding paragraph, have the following

form: u

2

(r - V’ V

= TH Mfl)

ikr cos(z-0)

f

^ (z)d!

and the complete solution of problems 1 and 2 will be given by the

functions

1) v(r , 0, 0Q

, -0

o

) = u

2

(r, 0, 0Q

) - u

2

(r, 0, -0

Q

) , (S)

2) v*(r, 0, 0o

, -0

Q

) = u

2

(r, 0, 0Q

) + u

2

(r, 0, -0

Q

)

. (6)

In order to satisfy ourselves that the functions v and v* do

indeed satisfy the necessary boundary conditions, let us set

JL<

dr

" (1) (2) u

2

7 -u

2

k -2i kr oos

2

[i(0-0o )] f sln

2ikr oos

2

[ J(z-0)] —1 d

f

e Mfl) 2

Make the substitution z = 0

- IT + i^; z - 0

+ Tr + i£ for

- oo < £ < + oo ; carrying out the integration we obtain

110

dr

(1) (2) 13 - 11

v 2 2 2 2 F'/Zkr cos [|(0-0O )] -i£

2 ^ ^ e d& -

On the other hand

and accordingly

(1) (2) u

2 + u

2

*1

= 1

( 1 ) _ 1 u

2 2

U

1 U + —L >/=5r

u,

( 2 ) _ 1

2*1 1 + v^IrF

eos[i(0-0o )] -i£

2 e d£

“\Z2tar cos[|(0-0o )] 2

e”

1^ d£

If we take into account the fact, that in the first sheet 0

varies over the interval 0 < 0 <271, and in the second — 2ir < 0 < 0,

and that . v 2

-to

e

' xl d4 =

*

we shall at last have

u?(r,0 s 0o

) = =

2

'" >r »r q

where

i (ilr) _1 rV^ cos[|(0-0o )] _^2

u^e ’"rr co e

' dE

i kr cos(0-0o )

Substituting this result in (£) and (6) we are convinced directly that for 0=0 the required boundary conditions are in fact

satisfied.

§22 „ Diffraction near an angle . Let two half-planes intersect in an angle -¥ that is rational in IT; in a plane perpendicular

to one of them let us introduce polar coordinates such that the

straight lines 0=0 and 0

= —IT constitute the sides of the angle

in question.

We shall call the region 0 < 0 < the physical region B; in

it we shall study the phenomenon in question,, On the straight

lines 0

= 0 S 0

= —IT there must be observed the boundary conditions

1) u = 0 in case 1, Chap, 1, §1.

2) = o in case 2. Chap. 1 §1. 3n

< The solution of both problems is accomplished by a generalized process of "mirror images" whose basic idea was described

in the preceding section. Having reflected the

physical region B in the line 0

= 0 S we obtain

the region B s in which 0 < 0 < —TT

; uniting m

similar regions s we obtain an angle of 2nTTs which we afterwards reflect in the n- sheeted Riemann surface.

We shall now show that the complete solution of the problem

may be compounded of the n-valued solutions ( 3 ) of the following

form

Q (r, 0, 0 ) n

3 v 3 r o u (r 0 0 ) + u (r 0-0 ) + u [r 0 —nTT+0 ]

T u (r 0 —IT- 0 ) + u [r, 0 > i-^^-nTT+0 ] n

5 m o n

L * m

r o

+ u (r, n

3

m ' o

bn 2n,

m

TT -0J + u [r, 0 S + 0 1 n m

+ vl fr, 0, ^^nTT~0J n*~ * * m 'o

Obviously On(r^ 0S 0 ) as a linear combination of u

n

is a

solution of the vibrational equation; besides this s we know from

the foregoing that at sufficiently great distances from the angle
E,

.

I

.

.

9

'

.
112

this solution is a plane wave incident at an angle 0 .

We shall show now that Qn

(r, 0, 0 ) satisfies boundary conditions 1 and 2.

Indeed, we know from the preceding section that expressions of

the form

u (r 0 0 ) T u (r 0 -0 ) and u (r, 0, 0 ) + u (r, 0, ~~

n

» r, r0

> n

v , ^

0 / n

v , r, r0

' nv * m

- 0O

)

satisfy conditions 1 and 2 respectively on lines 0

= 0 and 0

= “TTj

moreover, u

n

(r, 0, 0Q

) = u

n

(r, 0, 0o

+ 2nir)

Having taken these remarks into account, we are satisfied without further ado that Q(r, 0, 0 ) actually satisfies conditions 1

and 2 at both boundaries of the diffracting region.

§23. The method of parabolic coordinates . Let the diffracting screen occupy half of the plane xz (x > 0).

\ The wave falling upon the screen normally is given by the potential

$

= F(ct + y)

We seek a solution of the equation

Ji + +

dt

2 \h2 3y2/

3$

satisfying on both sides of the screen the condition -^ = 0.

Obviously the function ^ dx

2 r which is a solution of the equation —- = c A(J) must reduce to zero

dt

'Proceedings of the London Math. Soc. , vol. 8, 1910, p. U22 ff.
.

• '

.
113

in the negative part of the plane xz, at the same time as the nor3x mal derivative -r- must be equal to zero on both sides of the screen.

Fig. 12 Fig. 13

Employing polar coordinates

x = r cos 0 , y = r sin 0 ,

we see that these conditions are satisfied by the function

— i “I X = f (ct - r)r 2 cos'! 0 .

Now let us introduce parabolic coordinates:

i 1 in

£ = r

2 cos^ 0 ;

?j> = r

2 sin^ @ ^ then x = £

2

- n

2

; y

= 2^ ; r = £

2

+ n

2

.

The curves £

= constant, = constant, form a confocal system

of parabolas.

The curve ^

= 0 corresponds to the section of the screen. In

the new variables we have

zl-iYtil + t ii i Px 2r "

d7) ) pj 2ryn ^P\ J

whence, since X

= and X

= f(ct - r)r 2 cos^-0, we have

Integrating this equation while choosing the supplementary

function so that for in = 0, ~ ~ 0 too. we have: 1 * dq ’

5 = j£°°f(ct + y- t

2

)&l - J0°D f(ct-y-l2)dl + ~F(ct + y) +

|-F(ct-y).

It is evident that for large negative values of x the (jj that

is found must reduce to F(ct + y) • but when 9 is almost equal toll

and r has a large value, the upper limits of the integrals just

written will be + od and -oo, and therefore under these conditions

we have:

$

= f(ct + y- l

2

)dl - ^°°f(ct- y- £

2

)d£ + |F(ct + y) + |F(ct- y)

and this expression for ^ can coincide with F(ct + y) only on condition that

^°°f(y - L

2

) d£ = |F(y) • - cd < y < +co . (8)

Thus the solution of the problem reduces to this integral

equation of the first type.

Let us consider the case when

F(y) = elky • ( 9 )

since

r cd -ik£2

£ 1 Vm -^iTT

lo e dp. = 2

,

the solution of the problem will be

f(y) /IJiireiky

V ir e e

This, the simplest case, offers us the possibility of solving

the more general problem as well.
*

'

.

.

.

O

.

-
115

Using the Fourier integral, we have:

I /-2

D 'w.Uc(^*c ,

whence, on the basis of the foregoing, we have

t(T) - -i- dk . (10)

vp

1

Equation (8) may easily be converted into Abel's equationj indeed, having made the substitution

£ = Vy - p , we obtain

F(y) = fj -f

- d(? , -oo < y < oo , J 00 V7^7

and, using the usual method for finding the solution of Abel's

equation, we find that

Tr d(3 J-cc

f((3) -/-(D- dt . (11)

V(9 - t

If F(t) = 0 and lim \/(3 - tF(t) exists, we have, from (11),

t-> -oo

f<?> - Tf & ti^co '/

^T + I /-I •

and in particular, if F(t) = V“t G(t) and G(-oo ) is a finite number different from zero, then

lim - t F(t) = G(-oo) , t-* —oo

and we at last have

f((j) = i r oiLdt

p T J

- 00

(12)

CHAPTER 17

THE INTEGRAL EQUATION METHOD

§ 2 2j «, General remarks . Auxiliary formulas . All the methods

discussed in the preceding sections for solving the boundary problems connected with the Helmholtz equation have a limited sphere

of application; the actual solution of the problem requires in

each separate case a choice of the special coordinates giving the

simplest notation for the given problem, and the construction and

investigation of the special class of functions with which it is

possible to represent fully the solution of the problem. We have

already seen, in §9“19, with what sort of difficulties one is obliged to cope; for many practically important curves and surfaces

these difficulties are still greater, and indeed for arbitrary contours, despite that they may be analytic and without angular points,

the difficulties are actually insurmountable, the relevant realms

of mathematical analysis being in their current state.

As regards the methods of construction described in Chap. Ill,

they may be extended to open polyhedral regions with comparative

ease, but for closed regions of arbitrary form they are inapplicable without considerable modification, or even a modification of

them in principle.

The method of integral equations has a considerable advantage

in respect of generality over the methods referred to above. This

circumstance is connected primarily with the well-known generality,

a generality in principle, that always attends the integral equations
•

. ,

• .

.

'

. ..

; >
117

of all problems of mathematical physics. On the other hand, in the

given case we have to do with a generalization of the familiar method of Fredholm for the solution of the harmonic problems of

Dirichlet and Neumann, which are solved in integral equations in

the most general form.

This analogy is particularly complete in the case of problems

in electromagnetic diffraction, i.e. , in the case where a single

wave equation is under study.

The integral equations that arise in this connection have regular kernels and the investigation of their solutions will reduce

analogously to a similar investigation in the theory of Laplace

equations

The matter is somewhat more complicated in problems of the

diffraction of elastic waves, i.e., in the case where there is a

system of wave equations.

Even in the simplest case, when the boundary conditions give

the value of the displacement at the boundary, the integral equations have singular kernels, and the applicability of the Fredholm

theory requires an additional preliminary analysis, which reduces

to the regularization of the singular system of equations obtained.

In the course of this we shall use the following assumption:

If M(x, y), N(x, y) and f(x) are holomorphic functions of the

real variables x and y, periodic, of period % with respect to each

of the variables and such that the first two do not have other singularities with respect to y except a simple pole y

= x, i.e., in

the neighborhood of y

= x they have the form
'

*

.

'

'

.

.

' . . .

' - •

•

.

.
118

ML (x)

M(xs 7) = M

q

(x, 7) + —- ,

/ x

* N

1

( x) N(x, 7) = N

q

(x, 7) + —

where M

o

(x, 7), N

q

(x, 7), M^x), N^x) are holomorphic functions

of x and 7, then the following equation holds:

V ' p * I/ N( x^)d7 V.p. JJ- M(7,z)f(z)dz = -^2 N

1 (x)M

1 (x)f (x)

(I)

+

Jo^ f ( z ^ dz * V *P« ^N(x,7)M(7,z)d7 .

V.p. (Valeur principale) indicates that the integral has meaning

onl7 in the sens'e of Cauch7’s "principal value."

This formula allows us to permute the order of integration in

a double integral with Cauch7 s s "principal value."

In particular , if one of the functions is everrwhere regular,

the product M^xjN^x) = 0, and instead of (I) we have:

fj- N(x,7)d7 V.p.

JJM(7,z)f(z)dz = f(z)dz. V.p.^N(x,7)M(7,z)dy

or

Y -V- f^(x,y)dyfJ-Vl(y } z)f(z)dz = f(z)dz V.p. f z)dy .

A proof of equation (I) can be found in an article of G. Bertrand:

"La theorie des Marees et les equations integrates. “ Ann. Ec. Norm.,

( 3 ), XL- 1923 , pp. 201“ 2)4.7

*§ 25 . Generalized logarithmic potentials of a simple and of a

double la^er. Let us recall the fundamental results of the classical theor7 of logarithmic potential.

Considering the attracting masses as distributed, with densit7

ji(s), along a given singly- or multiply- connected closed contour

(j(), which has a tangent that changes its direction continuously

everywhere, we have for the potential of such a contour layer:

119

v

( 1) r = 7(x - £)

2 + (y - 7)

2

j $ = S(s), >? = >}( s)

This potential of a simple layer is, as is known, everywhere

continuous, including even passage across the boundary (y). Its

tangential derivative is also everywhere continuous, on condition

of the existence of continous first derivatives of the functions

ji(s), but the normal derivative has a jump of +2ttji(ct) on passage

across the boundary at point P(cr)<>

Together with this, the following relations hold between the

limiting values of the normal derivatives, as the point of action

(i.e., the point at which the potential itself is being considered)

tends toward a point on the contour from within (i) and from outside (e), and their value on the contour.

Here n^ denotes the direction of the positive (inward) normal to (jf) at the point s = O'.

tribution of the magnetic moments with density y(s) along (y), is

defined by the integral

( 2 )

The logarithmic potential of a double layer, i.e., the disw = r

ds ( 3)
4'-

y .
120

cos(n ,r) « / _

(U)

where d0 is the element of angle under which the arc ds is seen from the point at which the potential is being considered. The

potential of a double layer is no longer everywhere a continuous

function of the point, since on passing across the boundary of the

magnetic masses it has a jump equal to 2irV(cr)

, defined by the following formulas;

p cos(n r)

( u

cf) i

= v(s) - ds

( Ve - -TM

(o') * If)

*s)

cos(n

g

,r)

(?)

ds

The normal derivative is continuous; the tangential derivative, however, also has a jump equal to 2ir^(o'), defined by the formulas

3u

ds

du

ds

'cr -a

= TT

= *TT dA

ds

ds

cos(n „r) W y(s) —r~ ds

_ cos(n r)

b)

*s) ds

(6)

here ?u

ds cr etc., signify that we have in view the limiting

value of the tangential derivative of the function h(s) at the

point cr from within, etc.

We shall now construct potentials analogous to those just now

adduced, and playing the same role in the theory of the equation

2

Au +. k u = 0 as do the harmonic potentials adduced above in the

theory of the equation Au = 0»

For clarity we shall explain once more all the most important

symbols.

121

Let ( 1f) be a continuous, closed (for simplicity : simplyconnected) curvex = £(s) , y

= >?( s) , / , having no singular points.

Furthermore, let ^(s) and V(s) be functions that, together with

their first derivatives, are continuous along (y); let P(x,y) be an arbitrary point of the plane; let H(^ ,^) be a variable point on (y)j let n

g and n^ be normals to (-y) at the points where the arc

equals s and o respectively; finally, ^ let

H^J(kr) = J (kr) - iY(kr)

o 0

be the Hank el function of the second type of the zero-th order.

Let us define the functions

V(x,y) - / jx(s)

and

¥(x,y)

X

V(s)

k Ho (kr) ds

7U H

ds ;

( 7 )

( 8 )

because of the evident analogy of ( 7 ) and ( 8 ) to ( 1 ) and ( 3 ) respectively, we shall call ( 7 ) the generalized potential of a simple

layer and (8) the generalized potential of a double layer. However

for brevity of diction we shall drop the word "generalized" where

this will not occasion misunderstanding.

2 Since the special function H (kr) is a solution of the equa2 tion Au + k u 0 everywhere outside the attracting masses, the

integrals written above will naturally also be solutions of this

same equation. In order to satisfy ourselves of this, let us

1 ( 2 ) In the rest of this chapter H

v ' (y) is indicated by the somewhat

o

2

unorthodox, but more compact, symbol H

Q (y) . For the sake of simplicity,

the author’s notation is retained. Editor.
‘

.

.

.

.

.. ...
122

seek a solution of the vibrational equation in the form u = u(kr),

2 p r = v /r

x

- 2r

x

r

2 cos(e -6

o

) + r

2

From the equation Au + k u = 0 we obtain

d u 1 du , . 2 A

T~2

+ rdr + ku = 0

dr

or the Bessel equation

for the case n = 0 with argument y

= kr. The Hankel function that

we used above has the form:

H^(y) - J

0

(y) - iY

0

(y)

riy cos i

d0

Proceeding from this representation of

the Hankel function, it is easily shown that

the generalized potentials (7) and (8) at sufficiently great distances satisfy the EmisBy means of the familiar formula

2 2i o-? /'-I H^(kr) = Jo (kr) - & j^kr) ig ( kr ) - £i 2 Ud J (kr)

p=l

the generalized potential of a simple layer, (7), may be written

in the following form:

sion principle.
.

'

*

'
123

V(x,y)

-I X^ykrUgids

l X^(s)ig ff ds + 7 ^(s )[ y

0 (to)-i]igidS

GD / nP+1

«t J (kr) 2 J (kr) 2i o _ p 2p v p=l ^ ^ ds

H /.%<>

Pi 00 / xp+l

J (kr) - #~ 2 -U J 0 (kr) k _ p 2p v ' p=l r r ds

Comparing this expression for V(x, y) with (1), and taking into account the continuity of all the summands, except the first, on

passing across (^), we obtain from formula (2):

(MX - - w + fa

^ M

^

(<r) + ir

) ^

(s)

_0v

?n

H Ho (kr)

H Ho (kr)

ds

ds

(9)

where denotes differentiation along the normal to the arc (().

The generalized potential of a double layer, determined from

(8), also has properties analogous to those expressed in formulas

(5) and (6). Indeed, we know that

3 tt (2}„ * , , v,2/, \ 2i cos ( n

s

> r ) ar o

(kr) = " k c°s(r - n

s )H1

( k1') • - —?

s k cos(r,n )

* s

xjj

o

(kr) - — J

1

(kr) lg

l+2p

(10)

i “ (Xf) ' T

p

fQ p!(pU)

-

! — thP+1) * ?(P«)]

Substituting this expression for the kernel under the integral

into (8) and taking into consideration the continuity of all the

12U

summands , except the first, on passing the boundary ( y) , we have

(3), (<) and (6)s

(Vi = + jy)

V ( s ) 22n

<Ve = - ^ + i T

) ^s

> IF

A H^(kr ) 2i o

v scr'

A- Vkr ) 2i o

v sods

ds

> ( 11)

3s

3W

3s

'<r-J e

3V

0s

2V

?s

crJ r

r

+ ir)

y(s)

'Jr

/(/)

y(s) IT

A H^(kr ) 2i o

K scr

A H^(kr ) 2i o' so-

ds

ds

( 12 )

3 in these formulas everywhere denotes differentiation along the

tangent and

scr = / U(s) - «cr)] 2

[7|(s) -iKo-)]

2

In addition to this, because of the continuity of the normal

derivative of the potential of the -double layer,

lM_ = 2w _ Dn. hn 3n

x

Lastly, it is not hard to verify by direct computation that

the generalized potentials (7) and (8) asymptotically satisfy the

Emission principle, because of the special selection of the integral kernels.

§26. Fundamental boundary p rob lems and integral equations .

The first fundamental problem -- the generalized Dirichlet problem — as has already been noted in other places of this book, is

/
-

.

'

.

'
formulated thus:

To be found, a function u(x, y) , single-valued over the plane

(xy) , that satisfies the equation

2

Au + k u = 0

the boundary condition along (|)

au + bu. = co(s) e 1 ' 1

(where a and b are given constants, and co(s) is a function given

along ('jf))g

and the Emission principle at infinity.

As regards the boundary condition (11) we shall remark that

for a = 0, b = 1, we shall have the internal problem (D^) of

Dirichlet, and for a = 1, b = 0, the external problem (D ) of

Dirichlet

As in ordinary potential theory, the problem reduces to finding a distribution (the law of distribution) of radiating masses

such that the potentials defined by us satisfy all the conditions

set above.

We have already seen that the functions defined by (7) and

(8) satisfy the first and the last conditions.

It remains to choose the arbitrary functions ^i(s) and i>(s)

so as to satisfy the second condition as well.

The first fundamental boundary problem may be solved in the

general formulation set above, by endeavoring to find the solution

in ihe form of the generalized potential of a double layer, i.e.,

in the form of a function defined by (8).

Substituting it in (ll) and taking (ll) into account, we
*

.

.
126

obtain for a = 0, b = 1 5

(Eb ) p(cr) + JJ-

)j{ s)K(s, cr, k)ds = <o(cr)

;

for a = 1, b = 0,

(D

g

) v(cr) ~ JJ’

i>(s)K(s, cr

# k)ds = - w(<r)

,

where

h. TtT

H

o (kr) = ~ H

~ kH^(kr) cos 0

= K(s,<7,k) , s s

and 0 is the angle between the direction of the inner normal at the

point s and the segment r from the point s to the point s = or.

The second fundamental boundary problem the generalized

Neumann problem, is formulated just as the first problem is, but

instead of boundary condition (Hi ) , a condition is set here that

relates to the value of the normal derivatives at the contour, to

wit:

du

3n

+ b 0u

dn. = XT(s) e l

where a and b are assigned constants, and t(s) is a point function

given along (y).

In solving this problem it is convenient to use the generalized potential of the simple layer.

Introducing into (19) in place of u

g

and u^ their values from

(7) and taking (9) into consideration, we have: for a = 0, b = 1,

(ISO

(N

i

>

ji(cr ) - JJji(s)K(cr } s, k)ds “ - Tr(cr)

,

and for a = 1, b = 0,

(N

g

) + £^yu(s)K(cr, s, k)ds = x(cr) ,
i-:

.

.

'
where . as is obvious

J± 0^ H

o(

kr ) = “ kH^(kr) cos }

= K(cr, s, k) ,

and f is the angle between the inner normal at the point s = &

and the segment r(s,cr).

Apart from the boundary conditions of the Dirichlet and

Neumann problems, there may be still others, to which one physical problem or another leads. So, for example, the problems of

diffraction considered in §U lead to the following conditions on

the boundary:

u. = u + u , 1 e

*

2u. 2u . * 1 _ e ?u

dn 2n 3n ’

here, as there, u^ is the sought potential within (^), which in

this case confines a material medium of conductivity different

from zero, in consequence of which there are also obtained refracted waves; u

g

is the potential outside ( ^)

and u* is the

fundamental, given potential outside of (y), which upon meeting

(^) generates the diffracted and refracted rays.

In solving such a problem it is natural to seek the potentials within and outside of ( y) in the form of generalized potentials of the simple and double layers.

Let

X u. = /

' u(s)

1 J 0 1 k H

o< kr > ds

u = L

x v(s) dn k

ds

Then we obtain from the boundary conditions a system of

Fredholm integral equations:

128

•z— denotes a suc- dndn

s

( 18 )

cessive differentiation along the normals at the points s = cr and

s = s.

It is scarcely necessary to show that a number of other boundary problems as well, connected with the Helmholtz equation and important from a practical point of view, lead to the Fredholm equation; for example, the problem of the stationary thermal state in

which a definite combination of the sought function and its normal derivative is given on the contour, a problem of tidal theory where the boundary

condition relates the value of the normal derivative on the contour to the value of the tangential derivative, etc.

§ 27 . Investigation of the existence and uniqueness of the

solutions . Let us rewrite the integral equations (D

g

) and (N ),

employing the parameter X in them:

Here one must keep in mind the fact that the physical problems of diffraction under our consideration do not lead to equations (D ) and (N ) strictly, but to particular cases of them,

0 0

namely those for which X equals -1 and +1 respectively. We shall

prove the following fundamental theorems:
'til

'

‘

'

.

'
129

Theorem 1. The necessary and sufficient condition that X = -1

be a characteristic number of equation (D ) is the equality of the

frequency k to one of the proper frequencies of the internal Neumann

problem

Au + k u = 0 j

d u

dn.

= 0 (19)

r The rank of the characteristic number X = -1 is equal to the

multiplicity of k, and the fundamental functions of the equation

(D°) v(or) - JJt»(s)K(s, cr, k)ds = 0

are the contour values of the fundamental solutions of the internal

Neumann problem (19).

Theorem 2. The necessary and sufficient condition that X = +1

be a characteristic number of equation (N ) is the equality of the

frequency k to one of the proper frequencies of the internal Dirichlet

problem

Au + k

2u = 0 , [u

±

] = 0 . (20)

The rank of the characteristic number X = +1 is equal to the

multiplicity of k, and the fundamental functions of the equation

(N°) ji(cr) + ji(s)K(cr s, k)ds = 0

are the contour values of the normal derivatives of the fundamental

solutions of the internal Dirichlet problem (20).

Proof of the necessity. The necessity of the conditions formulated in Theorems 1 and 2 follows from the following proposition:

If k is not a characteristic number of the internal homogeneous Neumann problem

130

Au + k u = 0

'3u'

2n.

1

= 0 (19)

•-1 /

then A = -I cannot be a characteristic number of the equation

y(cr) + A /

?- < D y ( s)K(s, cr, k)ds = - u>(cr) e>

Indeed, assuming the contrary, we are led to the conclusion

that the equation associated with (D°),

» (N°) 9(cr) - v(s)K(cr, s, k)ds = 0

must have non- zero solutions.

Comparing this equation with equation (N^), we see that with

the aid of the solutions (N°) just indicated, we shall construct

the generalized potentials of the simple layers, which will be solutions of the internal Neumann problem (19).. But since, according

to the condition, k is not a characteristic number, all solutions

of (19) are zero within (^ )and by the continuity of the potential of the simple layer the solutions obtained by us will also

have zero limiting values for an external approach to the points

of the contour. But on the other hand, the potentials of simple

layers, constructed by us in the role of solutions, satisfy the

Emission principle; and as we know from §3, Chap. I, such a solution is unique, whence it follows that our solutions are identically zero.

Denoting these solutions by u^ (r =1, 2, •••, n) we have

u

r

= 0 everywhere.

and

/ 3u

b

1 \ \
.

<

*

'

V

.

-

,
131

whence it follows that

/i (°0 s 0 ,

i.e.all solutions (N°)are zero.

From this it follows that the assumption made was incorrect

and the proposition is proved in full.

Obviously a parallel proposition, relating to Theorem 2, may

be proved analogously:

If k is not a characteristic number of the internal homogeneous Dirichlet problem:

Au + k

2u = 0 , [u.] = 0 , (20)

i then X = +1 cannot be a characteristic number of the integral equation

(N

g

) +

kJ'J- jx(s)K(a f s, k)ds = + t(<r)

The proof of the sufficiency obviously reduces to a proof of

the following proposition:

If k = k is a characteristic number of the internal homogeneous Neumann problem, (19), then X = -1 is a characteristic number of the integral equation

V(cr) + A/? ^(s)K(s, cr, k*)ds = - u(<r)

and the fundamental functions of the corresponding homogeneous

equation (D°) are contour values of the solutions of (19).

Let k = k* be an n-tuple characteristic number of problem

(19)*, and u^ (r = 1, 2, **', n) the corresponding fundamental

functions of the problem.

Let p. (p = 1, 2, •••, m) be linearly independent fundamental

3? functions of the integral equation
1

.

,

.

'

V
132

So

s, k* )ds = 0

We shall show that m n. Assuming the contrary, let us define the potentials:

u = Jo* Vs) H H

o

(kr > ds (p = 1, 2, m) ( 21 )

These u* are obviously solutions of problem (19), but since

Ji the latter has for k = k only n linearly independent solutions,

a relation of dependence exists,

m

Z a u* = 0

p-1 PP

or H So

H

o

{kr > ”, Vp(s)ds = ° • P=1 ^ ^

Employing with this potential of a simple layer tne reasoning

that we used in the proof of the necessity, we obtain

m

Z

P-1

0 9

which contradicts the assumption concerning the linear independence of the solutions of equation (N?). Hence follows the incorrectness of the assumption that we made, m > n, and our proposition

is proved.

We shall now prove that m<£ n.

Let the fundamental functions of the problem (19), u. (j =1,

3

2, •*•*, n) have on (^) the respective values

1

+2h.(s).

3 Let us define by means of these latter the following n potentials

1The linear independence of these values is readily shown.
. .
133

of double layers

W. =

3 1° h

j

(s)

3rT H H

o<to > ds

,, (3 = 1, 2, •••, n) ;

and on the basis of (11) we write

fdW.\ AA

[W.(CT )3. - [W. (0-)] e

= 2h.(CT) ,

.

» Let us now introduce into the reasoning the functions u

defined as follows:

(j)| = Uj within (y)

(j)

u

and define the difference

= 0 outside (y)

X- = W. - u^ * 3 3 obviously

[ (cr)

e

- CX^cr)^ = {[W (<r)] e

- [W (cr)].} + j[u^ ^ (cr)] ± V)] e]

/3 )(

= -2h . (cr) + 2h„(cr) =0 ; 3 3

’

vS)'e -ffi). v 1

3u ( 3 ) 3u

(3)'

dn »n

= 0

since 2u (3 )N

?n

are the limiting values of the normal derivatives

of u. and accordingly equal 0 in accordance with (19).

3 On the other hand.

X. = W. - u

(j)

7 3 3

satisfies the Emission principle and by the uniqueness theorem (§3,

Chap. I) we have

V . 3 W. = u^

Consequently

13U

w.

3 L

x h (s) JL

dn H H

o<k

* r > ds

u. within (^)

0 outside

(f)

( 22 )

in view of the discontinuity of the potential of a double layer in

passing across ( , we shall have, from (22),

h^(o-) - J/ h^. (s)K(s, o', k*)ds =0 ( j = 1, 2, • •

•

, n) ; (23)

but this is just equation (D°), which thus has, for k = k*, n linearly independent solutions; on the other hand, (23) shows that the

fundamental functions of equation (D°) are, accurate to a constant

coefficient, contour values of the fundamental solutions of problem

(19).

Passing on to an investigation of the characteristic numbers

of equation (N

g

), we shall prove the following proposition:

If k = k* is a characteristic number of the internal homoge-

* neous Dirichlet problem

Au + k

2u = 0 = 0 » ( 2°)

then X = +1 is a characteristic number of the integral equation

(Nj ji(<t) + |i(s)K(cr, s, k*)ds = T(cr) .

Let k = k* be a n-tuple characteristic number of problem (20)

and u (r = 1 2 n) the fundamental functions corresponding ' 3 y 3 '

to it. Furthermore let V (p = 1, 2,

• •

•

m) be linearly independent solutions of the equation

V(cr) + v (s)K(s, cr, k*)ds = 0 .

We shall show that m nj assuming the contrary, let us define the potentials
. * 1 ' • .. '
135

* u

P

V (s)

P

ddn kr H

2

(k*r) 2i o

v *' ds

Obviously a relation of linear dependence must exist between

these m solutions of problem (20) in consequence of the assumption

that m > n; hence, as above, follows a linear dependence between

the Vp(s), which contradicts the assumption concerning their linear independence, and accordingly m n.

We shall now prove that m <j^ n. Let there be 2r

r

(s)^ (r = 1,

2, n) contour values of the normal derivates of u^ ^(r = 1,

2 ,

* *

* , n) ; let us define the n potentials of simple layers

V.

3 ds (3 = 1, 2, •••, n)

On the basis of formulas (9) we have

n.

Tt) dn * 2r», (V„)„ - (V.) . 3 e

tions u

Now, as above, let us introduce into the reasoning the func-

(3) defined as follows:

u

(3)

= u. within (^)

= 0 outside (y)

and define the difference

Then

1

(3 = 1, 2, n)

X. = v. -

J 3

. Wj) e

- - [(^)e

- * I(u (3)

) i

- (u (j)

) e] - 0 ,

= 2r .(<r) - 2r (cr) = 0

J J
- •

.

:

'• I

1;
136

and therfore, by the uniqueness theorem.

V.

o

V

We finally arrive at the conclusion that

u . within ( j) = L

x r

i (s) ds = 3 /' ( 3 - 1 , 2 . 0 outside (y)

In passing across (^) we shall have, in view of the discontinuity of the normal derivative of the potential of a simple layer,

rj(°') +

fo

r

j( s ) K(°r

» s, k*) ds = 0 • (2U)

But this is just equation (N°), which, for k = k* thus has n lineare * ly independent solutions.

On the other hand, it is evident from (2)i) that the fundamental

functions of equation (N°) are, accurate to a constant coefficient,

contour values of the normal derivatives of the fundamental solutions of problem (20), which thus proves the proposition enunciated

above

Combining the results obtained, we have a complete proof of

Theorems 1 and 2.

§28. Exceptional cases . Let us once again give a reformulation of the gist of the results obtained above:

a) The problem of finding a solution of the vibrational equation

to + k*U - 0 ,

such as to satisfy the Emission principle at distances infinitely

remote from the contour, and such as to satisfy on the contour the

boundary condition u» =co(s) oru = co(s) leads to the consideration

1 ^
; 5 ' '
137

of the Fredholm integral equations

(D^) V(o-) + y(s)K(s, cr, k)ds = co(<r)

,

( D

e

) ^(°”) ” t»(s)K(s , cr, k)ds = -

6)(<r)

5

moreover , if k is different from the characteristic numbers of the

problem

Au + k u = 0

du

tin, -

= 0

then equation (D ) has a unique, continuous solution, which is

found with the aid of the Fredholm resolvent;

= F(s) lead to the integral equations

b) Problems with the contour conditions^ y^-') = f(s) or

dn.

(N

±

)

(N

e

)

pCc) “ J/ Ji(s)K(o'} s , k)ds = - T(o*) ,

^(°*) + /i(s)K(<r, s, k)ds = t(cr) j

( 19 )

here, if k is different from the characteristic numbers of the

problem

Au + k^u = 0 , [u„] , = 0 , (20)

-L 0 then equation (N ) has a unique, continuous solution, which is constructible with the aid of the Fredholm resolvent.

Thus problems concerning the diffraction of incident disturbances with frequencies different from the frequencies of the proper vibrations of the diffracting contour have been solved in full.

In this section we shall indicate the solution of those problems in particular for the case of the frequencies excluded from

the general theory set forth above.
.

• •
138

Ao The Dirichlet problem . Let k = k* be an n- tuple characteristic number of problem (19). In this case the associated homogeneous equations

p-(<r) - p.(s)K(cr> s, k*)ds = 0 ,

y(cr) - y(s)K(cr, s, k*)ds = 0

have, in accordance with Theorem 1, the n linearly independent

solutions u? and V° (j =1, 2, ••*, n) , which we may consider to

0 J have been rendered bi- orthonormal without restricting the generality.

In this case equation (D

g

) has a definite solution only on

condition that

JJ~

co(o-)ji°(cr)dcr = 0 . (2Ua)

for the observance of this condition we shall introduce a new

function, defined in the interval (0, X)

co(<r) = - co(cr) + Z a.y°(cr) , (2^)

j=l J J

where

a.. = fjo(ar)ji°(cr)d<r .

Then condition (2lja) will be fulfilled for the function co(cr) and

the integral equation

(D ) V(cr) - [/ y(s)K(s, cr, k*)ds = - cb(cr) + Z a v°(cr) e Jo

^ J J

has for its general solution

r

, n

p>(cr) = oo(o-) + / ^ P (s, O', k*) (o(s)ds + Z c.f.(cr)

Jo J J

where p(s, cr, k*) is the suitably modified Fredholm resolvent for
-

,

.

.
139

n the equation (D ) , and Z c.i/?(cr) is the sum of the n linearly in0=1 J J dependent solutions of the homogeneous integral equation (D°).

Putting this value of (cr) in (8) and taking into account that

Jo

S3 (s)

u s

is equal to zero outside of ( y) , in accordance with (22) and (23)

= h.], we will have a single-valued solution W(x,y) completely

determined by its value on ( y) , to wit cj(cr) .

Now let us note that the solutions of problem (19) may be expressed thus:

c^(x, y) = ~ JJ-/i°(s)H

g

(k*r)ds

y ( j = 1, 2, • •

• , n)

and, as was proved in Theorem 1, the limiting values of these functions on the contour are linearly expressible in terms of the

Let ||d^|| be the matrix inverse to this linear transformation.

We shall show that the sought solution of the equation

*2

Au + k u = 0, with boundary condition [u L = 6^(s), which solution

0 0 satisfies the Emission principle at infinity, will be:

u(x, y) = - W(x, y) + Z Z a d ^(x, y) , (26)

j=l k=l J J

where

w< x. y) - h Jo k H

o<k

* r > ds •

y(s) being any solution of equation (D

g

), and

r = 7[x - £(s)]

2

+ [y - ^(s)]

2

.

In order to prove this statement it is sufficient to show that

H^(k*r) ds
-

.

.

:• '

::

•
mo

the function defined by (26) fulfills the necessary boundary condition, since it is perfectly obvious that it satisfies both the

other conditions directly.

From (26) we shall have, for points on (^),

n n

u(o") = o(o-) - 2 a.y°(cr) + 2 a .v?(cr) = oj(ct)

, j=l 03 j=l 3 3

Q . E. D

The second fundamental boundary problem may also be subjected

to analogous reasoning in the case of the exceptional values for k.

Let k = k* be an m-tuple characteristic number of problem (20);

*2 then the solution of the equation Av + k* v = 0 with boundary condition £v_

dn

- x(s), which solution satisfies the Emission principle at infinity, will have the analogous form:

m m

where

v(x, y) = v(x, y) - 2 2 b r tt (x, y)

j=l k=l 3 3k k

v(x> y> “ h Jo H

o (k

* r) ^

(s) ds >

(27)

p.(s) is any solution of the equation

pi(cr) + ju(s)K(cr, s, k*)ds = tr(cr) - 2 b^1

) (a)

3-1

i/

1 ) and are bi-orthonormed solutions of equations

0

pi(o') + ft Ms)K(v, s,k* )ds = 0 ,

y(cr) + y(s)K(s, cr, k*)ds = 0 ,

and b. and 7r. are defined by the formulas:

3 k

b = ft x((j)v^ 1)

(cr)da' , Vx> y) - H Jo

y

k

1)(s) Ta H

o (k

* r)ds
;•

'

a'

*: :•

' i ..

A

• : V- ,
iia

The constants form a matrix inverse to the matrix of the

linear transformation which, in accordance with Theorem 2, connects

the limiting values of Tr^(x, y) with ^^(cr).

Formulas (26) and (27) solve in full both the fundamental boundary problems for the exceptional values of k.

§ 29 . The integral equations of the stationary theory of elasticity . The problem of the diffraction of established elastic disturbances,, near an immobile,, walled-up obstacle of form (y), as has

already been remarked in §2, Chap. I, leads to the following boundary problem of a system of vibrational equations.

To find: solutions of the system

+ k-^fj) = 0 ,

A? + k

2

2

$

= 0 ,

(28)

such as to be regular everywhere outside of (y) and such as to sa'

tisfy the conditions on (y):

zf + zt)

,2n dsJ

(29)

(ik -r 2h

\2s 3n) ds

l 7 e

(where J* are the values on (y) of the assigned fundamental "falling

potential") and such as to satisfy at infinity the Emission condition:

lim ij) = limVr

oo r-» oo

lim f

= lim Vr [~ + ikgf) = 0 ; oo r-»oo

r

2 = (x - £)

2

+ (y - 7)

2

•

( 30)

Ih2

If the potentials that are sought be constructed in the form

of generalized potentials of simple layers

s) - y)

2

)ds $ , (31) y) - Ae(s) - x)

2

(t,(

t < x > y) - f( ^(s)H^(k2 V(6(s) - x)

2

(7,(s) - y)

2

)ds , (32)

then on the basis of (9) and (11) it is immediately obvious that

conditions (29) will be fulfilled if the ''density1

' s) and p(s)

be found from the equations

fo* 7°(s) IH; Hot k

i

r ( s ^)] ds + 2iy>(o-)

+ V ’ p * lo^

H

2

[k

2

r(s,o-)]ds = f(<T); (*)

V °P* lo Ts~ H^CV(s * <r ) ]ds “ 2iP^

lo ^s) 3“ H

2

[k

2

r(s,cr)]ds = 0(<r); (**)

(33)

0 in these equations y— denotes a differentiation along the tangent

(T at the point s = cr

} and f(s) and 0 (s) respectively equal

f ( s ) = - 2 i

dn

0 (s) = - 2 i

. af (s)

V.p. in front of the integral signifies that the integral has meaning

only in the sense of Cauchy's "principal value" ("Valeur principal").

Let us now occupy ourselves with an investigation of the character of the singularity that figures in equation (33)*

If we take into consideration the fact that
T

' f,

1

'
1U3

H

0 [kr(s,o-)] = (- ^ Jo

(kr) lg

-2r, / si J 2i T /, N „ kr

oo + J_(kr) (1 - # C) - |i l

(-)V2n

2 i

s-1

3

(3li)

tr H" n=l 2

2n

(n!)

2

it is obvious that

it Ho[kr(s,tr)] *u(s,o;k) - j -|f - uU,o;k) - ^

^isl > (35)

tr

where

<°( s , cr, k) = {- ~ / (kr)k lg ~ ~ [J Q

(kr) -

1] |

n

2i . oi 00 (") n

2n(kr)

2n" 1k 2 ^

+ (1-T C)kJ0 (kr) s=!

n=l

2

2n

(n!)

2 3r

aS

CT

# (3^')

and

^ X* = cos(r , s

CT

) = sin(naS r) = sin f

Since for s = cr the angle (n^ r) •* and r -> 0, we see that

3 2 the kernel -r— H (kr) at the point s = ct has a pole of the first

0 Sq- o 2i order with residue equal to - -p .

In expressions (33) the differentiation along the tangent under the integral signs has as yet only a formal character.

In order to show the legitimacy of these operations, we write

-k-aSo >> (s)H

o (kr)dssr Ms)Hot kr ( s > <7 >i ds

+ 7Te Xlf2(s)H

o[kr(3 > Cr)lds + TTa £T ^(s)H2 [kr(s,<T)]ds .

In the first two integrals, the differentiation can obviously

Hill

be actually accomplished; we then obtain

W; L

X

M(s)Ho[kr(s,<T}]ds

= X'^>l(s) j|y H

o (kr)ds + H

2

[kr(o-fc,o-)],i(o- t)

~ H^[kr(cr+s,a)ji(cr+£.)]

(36)

+ X

X ^

(s) -^; I^(kr)ds

To compute the integrals figuring in this equation we utilize an ex- pansion of the square of the chord in powers of the increment of arc

(Fig. 15) s

[r(s,s+As)] 2

= (As)

2

- 12 R 12

^

R^

as

R is the radius of curvature at the point s, hence

2 Ig f- - 2 Ig £&

Fig. 15 12 R‘ 2 Ig f - 2 Ig MS . Ii2L (i + a4s) >

where a is a bounded number, or

, kr 1 , ig y = ^

ig Ils4sL - (1 + ais) (37)

2 k R

£

On the basis of this dependence of the chord upon the increment of arc, one is directly convinced of the validity of the equation

2 2

1

H

o

[kr(cr- £,cr)]ji(cr- t) - H

o [kr(cr+ e,ct)]u(o-+&.)

. 2

12TTR

+ i-

_2i

”

TT

2i kr(°"' cr- £) 2i ^

(J o [kr(a,cr-t)]-l)lg — + J

0

(kr(o- f cr- £.) ) (1-— C)

kr(crtQ--£.)

j pipr-t) +‘Y

: [J

0

(kr(cr,<T+ &))-!] Ig

J 0

(kr(o-,o-+t))(l -^C) + ^

kr(cr a+6)

ji(cr+t) (38)

"Hfe designate As. by t for simplicity.

il£

As t •* 0, r tends to zero and the expression standing in braces, as a regular function, tends to zero; moreover, if ^i(s) has a

derivative, then

[u (s+t) - }i(s-6)] Ig ~

&'-*0

tends to zero as lg bt and accordingly the whole expression (38)

s •* 0

thereby tends to zero.

It remains to consider the last summand in the right part of

(36).

For this let us make the substitution

s = <r + t

then

_2_

3s_.

<T+t h— £ ju(s)H^(kr(s,cr))ds = — pi(o'+t)H^[kr(a,0'+t)]dt

T~ h2

3scr O

cr+£

p.(cr+t) -5!- H^[kr (cr,cr+t) ]dt

+ H^[kr(o-,CT+t)] p.(o-+t)dt .

Having taken into consideration the equality

kr

2 2 kV

lg

2 2 lg

2

2h R‘ (1 + at)

after elementary transformations with the aid of (3U) a-n<3 (35), em_

ploying the first theorem of the mean, it follows that as e — 0 the

integrals in hand tend to zero.

We thus have, finally,

jf- // ^( S )H^(kr)ds = Y.p. ft F

(s) H^(kr(s,o-))ds

' V -P- Jot . \ 2i sin f

cn(s,0-,k) - — p.( s) ds
. -

,

'

*

.

'
116

§ 30 . Reduction to a regular system. Each of the equations

of system (33) contains an integral with a singular kernel; as is

obvious from the section preceding , the character of these singularities is such as we have already encountered in § 2 U- In addition, all the functions figuring in system (33) are periodic with

respect to s and cr, of period and are holomorphic when s and o'

vary along the real axis, except for the term -

5

-^^ ^

, which

has a simple pole at the point s = <j, with residue equal to

21

Tr

*

Erom this it follows that to regularize system (33) we must

utilize formula (l) from§2U, Chap. IV.

Erom system (33) we find;

/( s> f(s) - H V -P- {LX

W- Ho[kir (z - s)ldz

s

+ Jo JT H

o[k2r(z ’ s) H ’ (39)

s

ps) = - i- 0(s) + V.p. {///*) JTs Hot k

l r(z > s)]dz

-

/

0V<z) H

o [k

2

r(z - s)ldz

]

• m s

Substituting (39) in (**) of system (33), and ( U0 ) in (*), we

obtain

X

X

/

>(s) I^ Ho[V(s -°')lds + 2i/M + V - p - l

X

[• H 0 (s)

+ Ij V.p. jsUjH^kjpz.s^dz - -jr V.p.

(33 bis) H^[k

2

r( z ,s))dz]^- H2[k

2r(s,o-))ds = f(cr) . s ^

/oVs) ^[k2r(s,<r)]ds + 23^(0-) - V.p. f0

f {^-l(s) - i-

* V - p ' Jo P

{z) Hot k

l

r(z ’ s > ]dz - i V.p. // jx(z)

s *

x IT Hot k

2

r(z » s)]dz

) TT Hot k

ir ( s ,°')] ds = - 0(&) . S &

Let us apply formula 1, § 2U, Chap. IV to the double integrals

encountered in these formulas:

jo TJT^

H^[k

1r(s,o')]ds + 2±p(ar) - ^ V.p.

x Jf TsJJ ^o^

k

2r ^ s ,cr^ ds " 2±P

^ + H f0

XAz)dz V * p *

*!o JT H^ kir(z,s)] H^[k

2

r(s,o-)]ds - ft Ji(z)dz V.p.

s cr

*Jo 5H” H

o[

k

2r(r - s)] H

c

(k

2r)ds = f(<r> ! s

JJ P-M H^[k

2

r(s,cr)]ds + 2in(cr) - i- V.p.

xJo

X f(s > Holh.rtB '°')lds 2i^

(<r) + h SJ P

(z)Az v -p-

^

?r Ho

(k

2

r) 75- H

o

(k

ir > ds + a Jo

v -ps cr *ft H

2

(k-,r) H

2

(k-.r)ds = - 0(cr) . Jo ?n o

v 17 2s,_ o

v 1' s & J

Now introduce the designations

KTa

H

o[kl

r(z >°')1 S TT Hot k

ir(z - s)1 ll; ^[k2r ( s,cr)]ds

- K(z,cr,k^,k

2

);

21 V ' p - If db H

ot k

2 r ^ z ’ s)] IT" H

o[

k

2

r(s,o-)] ds = L(z,<r,k )

•

f(or) + H V * p * Jo" ^ (s) ^^[k

2r(s,o-)]ds - V0*)

s

3 tt 2

^o*

Ho^2r ( s,Cr) J + 21 V,p ” f/ TT H^[k

2r(z, s)] H^[kLr(s J o-)]d£

= KCz^kg,^);

ll

2

fo* dJr U

o[k

lr(z ' s)] H

o[

k

lr ( z »°’)]ds =L(z,cr,k

1

). s

- 0(0-) + i V.p. ^ f(s) ^ H^[k^r(s,o-)]ds = 1?2

(cr) .

System (I4I) then acquires the following form:

Jq /(z)K(z,a,k

1

,k

2 )dz - ;i( z)L(z,cr,k

2 )dz = ^(o-).

fo ^(z)K(z,cr,k

2 ,k1

)dz + JJ- yo(z)L(z ,cr,k1

)dz =T2

(cr).

We thus reduce the singular system ( 33 ) to the system (1*3) of

equations of the first kind, which are now regular, as will be

shown below.

We obviously must now show, on the one hand, the equivalence

of system (I4

.3 ) and (33), and on the other hand, we must satisfy

ourselves that system (h3 ) does in fact contain no kind of singularity.

§ 31 . Proof of the equivalence of systems ( 33 ) and ( )j 3 )

• Obviously any solution of system (33) is also a solution of ( U3 ) too.

We shall show that, conversely, any solution of system (h3 ) is

also a solution of system (33).

11*9

For this let us introduce into the investigation the two functions

y(s) = J>(s) - i f(s) + iJ/ p(z) H^rU, s)]d z

s

- V - p - H 1/ ?

(z) TT H

j)

k

2

r(z . s) ] dz . O

X(s) ‘ Jl(s) *i-0( S ) + itZ/rW jfr H2[

2

r( z ,s)]dz

S

' v - p- h ZVz) ar Hot k

i

r ( z . s )] dz • s

Moreover, from (33 bis) we find

/( s > a

f(s) - klo kr $V( **«> ld* + v

s -p(h/^

2

X

/o

Z

^ (z > ff $k2

r( Z>S ) ] dZ - Qj) v.p.//^-H2[k

2

r( z , s )]d z

2

* ? -P-I*/>(t

> TT H

o[kir(t - z)]dt + (k) v -pZ V y

y

%

fo Js~

H

o[

k

2

r(z - s)ldz •/>>(*) air H

o[k2r(t - z)ldt - S o z

/l(s) -

- JT 0(b) - i- J

Xp(z) H

2 [k

2

r( Z ,s)]dZ + (?£) v.p.

s

N

* X* f(z) aT H

otkir(z ’ s)]dz - H v - p - X* ar H

o[kir(z > s)ldz

s s 2 * V -P- // /<*> W- HXr(t - z ’ ]dt -(h) t -pz x x

• * S TT H

0[kl

r(z > s ^dz V -P- S ?(t) aV H

2

[k

2

r(t, Z )]dt . s z

Let us put these expressions for p and p. intoY(s) andX(s);

carrying out the obvious reductions and simplifications, we obtain,

for jK s):

ISO

f(s) - - i V.p. j* H2[k

2

r( z>5 )]dz V.p. jP/CWjf- H^k-f (t,z)]dt

s z

' HI v - p - J? if H^[k

2r(s,z)]dz V.p.^f(z)7|-

S 2

+ HI

V - p- H if H2[k

2r(s, z )]dz V.p. J/

jL. H^[k r(z,t)]dt

s z

* V - p -

c Hot k f(t^)]dy + ^ V.p .f/jj- H

o [k

2

r(s ’ z)]dz

t- s

* V -P-i

X if H

o [k

lr(z - t)]dt?-P-Xf ^if $k2r(t,y)]dy . z t

Changing suitably the designation of the variables of integration, we obtain, as the result of elementary transformations:

Hs) - \

v.p.

-fH^k2 r(s,t)]dt V.p.

s Is” H^rCt.zJMsOdz,

and, once again applying here the equation [

( I ) ,

§

2b] for the successive integrals with principal values, we finally obtain

f(z)R(s,z)dz = 0 , (Ui)

where

R(s,z) = V.p.

JJH^[k

2

r(s,t)] y— H^[k

1r(t,z)]dt . (!£)

s z

Integral equation (Uf) must be satisfied, on the one hand, for

all values of s and z, and on the other hand, for any values of

and kg.

Taking into account the properties of R(s,z) and f( z), we obtain from (UU) that

f

‘ f(z) = 0 . (U6)

Repeating all the foregoing reasoning forX(s),.we show analogously that here too.

iSi

/

X(z) = o . (1+7)

Equations ( U6 ) and (b7 ) prove the equivalence of systems (33)

and (I4.3).

§32. Investigation of system (I43 ). It has been shown above

that

if- H^[kr(s,o-)] = u(s,<r,k) - ^

SiSl

where co(s,cr,k) is a regular function determined

fB(r) from formula (350 (§29).

Using this formula, we shall show below

that system (U3 ) contains no kind of singularity.

Let

£ = f(t) , 7i = 0(t)

be the equation of the curve (j'), and

f(t) = f(t+X

1

) ; 0(t

= 0(t+^) . (U8)

In particular, the variable t may coincide with the arc s,

reckoned from an arbitrary fixed point on (y). For generality,

however, we shall not assume this at present.

Imagine that point A, with coordinates £(t), ^(t), is attracted

by the mass />(t), concentrated at the point B with coordinates £(t),

yj(x) (Fig. 16). As regards the attraction, we have everywhere in

view a gravitational law such as the generalized logarithmic potentials constructed in §2£ will correspond to. In accordance with

this, the attraction exerted on point A by the element ds at the

point B is expressed by the quantity
*

.

'

.

.
15'2

/

) ( T )H

2

[kr(t ,T )]ds ,

and the tangential component of this force of attraction at point

A will be

co(t, T,k)ds /(x) • - 2i sin f

IT ds \ , 2i cos(r.s) , co(t,T,k)ds - — ds

Let us express these functions in terms of t and T:

r = y[6(T)-E(t)] 2

[t? ('c)-^(t )]

2

; ds = vV 2 (t)^' 2 (t)dt ,

cos ( r . s ) = y<t)[£(T)-Ut)l * VwDiM-nWI

y[?(t)-s(t )]

2

+ w(r)- r,(t >]

2 .yy 2 (t)^

, 2

(t)

and consequently

(T) cos^sl ds

(1

= ^(t)[t(T)-^(t)] + ^?

l (t)[7?(T)-7? (t)] Jt' 2 (x) + y} 2

( x)

o(r)dx

[£(x)-£(t)] 2 + [7?(X)-7j(t)]

2 y^ 2 (t)+7j

2

(t)

We shall assume below that S = f(t) and ^ = 0(t) are holomorphic functions of t and that f'(t) and 0

1 ( t ) are not simultaneously

equal to zero.

As a function of T the denominator of the fraction (U9) reduces to zero only for T = t, and in the neighborhood of the point t

x = t + At

We write the expansions:

oo

- Z » *•<*>

v

-1 (At) n y

nti)

(t

)

/ nl n=o n=o

oo

153

by means of which the following expression for .-°-?X-s. s

.) ds

readily obtained:

is

2

(At )(£’ 2 +V 2

) + (Vt"+y,rl") +

••• J^'

2

+r\ 2

+2At(Z , ^

, '+Y}'y) »)+•*•

[(At) 2 (^' 2

+V

2

) + (At) 3

(S»?"+Yi»yi") +•••] JV2 +i 2

1 2 1 2 separating out the (At)($ +rj

) $ after elementary transformations

we obtain

cos(r.s) — , 1 1 i i Ho = + !_ + ... r QS r-t 2 ^’2 +

t|'2

m

and since

-r 2/^x . >2, ^^(T) +7j^(T) =

^' 2 + ^j' 2 + 2 ( At ) ( £

1 ^

" +Y|»Y

l'') +

ds

£

l2

(*) +1

t 2

C'0

= 1 - (£'£" MV) + •••

. s'hn' 2

°°s ( r . s ) , iV.un

'f. ry.2

. i j t ... (5D r T - t 2 77^727572 ^ ? "1 j ^ ' (S +7? )

In the right parts of (50) and (50 let us for brevity denote

the sum of the regular summands by 2 ^ and S 2

$ ‘then

cos(r.s) $-*-* ds•, = I.® =g_T + a

± , (50)

/ x it ' 2 »2x-l/2 cos(r.s) _ (g +7) )

' (51) + T - t

0

2

*

These formulas permit us to establish the character of the integral kernels of system (h3).

In accordance with (1*2) and (50)- (51), we shall have

15U

K(a,o-,k1.k

2

) - ^H^rU.o-)] i V.p.

XX^{~27—Xt

“ + 21

,S,+ ———E, + E(z,s,cr)]ds

J° (T^(z-s) (s-cr) TT(s-z) 2 ir(a-sj 1

J

+ O

1

(k1

,k

2

,z,or) , 2 1 n

. 2 z -O' lg nr

1 . X(z- CT) oTz-TT

(52)

where the values and 3 1

and E are obvious from the notation

adopted above and used in the transformations of the expression, and

Q l(ki,k2 ,z ,cr)

fln. H*[kjr(z,o)] *iv.p. ^ t-3 ’ + —S’ + E(z s a) s-z 2 cr-s l

^ > j ds

is a regular function for z = cr.

Thus we see that the functions K(z,cr,k^,k

2

) and K(z,cr,k

2 ,k^)

are regular functions; as regards the functions L(z,o-,k

2

) and

L(z,cr,k^), their regularity for z = cr is obvious from the integral

representations (lj2).

Hence it follows that system (b3) actually has no kind of singularity; let us rewrite this system in the following form:

+ O-L(k1

,k

2 ,z,a)y0(z)jdz

- JJ-

^(z)L(z,cr,k

2

)dz = ^(cr).

M53)

+ Q 2

(k1,k

2 ,,z,CT)^i(z)jdz

+ ^ /(z) L (z,a,ki)dz = F

2

(cr).

§ 33. Reduction to a regular Fredholm system. Let us define

the function, regular for z = cr.

= R(z,cr) .

' 1 , z(l-cr) 1

" + 5F-?

y

Differentiating system (53) with respect to <r term-by-term,

(one can readily satisfy oneself of the legitimacy of this operation),'*' we obtain:

I? V ' P ‘ fo^^z^tR ( Z ’ Cr

') + ^{(kls k

2 ,z s a)]dz - Jq

^(z)L '(z,cr,k

2 )dz = F^(cr),

IT

r

7

V ’ p -fo^

J

^

[R(z

^ + ^2(k

l^

k2’^ a)]dz + /0^o(z)L

! (z,o-,k

1

)dz =F

2

(a)j

the sign 1 in these equations denoting differentiation with respect

to cr. Lastly, employing the notations

-|-[R(z J a) + = M(z,<r) •

-|-[R(z ? or ) + Q 2

(k1

,k

2

,z,o-)] = N(z,cr)

j

(5U)

we have

V.p. J/ “4^ ^(z)dz - /i( z)L'(z,o-,k

2 )dz = T^(cr) ;

V ‘ P ' Ic

/

1 ( z ) dz + /(z)L (z J (T

J k

1 )dz = ^(cr) . >(55)

Let us remark once again that the functions M(z,cr), N(z,cr) }

d. d. L'(z,cr,k

2

) = ^L(z,(T,k

2

) and L’(z, cr,^) = ^ L(z,cr,k1

) are regular functions of z and cr„ Besides this, not one of these functions

vanishes for z = cr. We shall not adduce here proof of this statement, which reduces to a simple calculation of the values at the

point z = cr of the functions mentioned.

1Hobson, Functions of a Real Variable, p. 599.

156

Multiplying all of the elements of equation ( 55) by an(} cr

integrating with respect to cr across the interval from 0 to we

obtain, on the basis of formula (I),§ 2h, which we utilize repeatedly,

- Au

s )p(s) * V.p. ///*>dz dcr - v.p.

*[/}>( z)dz [X — ' dcr = V.p. Ilisl

Jo / Jo cr - s ^ Jo cr - s

dcr

- V

2

N( S , S ^( 5 ) * V.p. I' i.Vi l' p

; dv V;.

'// j? diifAi PgU

Reintroduce the designations

tt f? 1 M(z.cr) , , \ v.p. / ^ - —pr trr~^—r dcr = a(s,z) : Tr 2 (cr - S ) ( z- cr) M(s,s) 5

V.p. 7

L’(z,a-,k2 )

cr - s

dcr = c(s,z) ,

V.p. f - —z — —— dcr = b(s,z) 5 TT 2 (cr-s)(z-a) N ( s > s )

TT fX L, (z, ff ,k!)J ,, , V -P- Jo — da * d(s > z) -

v.p.

n nisi . e(s) v.p. [X d0- . g(s) * Jo <? - S

’ r Jo cr- s

Then at last we have

/s) f* + JoA y)( z)a(s,z)dz - JJ~

;i(z)c(s,z)dz = e(s) ;

/i(s) + J/ ;i(z)b(s,z)dz + fj~ f( z)d(s,z)dz = g(s) .

^ (56)

And system (56) is in fact a regular system of Fredholm equations, the obtaining of which constituted our final task.

15 ?

EDITOR'S NOTES

Note 1. In equations (l8), (19), and (20), page 38 of translation,

Kupradze failed to state the following relations, without which the

coefficients are not completely defined:

(a) Equation (19) is incorrect for s » n - 1.

Write instead

If s - n - 1, n * 1, replace o by 2 .

°° 0 If n 0, then ce

Q (l > ^

m Z f

3 s cos 2 s 1/ I s*o

(3 a/d°

0/8 qx | (5% " |

(-a + b)/8q 2ft

P s " 2P Ps * &h.(/^s~l

+ ^s+1^ > s * 3

Note 2. In equations (22), (22"’) and (23), pages L£> - U8 of the

translation, the author gives bounds tor/3:, » It has not been shown

o that A

0 / 0 o On page U6, immediately following (22") it is stated that

A

q

is different from zero for small values of qs this is true „ No

corresponding statement can be made for general values of q. It has

been pointed out in the Editor 8 s Preface that Kupradze used Whittaker's

normalization, and in that normalization (3 can in fact become

s infinite for a certain set of discrete values of q. This was shown ty

S« Goldstein in 192 7

Note 3» It is not clear what the author meant, in the statement

after equation (50) , page 65 of the translation, about replacing GT) by

-Y|5 such replacement is unnecessary e Moreover, the Russian text has
.
158

( 2 ) the -wrong sign in the expansion for H

2k '^l

1^ sin + 7

y 0, 1, the first two equations of (5>CT

c

) . This was corrected in

the translation,, Similarly the sign was wrong in the equations for

Se2n^ ’ ^ anc^ Ze2n^ 9 ql^ an eclUa^ions (£l) on page

‘ 67 . While it

is true that the change in sign merely affects the normalization of

the solution, it is best to retain the sign required by the fundamental

definitions I, II, III, IV, on page !?5> of the translation.

Note U. A slight liberty was taken in translating the second

sentence on page 68 of the text* A literal translation reads

"utilizing the well-known asymptotic expressions of

the Bessel and Hankel functions for large values

of the parameter.”

The author obviously did not mean that.

Note Page 60 of translation, immediately following ( U0) s The

phrase "dropping the factor (-l) n

exp in 1J" does not appear in the

Russian text. A literal translation reads: "Let us study the expansion in a Fourier series of the following solution of the vibrational

equations

H

n

(2) (\A

2 + - 2R cos 00 )

The above function by itself is not a solution of the vibrational

equation, and it is believed the author really meant what the altered

translation implies

„

in©

e

159

Page

15

17-18

17

27

28

30

35

36

39

LiO.

ERRORS IN RUSSIAN TEXT

Footnote, Debye (spelling)

Summation index and independent Variable both denoted by r in

the same equations. Correct as in translation.

Line lit - Replace kr in the denominator by Vk r

Footnote 1, E. L. Ince

Eq, 19, Superfluous factor 2$ replace c^^n) by . S 1 S+l

Correct as in translation,

Eq. 20, Superfluous factor 2,

Also see Editor's Note 1.

Third line from bottom. Replace b^3^

by b^s+l^ .

Eq, 25. Replace k-^ cos yj cos 9 by i k^ cos 07 cos 0 in

exponent

Equation inpnediately following eq. ( 27 ) and eq. (28); signs

incorrect, superfluous factors, and missing factors. Correct

as in translation.

Last equation on page, add factor (TMl) r

.

See Whittaker and Watson, Modern Analysis p„ I4II for comparison

Equation immediately preceding (l)s Read 9 * arc tg

Eq, IV. Read ce2n+-^ (V .qf under the integral, in place of

se

2n+l

(7f ,ql^

First equation. Read + O(^)

Line 10, Replace Se2n by Se2n (€. ^q^).

Last line and 5 lines from bottom. Replace o(l/cosh )

by 0 (l/cosh^ £ )
*

.

-

f

-

..... •

'

:

'

'

'

' ' "

J.

*
160 Page

111 Line 3. Sign incorrect. See translation.

U3 Top of page. Add ''Dropping the factor (-l) n

exp i mj" .

See Editor’s Note.

kk Nine lines from bottom. Right-hand member of equation for

j

•- C (R) J (P); Exponential factor wrong. See translation.

S'TMII s *

U6 U lines from bottom. Replace (-l)

S ^ by (-l)

s as in

translation.

bl Signs wrong in first line of page, and in first terms of

Se^Of ^q^) and ^®2n^l^l^ * Also notation errors in coefficients. Correct as in translation.

U8 First line of second paragraph. Replace Selmby Se

2m

.

9 lines from bottom. Replace N - n by N - m.

£0 Equation immediately preceding (£ 7 ) . Symbols wrong. See

translation.

£2 Replace ce^

g

by cej^ in first eq., as in translation.

66 Equation preceding (92). Symbol should be^-^.

68 Eq. (96) . Replace a ~ by X

s ji s

}

r 77 Second equation. Add = 0.

'78 Equation for . Replace F(cty) by F(ct - y)

108 Last equation. Replace S by S .
• • •

'

"i
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act

of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

These include the development and maintenance of the national standards of

measurement and the provision of means and methods for making measurements

consistent with these standards; the determination of physical constants and

properties of materials; the development of methods and instruments for testing

materials, devices, and structures; advisory services to Government Agencies on

scientific and technical problems; invention and development of devices to serve

special needs of the Government; and the development of standard practices,

codes, and specifications. The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services

and various consultation and information services. A major portion of the

Bureau’s work is performed for other Government Agencies, particularly the

Department of Defense and the Atomic Energy Commission. The scope of activities is suggested by the fisting of divisions and sections on the inside of the

front cover. Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and

devices or published papers and reports. Reports are issued to the sponsoring

agency of a particular project or program. Published papers appear either in the

Bureau’s own series of publications or in the journals of professional and scientific

societies. The Bureau itself publishes three monthly periodicals, available from

the Government Printing Office: The Journal of Research, which presents complete papers reporting technical investigations; the Technical News Bulletin, which

presents summary and preliminary reports on work in progress; and Basic Radio

Propagation Predictions, which provides data for determining the best frequencies

to use for radio communications throughout the world. There are also five series

of nonperiodical publications: The Applied Mathematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can he found in NBS Circular 460,

Publications of the National Bureau of Standards ($1.00). Information on cali- bration services and fees can he found in NBS Circular 483, Testing by the National

Bureau of Standards (25 cents). Both are available from the Government Printing Office. Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications, National Bureau of Standards,

Washington 25, D. C.
NBS