Integral representations of solutions of the Helmholtz equation with application to diffraction by a strip

INTEGRAL REPRESENTATIONS OF SOLUTIONS OF THE HELMHOLTZ EQUATION WITH APPLICATION TO DIFFRACTION BY A STRIP

PROE FSCHBIFT

ter verkrijgthg van de graad van doctor In de Technische Wetenschap aan de

Technische Hogeschoól te Deift op gezag van de Rector Magnificus Dr. R. Kronig, Hòogleraarin de Afdeling der Technische Nátuurkunde, voor een Commissie uit de Senaat te verdedigen op woensdag 26 april 1961 des namiddags te 4 uur

door

RALPH ELLIS KLEUMAN geboren te New York

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR. PRÖF. DR. R: 'I'ÏMMAN

The research presented in this thesis was supported by various agencies of the Unitód States government. The Wörk was begun when the author was a Fuibright grantee at the Technicäl Unlversfty1 Delft, In the

academic year 1958-59 while on leave of absence from the Radiation Laboratory of the University of Michigan. During this time the Veteran's Administration also contrthuted support wider the "G. I. Bill". In the subsequent fifteen months, the work was carried out under ontractual support -of the Radiation Laboratory with the Department of the Air Force

CONTENTS

Chapter 1: INTRODUCTION i

Chapter 2: A CLASS OF SOLUTIONS OF TÌIE HELMHÓLTZ EQUATIOÑ 6

1. Some Remarks ön Superpcsiton - 6

2. Derivation of Non-Triviai Solutions 8

3. An Alternate Proof 12

4. Some Properties of the Solutions 14

5. A Limiting Case 15

Chapter 3: REPRESENTATION THEOREM FOR WAVE FUNCTIONS

SATISFYING D]BICHLET CONDITIONS ON A LINE SEGMENT 17

The Representation Theorem 17

Some Remarks on the Application of the Representation Theorem 22 Integrai RepresentatiOn of a Class of Cylinder FunctiOns 2

4 The Integral Representation of Line Sources 33 5 Asymptotic Behavior of the Line Source Representation 43

6. Double Integral Representations 46

Chapter 4: DIFFRACTION BY A HALF PLANE 54

1 Important Relations in Parabolic Coordinates 54

Parabolic Fórm of the Representation Theorems and Applications 57

Green's Ftthctïon for the Half Plane 63

Chapter 5: DIFFRACTION BY A STRIP 72

important Relations in Elliptic Coordinates 72

Derivation of the Basic Solutios 74

4 Limiting Case: Laplace's Equation 82

Elliptic Form of the Representation Theorems and Applications 83

Green's Function for the Strip 90

6 Analyticity of the Diffracted Field 95

7. Properties of the Solution 01

8. Some Further Properties 108

Chapter 6: CONCLUSION 1 1.1

References 118

Chapter 1

INTRODUCTION

Exact solutions of diffraction problems are rare an until recently there was only one method of solution applicable to more than one particular pröblem. By diffraction problem is meant the génerai problem of deter-miniug a solution of the Helthholtz equation 'which Satisfies homogeneous boundary conditions, either Dirichiet or Neumann, has a prescribed character both at Infinity and In the neighborhobd of any edges, and which may be singar only at points correspóndiig to sources. The physical problem then is the determination of a spatial field when an obstacle is in the presençe of a time harmonic source of wave motion. Although the term

diffraction is usually applied only when the boundary, m the language of geometric optics, has an "illuminated" and a "shadow" region it will not prove inconvenient to include the limiting eases öf no shadow, e.g. when the

boundary is an infinite plane.

The classic teöhnique produces solutionS as infinite series of eigen-functiòns and is limited to coordinate systems in which the Helmholtz

equation is separable and boundaries which aré level surfaces of these coordinate systems. Ifliporta.nt as this method is, the usefulness of the solutions thùs obtained aiost invariably suffers because of the slow con-vergence of the series.

.Exabt solutions obtained in "clösed form" are even

_{rarer. What}

constitutes a closed form solution is subject to debate but. it is generally agreed that infinite series do not qualify. When Sommerfeld (Ref. 25) intro-duced his many valued wave functions to solve the laif plane problem in 1896, he thought this idea could be extended to solve other diffraction problems, notably that of the strip. _{However, all subsequent atteits at extension,} exöept to wedges of which the half plane is a special case., have been unsuc-cessful. Thus the Sommerfeld approach remains a technique, indeeda remarkable and elegant one, for solving a particular problem rather than a general method of solution.

More iecently, Wièner Hopf techniqués have been Successful in treating certain problems involving parallel half planes (see Bouwkamp, Ref. 3, for a discussion of Wjener Hopf techniques in diffraction theory). Exact solutions by th.is teähniqué are, to date, limited to boundaries of infinite extent.

A simpler Integral equation formulation than that previously used in the Wiener Hopf treatmeit has been given by Ciemmow (Ref. 8) The scattered field is considered as a superposition of plane waves of complex angles of incidence and in this sense is the method comparable t Sommer-feldTs. technique. This approach does not èlimiñate the restriction to

n the present work, howevér, we shall concernourselVes with a method for obtaining exact solutions in closed (integral) form even when the boundary is finite. A general class of solutions of the Helrnholtz equation is derived which resemble Clemmow's and Sommerfeld's functions in that they are superpositions of elementary solutions except that the superposition is accomplished in a manner such that the solu,tions assume particularly simple

form-on boundaries which are levél surfaces of thé coordinate system used. The usefulness of these solutions is demonstrated by employing them to constrüct an fxact integral representation of the field diffracted by a strip.

ThIS problem of diffraction by a strip Ms occupied a prominent place in thé Uteratwe since the work of Lord Rayleigh(Ref. 2 i)who found an approximate solution for long Wavelengths in 1897. Since that tme exact slutibns in the form of infinite series have been found by Schwarzschild (Ref. 23)who used the Sommerfeld half plane solution as a basis for calculating

successivé interactions between the two edges, an Sieger(Ref. 24)who

found the Solution in terms of Mathieu functions as suggested by Wien (Ref. 34). We shall not attempt to give a complete bibliography but refer the reader

to the. treatments of this problem given by Sommerfeld (Ref. 26) Baker and

Copsàn (Ref. i) and Buwkamp (Ref. 4). Bouwkamp's exhaustive: survey just cited covers the many attempts to find long wavelength approximations and summarizes the numerióal results. Short wavelength approximations have also been sought and, again without claiming completeness, we call

attention to the work of Clemmow (Ref. 9), Karp and Russek (Ref. 12), Levine (Ref. 14) and, Millar (Ref. 18), whose treatment is based In parton

the Schwarz schild approach of successive Interaction, and Burger (Ref. 6) and Timmàn (Ref. 30) who applied techniques of supersonic airfoil theory to the hyperbolic (time dependent) wave equation.

The success of the present approach depends on the fact that the problems of diffraction .by a hail plane anda strip are not really independent and IIi fact the method consists, in part, of transforming the solution of _one into the solution of the other.

This Is accomplished by using the new sOlutions of the Helmholtz equation to derive an integral equation för the wave function satisfying a

Dirichiet cönd tion on a line segment. We write these integral equations for both the haf plane and strip problems, one In parabolic coordinates and the other in elliptic coordinates, and then assume that the unktiown fuiict1ons in the integrand are related via the same transformation relating the two coordinate systems. Since the half plane problem has been solved, we are able, to obtain an explicit representation of this function, hence, if the assump-tion is valid, we also obtain, using the strip integral equaassump-tion as an integral representation, the solution of the strip problem. The validity of the

assumption is established by demonstrating in detail that it does indeed produce the solutlön of the problem of diffraction by a strip.

As will be evident, the conétruction of the solutions depends vitally on the fact that we consider line sOurces rather than plane wave incidence.

The plane wave solutions may be obtained but will involve a rather compli-cated limiting process. Since it is almost standard procedure in diffraction theory to consider the plane wave case first, the fact that line sources are apparently more appropriate in this case may help to explain why the strip problem has resisted closed form solution for so long.

We shall confine our attention entirely to two dimensional problems which are particularly appropriate to the present approach. The possibility

of extension to three dimensional problems is not to be excluded but will not be treated here.

 óLss OF SOLUTIONS OF.THE HELMHOLTZ _EQUATION

In this chapter we shall derive a class of: Solutions of Helmholtz' equation which provides a basis for afl that f011ows. These söltitions can be öharaóterized as a non-trivial superposition of élementary solutions Where, as will be seen, non-trivial denotes that we integrate elementary wave fudctions between variable end points..

1. Sóme Remarks On Superpostión

±jIo ±fky

Although plane waves of the form e ana e are the elementary solutions f the homogeneous Hèlmhóltz equation (2. 1. 1), in rectangular coordinates,

2

Chapter 2

}

we shall consider line sources, the elementary s lutions of cylinciticai coordinates.

Following convention, we will use three dimensional terminology to describe two dimensional problems; thus we shall speak of line soirees rather than two dimensional point sÖuces, diffraction by a half plane rather than a half line, etc.

In cylindrióal coordinates the non-homogeneous Helmlioltz equation,

for a source at r = r-

-.

, e = e

o o

- _

J2

.2

R=r-r =Vr+r -

_{o o}

wIth a suppressed time

2) (kR) H(2) \/(x

-

X =47r

2rrcos(6-9 )

= \I(x-x o

±(y-y)

has soIutlóns 4?

l7rH(kfl), iir

H'(kR),

and -r N(1R) where

(2)

dependence of e , -iir H kR) represents waves diverging from R = O. Neg1ectin the constant faötor we write this as

lii keeping

)2)

= H(2

\jx+

_{iy) - (x.+ ry5 [(x_ iy)- (x}

_i})

and observe that the function obtained by setting y O, replacing x by some óomple a, multiplying by an arbitrary function of a, and integrating over a, viz.

H(x+iy-a)(x-iy)) fa)da,

(2.1.3)

o

will stifi be a sòlution of the Helmholtz equation If the contour ë is independent of x and y and f(a) is sifficient1y well behavçd The points of the path of

integration represent sources of strength 4i«c). Siniilai'ly, letting x O

and y = io' we obtain

'F

(x+iy+a)(x-i) ) f(a)da

(2. 1. 4)

whiëh also remains a solution.

bi a sense we have reversed the usual superposition where the elementary sólution f rectangúlar coordinateS (plane wave) is written iii cylindricál

coárdinates and thtegratd over complex angles of incidence of the plane wave, i.

ç' ikrcos(O-a)

\ f(a)e da

sic

We have writtén the elementary solution of cylindrical coordinates and integrated over complex positions of the line source.

2. Derivation of Non- Tivil Solutiöns

At first this reversed sùperposition may appear wmecessarily

complicated since now we must worry about the branch points of the integrands. HoweVeï', by òhöosing a particÙ1ai path we are led to a .ather surprising and interestin résu1t naniely: if «a) is analrtic in a simply connected region containing the path ofintegration and A is constant, then the expressions

x+iy -

-(k\/(x+ia(xy.a)f(a)da

(2.2.1) A

X]3

_j

J (k

(xl-iy-a)(x-iy-a) )fá)da

(2.2.2)

o\

f

S-x-iy

A

J(k(x+iy+a)(x-Ïy-a))

f(a)da. (2.1.5) (2.2.3) and

-i

JV(x+iy±a)(x-Y-a)) f(a)da

(2.2.4)

are all solutions of thé homogeneous Helmholtz equation in this region. J denotes the ordinary esse1 fi.tnction of order O.

To see that this is true, first for (2.2. 1) we proceed as follows. In formula(2. 1 3). where the integrand has branòh pQintS at = x± iy, take the path of: integration to be a loop enclosing the point a = x+ iy as shown in figure 2.2.1.. 9ma. Rea -jy

j

FIGURE 2.2.1: a-:PLANE

Béfore discussing the branch cuts, let us note that, despite the square root of its argument,

J4c V(x+

iy - a)(x - iy a)) is an analytic function Of a in the entire finite a-plané, so we 'need Öuly Oonöèt OÉr sòlvéé With the logarithmic branch points of the Hankél function. These can be separated

-2i

J0(k\/(x+iy_a)(x_iy_a))log\J(x+iy_a)(x_iy_a) )

+

F(\f(x±iy_a)(x_'iy_a'))

(2.'25)

where F is an analytic functiOn Of a. This is evident on looking at the serlés representations Using the shorthand p

\j(x+iy_a)(x- iy-a)

, formula (2 1 3) can be written as

out as follows:

r2Qcp)f(a)da

=

J(kp)logp2f()d

_F(p)f(a)da.

_(2.2.6)

Since «a) is assumed analytic throughout a region containing the contour

e and F(p) is also an analytic fu ction of a, the contour shown in Figure 2.2. 1 is really closed for the second term on thé right hand side of (2. 2 .6) hence, by Cauchy's theoreni, this term ianishés. We have written the logarithmic part as log p2 to eliminate, when defining the branch cuts, any cornlications due to the square root in the definitiOn of p.

We choose as a branch cut the negative real axis in the pTh.nè. To

see what this maps into in the a-plane, we write a = + irj and examine

p2=(x+ly-a)(iy-a)=(x+iy--ir)(x-iy=-lr)

(x-)2-fl2+y2+ifl(x).

(2.2.7)

The condition, p2 real

> 2r(-x)

O and

p2<O(x-)2-?72± y2<O.

If rj =0 the second of these is violated.hence _{=x, Tl>IyIand} x, are

the branch uts in the a-plane, as shown in Fire 2,2. L

To keep track of Which sheet of the Riemann surface óf logp2 we are

dealing with, wé employ the following notatiön: we defie the n sheet to. consist of all values of logp2 where (2n- l)7r <argp2< (2n+ l)ir and indicaté this explicitly by writing lOg _p2. Finally, by requiring that arg log p2 = O,

we remove ail ambiguity from the definition. The values of log p2 on tVo

succesSive sheets are related by

Now we can make precise the meaning of the contour shown in Figure 2 2. 1 Starting at the point A, we choose the principal value of the logarithm,

log p2, and let the function vry cöntliiuously along c. Thus thè solid pórtion of c indicates points on the sheet of the Riemann surfaóe o the logarithni,

st

but continuous variation açross the branch cut takes us onto the -i sheet which is indicated by the dctted portion of e.

Since w have asumëd f(a) to be nlytic, however, we may deform

the contour tö that shöwn In Figure .2.2.

Re a' =

Using relation (2.2.8), this beöomes

X +_jy A

I

FIGURE 2.2.2: a-PLANE The integral in (2.2. 6) can noW be written

5

H(kp)f(a)da

=

5

wé find that

J@/(x+iy_a)(x_iy_

) L a = x- iy .2

ky

2 2

ky

2 (2.3.1) x+iy

H _{Oçp)f) da 2}

_{J(kp) fa) da.}

_(2.2.10)

Since the 'left hand side of (2. 2. 1Q) is a sölution of Helmholtz' eqûation, the iight hafld side is also. Hence (2. 2. 1) Is shown to be a solution of the

Heimholtz euatióñ. This same rocedtthe can be readily used to establish that (2.2.2), (2.2. 3), and (2.2. 4) are alsó solutions and it wOuld be needlessly repetItiouS to do this explicitly.

3. An Alternate Proof.

An alternate procedure, consisting of direct substitution În thé differ-ential equation, can be employed to establish that. the expressions (2.₂ 1) -(2.2.4) are Indeed solutions of Hèlmhóltz' equation. Contrary to the method used above, this gives no hint as to how the relations were found but does have the advantage of being somewhat simpler. This is illustrated by

demonstrating directly that (2. 2,2) is a solution. Whereas before Wé needed f(a) to be analytic in order to deform contours, now the aa1yticity forms a sufficient condition to permit differentiation according to the usual rule.

Thus, keeping in mind the. following easily verifiable relations,

J0(k\,Ax+iy_a)(x_iy_a)) la

= x-iy =

a

ax

X - jy

J0V(x+iy_a)(x_Y-a)) f(a)da

x - iy

f(x_iy)+f J(kV'(x+iY-a)(X_iY-a

I(a) da

(2.32)

f - - . --

\

af(x-iy)

ik:yf(x-iy)

±iy-a)(x-iy-a) jf(a)dc =

-X-iy2

r

± J

((x± iy-a)(x iya)) f(a)da.

'A

(2.3.3)

Similarly

r'x-iy

JV(iY-a)(x_iY_a))f(a)da

_1(x-iy)

+ f(x-iy)

.x-iy2

-z

J(k (x+

iy - a) (x - 'y -

) I(a) da.

..3.4)

Siflee iy) , upon substitutionof (2.2.2) in the Heimholtz

ay ax

equation we obtain, with (2.3. 3) and (2. 3. 4)

2

'A

a ax

2 2

J(

x-i-ly_a)(x_iy_a))

f(a)da

=

C'Y12

\

ii

+

2+k, J(I(+iy-a)(x-iy_a) )f()da. = O.

(2.3.5)

2

J

'A L

Hence (2.2.2) Is a sólution of the Iielmholtz equation and, of course, an almost Identical procedurecould be used to establish that the other relatiOns (2.2. 1), (2.2.3). and (2.2.4) are also Solutions.

4. Some Properties of the Solutions

Upon subt±act-Ìng (2. 2. 2) from (2. 2 i), the constant end point of

integration IS eliminated and we obtain a function, Ø(x, y), with some

remark-able propetles. Explicitly'

rty

Ø(x,y)= \

/(x+iy1

iy_a))f)da

. (2.4.1)

4

If the path of integration is entirely confined to a simply connected

regioni R, where f() is analytic, then with no other reStrictions on f(a),

Ø(x, y) is a solution of the homogeneOus Helmholtz equation and vanishes on the Segment of the line y = O lying in R.

As will be seen shortly, solutions of the form (2.4. i) and (2. 4.2) can be constructed for the non-homogeneous Helmholtz equatioñ by allowing fa) to have singularities. We shall make use òf thèSe eçpressions in the following chapters to find integral representations of solutions of some boundary value problems, fOr which purpose these functions are obviously well suited.

5. A Limiting Case

To end this chapter we call attention to One of the Îiost immediate consequencés of the particular form of the Solutions (2.4.1) and (2.4.2). With thé shnplèst (a stibjective but hopefully not an unreasonable judgment) non-trivial choice of f(a), namely f(a) 1,. (2.4. 1) and (2.4.2) become the.

Helntholtz equation generalizatlOns f the. slntplest nQntrivial solutions of Laplac&s equation, y and x.

y)

_(x,

yO

y). y=Q = 21f(x) (2.4.3) ax ay and ao(x, y) x=O =. 2f(-iy) x,y) x=O (2.4.4) ax ay

x-iy

çli(x, y) =

J0(k\J(x+iy+a)(x_iy_a)) f(a)da

(2.4.2)

-X- iy

where ç?i(x, y) vanishes on the apprqprlate segment of x = O.

Further, the derivatives assume particularly simple form on these boundaries:

Thus + jy

J\/(x+iya)(x.iy

x-iy

11m

k3 O

X-ç'x_iy

\ J\j(x+iY+a)(x_iY_a)) da

J-x-iy

.'x+iy

\

J(k \J(x+iy'ia)(x_iy_a)

x,-iy x-4)' «

cx_iy

o

\

J(k\/(x+iy+a)(x_iy_a))cla

\

da = 2x.

(2.5.2)

J-x-iy

The expressions for the wave functions can be simplified considerably. 'In the first case, with the substitution a = iy cos 6 + x, we find that

ira)(x

iy.a))

2 iy

J(kysinÔ) sinO dO.

This last form can be integrated explicitly, (see reference 16), obtaiiàing

2i

=

. sinky

(k )

2/2

Put

1/2 y

- (ky)h/'2

Similarly, the substitution a xcosO- iy in the second form enab1e us to write

While these forms, are, of course, among.the, most-elementary wäve functions it is noteworthy that when the expressions comparable to (2. & i) and (2. 4.2) are developed in other coordinates,, (see for example the discussion of the elliptic coordinates in Section 5. 3), it is possible to find wave equation

2.5.3) (2.5.4) (2.5.5) = siflkx. (2. 5 6) + iy da = 2iy _'(2.5.1)

Chapter 3

A REPRESENTATION THEOREM FOR WAVE FUNCTIONS SATISFYG DmICHLET CONDITIONS ON A LÏNE SEGMENT

Iti this chapter, we use formula (2.4.1) as the basis for an integral representatiOn theoìem for ceìtaln SôlutiohS of the Heimhóltz équation satisfying Dirichiet boundary cönditions. This theorem is then ethployed to obtain integrai répreséntatibns of combinations of cylindr funetios. Particular attention is devoted to the case of the line sourpé, 11(2)₍₁

1. The Represeùtation Theorém

With the uhdérst&ndfng that by an function of the real variablés x and y we mean that the function has a Taylor éxpansion in X and y but not necessarily in z = x+ ly; the fundamental result of this Section1 the repreSentatïon theorem, Is formulated as follows.

Theórém: II. Ø(x, y) is an analytlà solution of the Helmholtz equation in a simply connected region, <, containing the line segment y = O, x1 <x <x, and Ø(x, y) = O on this segment, then in this region, ', Ø(x, y) has the

intègral representation x+iy

Ø(x, y) =

_{J\j(x+iy-a)(x-iy-a) )}

a,v;'

Xiy

Tbeproof of this theorem, proving that (3 1 i) is not an equation but an identity, consists of showing that both sides of (3 1 1) have the same

value on the line segment and that their normal derivatives are also equal on this segment Then, by vitue of the Cauchy-Kowalewsky theorem which ensures that there cannot be more than ohe analytic solutiOn of the Helmholtz equation in a neighborhood of a curve n which the fufletio and 1t normal derivative are prescribed, the validity of (3. 1. 1) aS an identity follows.

Thus the left hand side of (3. 1. 1), Ø(x, y), is given to be an analytic sOlution of the Helnthoit equation, vanishing on the line segment, and whoSe normal derivative on the segment is given by

= The right hand side of

(3. 1. 1) is an analytic solution of the Helmholtz equation since it is of thç form (2.4. 1); it obviously vaniSheS when y O; and its normal derivative

at y:0 (seê (2.4.3)) is

"o Hence by the uniqueness cited above,

(3. 1. 1) is establish as an identity.

Sommerfeld, (Ref. 27), shows very clearly and constructively why tm.queness obtainS when the solutión of n élliptic equation iS given, with its normal derivative, on acurve. Hadamard, (Ref. 11), discuSses tbe more

general results of Cauchy and Kowalewsky which guarantee existence as well as uniqueness. He also cites the work f Holmgren which indicates that the requirement of analyticity niiht be weakened to a condition of stili icient regu]ariy derivatives up tö second order throughout the region, but we shall not consider this possible generalization at present.

ote that a completely analogous representation of wave functions vanishing on the line x=O can be obtained using the expreSsion (2.4.2), namely:

connected region, , containing the line segment X = O, y . y. Y2 and

1ì(,c, y) = O on this segment, then, in this region . çli(x, y) has the representation

i

çli(x, y) =

If çlì(x, y) is an analytic solution öf the lieiniholtz equation in a. simply

J (k \J(x+iy+a)(x_iY_a))

-x-iy

The proof completely parallels that given above.

The theorem of this section can be cons ered in tWo different ways On one hand it provides a method of obtaining Integral representations of solutions Of specific boundary value problems if the solutiQns are known. Qn the other .hand, if the solution is not known, (3. 1. 1) provides us with an

integral equation Whic1it must satisfy. This integral equation is akin to that obtained through the use of Green'.s functions except that here we have a Volterra equation where the path of integration des not have such a ready interpretatioi as a physical boundary.

This integral. represeittation can also 1e. derived, as the Bessel function.kernei suggests, by employing Riemann's method of integrating the linear second Order hyperbolic differentiâl equation (see Ref. 28). Intro-ducing the Characteristic coordinates

x+ly transfòrms the elliptic wave equation

± .

+k2}

Ø(x,y) _O

(3.l4)

(3. 1.2)

into the hyperbolic equation

4

)

ø[;

r7J (3.1. 5)

Following Riemaun, we express a solution, _0' of this hyperbolic equation in terms of its values, on a

.th the .r-plane in a manner

completely analogous to the use of Green's theorem in elliptic equations, where instead of GreenTs funötions we employ thè ph.aracterstic nction or the

Riemann- Green, function

o (3.1.6)

Requiring the solution Ø [, r,] to vanish on the line r, (which corresponds to the bOundary condition Ø(x, O) = O in the xy-plane ) and choosing the

region of interest to be bounded by segments of this line and two characteristics, (see Figure 3. 1. 1) enables us to express the function at

, r in terms of

its values

r,

the segment of = r,, as follows

í rio-_1 =

so that

o

Since r is the segment of the line lying between ¿ and =

this can be sthiplified considerably. With the normal drawn outWard, cos(n, ) = and cos(n, ri) = thus (3. 1. 7) becomes

> O . (3. 1.10)

Íí, _{- ri} <O,, that is if the point (, r) were on the other side of the line

= ri from that shown in FigUre 3 1. 1 both the seflse f r and the sign of the square root would alter. In either case (3. 1. 8) becomes, writing y explicitly,

fro' «]

-

\

o{(/)

(a

-Although it has..been convenient to consider , ri _' ri

a,sreal,it Is

of course true that the fuflction defined by (3. 1. 11) is still a solution of the hyperboiiò eqùation (3.. 1. 5) and vanishes when = ri0 even if , ri' , _{and ri0}

become complex.. In particula.r, with the tranSformation (3. 1. ¿) it is easily

(3.1.8)

(3.1.11)

=

The distance

_s

ds mi.tt be positive if and are real thus we thust

r

define the ime element along

t'

as

seen that

and

a

(3.1.12)

8T)

lay

corresponds to yO Thus (3.1.11) becomeS X + iy

-o o

y)

d.

(3.1.13)

1

2i

J0( \i-

Xo=. YóXX_Xo+ iYo

/ay

x -

y=o

o

Renam1g the dummy variables appropriately and dropping, the subscripts yields formula(3. 1.1).

Choosing r to be a Segment of the line -r rather. than = would

lead to formu].a (3. 1.2) by the same procedure.

2. Some Remarks on thé Application of the Represéntation TheOrem. Ït must be pointed out that some cautiOn should be exérciéed in the uSe of (3.1. 1) as a representation of sôlutiöns of boundary value problems valid for all valueS of x and y.

-First of all, it is often true that physicaily significant problems deal with functions that are not analytio at the boundary As an eaniple,

consider the problem. of finding tie field, Ø(x, ), of a single line source in the ptesence of a perfectly soft strip, a problem considered at SOme length in hapter 5. An attempt to immediately Write the solution of this problem In the form (3.1.1) for points in the regione (see Figure 3.2.1) might be

unsuccessful because Ø(x, y) is not analytic in y (when x1 x (X, y) is not continuous at y O)

It is still possible to use the integral form (3. 1. 1) to represent the field in this case Without relaxing the analyticity requirement, by considering an intermediate probleth, or more correctly, by considering an alternate but équivalent problem; i. e., the mixed boundary value problem depicted in

Figure 3 2 2 The advantage m treating this problem is that the physical

y

(x,y)

source, point R (x,y)

field point

d

(x, -y) ordinary point

çL'=O

(x,y) - source point

(s, y)

field point

-y0)- ordinary point FIGURE 3.2.1: A LINE SOURCE FIGURE 3.2.2: A LINE SOURCE

IN THE PRESENCE OF A SOFT IN THE PRESENCE OF A SOFT

STRIP STRIP N A RIGID SCREEN

problem is confined to the upper hálf plane.:and an analytic continuation of to the lower half plane in the region is possible, even if çli has no physical meaning there. Henc çlì can be expressed jn the integral form (3.1.1).

That the problems depicted in Figures 3. 2. 1 and 3. 2. 2 are really equivalent, that is, give the sölutioñ to One. of thém it is possible to construct the sOlution tO the Other, is easily Seen. Following Bouwkamp, (Ref. 5) , the solution to the problem of Figure 3. 2. 1 cari be written, save

With this relation it is clear that the function (x, y), defined as,

çO(, y)

H(kR) - H(kR') + 2Ø(x y)

- W <X <W

y>0

is a solution of the problem depicted in Figure 3.2.2. Thus1 1iowing ny one of thé functions Ø çii, or 0D' it is possible to construct the other two. Further-more, in the region , these fu.ctiofis can be expressed as integrals of the

form (3i.1)

(3.2.3) (3.2.4) (3.2.5) &H(2)(kRT) ay 0

- -

ay 0

it follows from (3. 2. 2) that

(x, y) aH2) j X <X1 ay y=O y=0 x>x2 Ø(x, y) = (lcR) - H2(kRt) _{+ ØD(xJ y)}

0.

ØD(x, -y) i _(3.2.1)

where ØD(x. y) is finite fôr - w <x <w,, y >0 and vanishes for x1 <x <x2, y =0. Moreover, must be continuous at y = O when there is no physical

barrier, i.e., When x <-x1, or x > x2 hence, for these values of x,

;

[u

-

H(2)(kRj! ₌

0ØD(x) _Y)

. (3.2.2)

y0

y-o

y0

Of course the next step is to attempt, by analytic continuation, to obtain an integral representation valid for the entire range of x and y. This step must also be taken with caution. The functiOn Ø(x, y) in the strip problem and çlí(x, y) in the mixed problem both have non analytic behavior at three

points of physical significa.nce; (x=x, y=y), (x=x1, y=O), and (x=x2, y=O). The first of these, the sOurce point, can be eliminated by considering

ØD(x, y), the diffracted field. The other two remain however, so when extending the definition of the integral representation of ØD(x, y) to the çase when x

or x > x2, the path of integration will have to vary as illustrated in Figure 3 2.3 in order that the continuation be analytic.

FIGURE 32. 3: CONTOURS FOR THE INTEGRAL FORM _{OF Ø(x, y)} Thus it is seen how the integral (3.1.1), which apparently vaxiishes on the entire line y = O, actüally can represent a fùnction which only vanishes on a segment of tnat line. It will prove mOre convenient, however, when

considering the strip problem, to obtain a representation comparable to (3. 1. 1) in elliptic coOrdinates where the apparent behavior and the actual behavior are the same.

c) x > X2

3. Integral Representation of a Class of Çylinder Functions

Having pointed out some of the hazards involved in using (3. 1. 1) to represent a function (solution of Helmholtz' equation) which vanishes on a line segment, .we now consider the representation of functions for which the form (3. 1. 1) is ideally suited; that is, solutions of Helmholtz' equation which vanish ón the entire line y = O. A large class of such functions is known to be given by

In the sense that cylmder functions result from solving the two dimen-sional Helmholtz equation by separation of variables, leading to circular cylinder functions In polar (circular) coordthates, parabolic cylinder functions in parabolic coordinates, etc., the expressions (3.3. I) can be cálled rectaigular

cylinder functions. In this sense, "rectangular" coordinates are misnamed and would be more correctly called "right angle" coordinates. However, regardless of the possibly offensive, nomenclature, the expressions (3. 3. 1) can be written, with the representation (3. 1. 1), as

X +_ly CoSVk2l,2çS.in

\

J(x_q)2+y2)coJk2_zi ad (3.3.2) 'X-\r2 2 cos\k-zì x 2 2

sinVk-v x

+ jy

x

¿ir-\ J(k\/(&)2+y2

)

2 q da. sthzìy. (3.3.1)

simple form, i. e.,

2

iyk-v

x lì.

e sinvy =

The representation of the sum of (3.3.2) and (3.3.3) has a particularly

jo

V22)

ej2

a

'X - jy

which, with the substitution x+ iy óösØ, ylélds

-a

Siti'y=

J(kysinØ)

e2c

sinØ dØ.

o

This IS a special. case of aförmula discovered by Gegenbauer (Ref. 31). We

have already encountered formula (3.3.5) for the special case i' = k In Seçión 2.5.

-Another, perhaps more fitful, application fthé representation theorem Invólves .the class of fuctiöns glveú by

Z(kR)

e4

-

Z(kR') (3. 3. 6)

where Z represents any circular cylinder functibn of order y, and (see Figure

3.3.1) .

R2 =

(x-x

)2 ,2 (x-x)2+ (y+y0)2

®

_tan1

tant y±y

_(3.3.7)

X-. X X-X

O o

Clearly if y = O, then R = R' and - ®' hence the expression (3.3.6)

vanishes. This expression is of course an analytic SolutiOn of the Helmholtz

(3.3.4)

FIGURE 3.3.1: GEOMETRY OF R;

ANt R', ®'

equatjón in any simply connected rçgion excluding the points x

(Tfldeed if Z

J, n=O, ±1, ±2,..

if y Û, it f011ows from the representation theorem that in a neighborhood of the line y = Q, Z (kR)_ii

Z (kR')e'

li -

2i

and Z kR) e

- Z (kR?)e'®

1

2i

then it is analytic everywhere.) Then,

x+ iy Jo X -iy

x-a

)2+2,)

f(x, y, a)da

(3.3.8)

2 2)

yy..

(x ,y ,a)da

₀₀

(3. 3. 9)

where

f(x .,y ,a) =

-p-. o o and. g(x3 y, a) cos .x- X o___

A [z (i)e"'

ay .iì

\(x x)2+(y-y)2

= cÒs®

_+iSifl®

In exactly tirne samè way we fjid tiat

r

dZ (kR) J. lì

ill®

f(x, y, a)

L k cflkfl) e

-z

X-' X0+ i(y+y) +iv (kR')e sin

prom the definition of ® in (3.3.. 7) we find that

y 0

x a

2 2

V

(x-x) ±(y-y)

where, as igure 3.3. 1 makes clear, the pòsitive square root is to be ernployed Thus

x-x+i(y-y).

(x- X)2 ( 3rd)

y-yo

With these expressions we may Write f(x, y, a) and g(x, y, a) elicitly in terms of

y, and a..

Thus, carrying out the indicated differentiation,

(3. 3. 10) beöornes

iii z (iR)

e''®

.

y ay C IcR')

e'-r- + i zìZ (kR') eT

y, = O x=a (3. 3. 10) (3.3.11) (3.3. 12) (3.3.13) (3.314) (3.3.15)

\J (x-x)2+

(y+ y0)

Upon introduciúg the familiar recursion relations and a® -a y=o aYJ

It follows from their definition that

R[0 = Rv0 =

+11;

Zv+i\J(a_x.

ie''

-

+jR'

rz

j

a Yo

r° - -

-\j

(x-x)2+y2

f(x,,a)=

_{ik ZvV(a_x)2±y2}

\

I.

-x-iy

o o

2

iv'-? +Y

04 2 2

-x )+y

o o (kR) + z

kR)le'®

L®

j

kR')+ Zi(kRt)ie1

Substthiting thesé expressions, togethçr with (3 3. 13) and (3. 3. 14), In (3. 3. 17) we obtaiñ, after simplification,

i*1

a-x

-iy

(3.3.19) ay

ro

x=a (3.3.17) (3.318) 2 dZ(kR) = (LR) and

Z1(kR) + Z1(

Z,(kR) we obtain

f(x,y,&)

kir

_{- Zv+ikR)Je}

lui

_Z

j.

(3.3. 16)

-J

A similar procedure yields

g(x, y, a)

1k Z

\J(a x)2+ 2

k

Z1J(a-x0

a-x +iy

o o - 2 2 V

(a-x)

2 &-x +iyo o

J

2

y(a-x0) +y

If, in (3 3.19) and (3.3.20). Z J and ii = 2n + i then f(x, y, a) and g(x, y, a) have no singilarities. In. all Other cases, however, they have branch points and possibly poles as well at n xtiy. In these cases we must specify the branch, cuts and paths of integration in (3. 3. 8) and (3.3. 9) in order fòr these repre-sentations to be rneningfül. This is accomplished by employing the same branch cut convention adopted in Section 2.2. ThiS enSures that when a is

real the argument of the cylinder function is real and positive. We number the sheets of the Riemann surfaöe of the logarithm exactly as in section 2. 2 and then reqUire that we remain on the 0th sheet along the paths of integration

in (3.3.8)and (3.39).

The branch of the factor (a - x + iy)11. is determined by requiring consistency on both sides of (3.3.8) and (3.3.9). That is1 in (3.3.8) the factors e+hh1® _{and e give rise (see (3.3.13) and (3.3. 14) ) to terms}

like [x- x+ i(y- y)]" and [_

_,-

i(y+y )j" respectively, which are equal, when y 0, tô

[x_

x-

V We must then require that

the factor [a.- x-

In f(x, y a) have exactly the ame value when a x,

then let it vary continuously as a becomes complex. Similarly, in (3.. 3. 9)

when y =0 the factor [x - x+ i3ÇJ" appears on the left and this determines the branöh. of [a x iy.]1'

in g(x,

?a). Figure 3.3.2 shows, the brançh cuts and a possible path of integration.

FIGURE 3.3.2: a-PLANE

While it is true that so fär we have only established the validity of the representations in a neighborhood Of y =0 it is clear that by' analytic. continuation these representations remain valid throughout the cut =plane. The points x x, y = ±y are singular and in general the functions

+. _+.L,tw'

Z (kR) e

_li.

and Z (kR') e are multiplé valued thus we must cut

'li

the xy - plane as well as the a-plane. However, in;the Special case when

v=n(n=0, ti, t2,...).the functions

Z(kR)e±m and.

Z(kRt)e_m®

are single valued throughout the xy - plane hence in this case the integrals (3. 3.. 8) añd (3.3.9) must also be single valued, even though the integrand. still has branch points. This means that. for x = X, y > y, the path of integration can be either that shown in Figure 3.3. 3a or 3.3. 3b depending on whether x - .x from right or left but both paths must yield the same value

of the integral. As long, as

_'

y' these integrals are defined and iust be

eqiat since they both. represent the same function.

H(2)(kR) 0 0 2 +

a)x=x

b)x=x

o o FIGURE 3.3.3: a-PLANE

The fact that (3. 1.1) ¿ticcèssfully represented a function with a

singulailty illustrates how (2. 4. 1) and (2.4. 2) can be used to represent solutions of the inhomogeneous Helmholtz equation.

4 The Integral Representation of Line Sources

Of special interest is the particular case of,(3. 3. 8) or (3. 3.9) when

(2) Z is chosento be H , viz.: II o X + jy Jó

J2+2)

X - jy +

(ax

)22

01

a-x -iy

_o o

V'

(a-

x)2+y2

da.

(3.4.1)

Further simplification yields :+iy

H(kR) - H(') = -il

J(k V(xa)2+Y2)

H2

2+2

x-1y o o

which can also be written

+ ly H(2)(kR) _{H(2)(kR,) =}

\

_Jo

\/_a)2+;2)

jy of (3.4.3) obtaining + iy X - iy (kysint)

;;

H(2) o

(2)V2+Y2)

d In addition to being of interest in itself, this last identity proves to be of considérable importance In the diffraction problems discussed In later chapterS It can be proven valid without making use óf the repreSentatiOn theorem of this chapter and although this dfrect proof is somewhat tedious, formula (3. 4.3) is felt to be of sufficient importance to warrant its inclusión here.

We proceed by firèt substitut.g a .iyçost±* in the right hand side

:X0X iy coat)2+y2

(3.4.2)

(3.4.3)

We re Strict y so that

2 2 2

-x0) +y >y

which Implies that

(x-x)2

+ y

> iycost

and then make use of the addition theorem fòr thé Hankel funótibn (Ref. 32)

2+p2 2rp cos and cos 9 which Implies cosnø Thus ( - 2 2

r= y(x-x0) +y0

iyôost

x -x

X - X+iy J. o _Q .

l

22

y(x0-x) +y0

ç

J (kp) 2(kr) ósiØ where =1,

= 2 (n= 1, 2, 3,...), and

_rI

>(p'. O rl

Iti Our case we choose

Vx0- )2 r n

i

r Lcosø+ismøJ + LC0Sø-18nø + ] (3.4.5) (3.4.6) (3.4.7)

L

H(2) &y o H(2) _\J(x

_-x-

iycost)2+y2)

1

[.1J1ukYcost)

+ X -X+ I o o I ' 2 2

\V'ö'

Yo + .H(2) n+1 X -X+jy o

r

iY

J(ikycos t)

I 2 =o + X -X-iy

io

o

Li

2 2

\\/(x_x)

_+Yó

\n-1

f X

-X-iy

to.

o.

[(x_:+iY)

n+1 / . f X. -X-ly o

r

Upon differentiating (3. 4. 8) with respect tó y the indices of summation so that

ii2

occurs

H2(kr)

To obtain' a useful form of the derivative 'With respect to

y, necessary in

(3. 4. 4), we first examine thè y0 dependent part of a general term of (3. 4. 8). With

(.xiY1J

}

utilizing (3. 4. 9), and adjusting

- th

in the n term, we obtain

\J(x0_x _iYcost)2+y,)

--

J(fkycos t)

H(kr)(''

'

r

-+

ço.o)

r

2 2

x-x) +y

_{o o} X -X-iy o o (3.4.9)

/

,.. IX -X-1g

_(

o o

r

the substitution r \J(x=x)2+y2

- - and the recursion formulas

(3.3.16) we find thät

)fl

But

À H(2)

ay o X

=x-iy

X

-xI-iy

r

r

-. o o o o

x-x+iy

x-x-iy

r

r

o O O O

thus, recalling the definition of together with the fact that =

-(3.4.10) becomes

L

n-¡ 2 2

(iky cos t)

_n+i'

cost]

which again employing the recursion formula for the Bessel fÚnction, beóomes finally,

H(2)(k\/(x_x_iycost)2+y2

)

>1 n= 1

nJ (ikycost)

(2) (X. -x+ly - H kr) °

ycost

n

\

r

ubst1tuting (3.4. 12) iii (3. 4 4) we obtain, after ihterchanging summation an4

integration, +iy (a_x)2±y2 _X ly a _H(2)

_{\,4a_x 22.}

ay o o o (Xo_X_iYO

r

[(XOX1YO) _-

(T:!O)

i

(x

x-

i)

n]

)

do (3.4.11) (3. & 12)

However

7

J (kysint)J(ikycost)tantdt

_{[i-. (}

o

-, jy

and this last form has been explicitly integrated by Rutgers (Ref. 22) who found that

7/2

J.(r sit) JUI còs t) tan t dt

J(y)

o

-Ç

J(ky).

(3.4.15)*

With this result, (3. 4. 13) can be written as

/

-x+iy

t -o- o

J(ky sin t) JUky cos t) tant dt

(3.4.14)

ix

-X-f o

r

which upon multiplying out the factors with th powers and reintroducing the

E

notation (/2

i) , whióh is possible sincethe n = ô term is zero,

becomes

ormula(3. 4. 1i)apears in Watson (Ref. 33), but is in error by a factór of 2.

n=o where Hence +iy

J(k \J(_X)2+Y2)

fl(2)

_{\J(a_x}

)2+2)

_da 'x.-iy J (ky) H(2)(kr)

-nfl.,

n 2 óösØ

(0_ i(x- x)"

r

i 2 yo r 'L

f +i(x -x)

+ ( o

H(kr)

xo_

₎

(+ o_

-i(x --x) -

O±

+

r

2 .2

(2)j/2

2

y ±r -2yrcosØ

j

H iy +r -2yy

Now we agam make use of the addition theorem for which the earlier restriction (x-

x)2+y

' r2> y2 stÌil suffices. For the fIrst thum on the right hand si'd

of (3. 4. 17) we wíite

(3.4.19)

cO

2

22

or, since r = (x-x) ±y

_o

o

H\jy2+r2_2yrcosØ) H(k\Jy2+(x_x)2+y2_2yy)

Jo \J(x_a

2).a

H(2)(k

SimilarIy. the sebond $urn on the right hand, side of (3. 4. 17) is found to be

H(2)kR,) _{consequently the relation we set out to prove, (3. 4.3), is established.}

In order to use the addition theorem, it was necessary to restÍict the values of x and y but this is clearly a restriction on the use of the addition theorem and-not on the validity of (3 4.3) since the functions on both sides of(3. 4.3)

can be analytically continued throughout any simply connected region excluding

the points x, ± y.

In the ambiguous case when x X and y > y two. integration contours

are possible, äs shown in Figure 3. 3.3. Thé two integrals thus formed must

have the same value since they both represent the same function. That this

is o may be directly demonstrated by showing that their difference vanishes,

i.e., that

where the contour, shown in Figure 3. 4'. 1, is to. lie 'efltirely on the

0th blade of the Riemann surface of the Hankel function.

(3.4.20)

7r

ay

FIGURE 3.4.1: tNTEGRATION CONTOUR

Rewriting the thtegrand with the help of (2.2.. 5) we may treat tè

singular part separately. Thus

Jo \/(x_a)2+y2

)

_.;,

H2)( \J(

)2.2

) da

(x-

)22

_{) log}

_- y

J

2y',

G(a)

+

_{(;-:0_ iy0)}

_{(a :*+ iy)}

where Ga) is analytic. With Cauchy's theorem then

and, since the residues t the twO poles are equ1 but opposite in sign',

ç.xo+ iY -22ri

J k

O\ S

\J2±2

) G(a) da O J

,\J( a))

da

(a-x -iy )(a-x +iy )

_{o o o o}

The logarithmic term is also easily evaluatèd. With the help of formula (2.2.8) we find that

\J(x- a)2+

2)

0V'

x)2±y2.)

1o.[a

- X

)+y2J da

Q .. (3.4.24)

The right hafld side can be siplffied, since i is independent. o,f X

(3.4.23)

(3.4.25)

X +iY o o

=

_22ri\

+ 2iri

j'

O\

22

)

-iy .

which is clearly seen to vanish on substitution of -a for a in one of the ntegrals but not in the other.

5. Asymptotic Behavior of the. Line Source Représentation

FoÎmula (3.4. 3) can be used to denionstiaté that for large values f x.

and. y, the circular cylinder function relations pass over intO the rectangular cylinder functloú relatiöns of sctïon 3. 3. . . . -.

Consider first the cáse of large y, whère . . da

Retaining the y in the phase and neglecting it in the amplitüde in the customary manner, we use HankePs asymptotic form (ref. .17) to obtain.

(2) ik(yy)+

-H (kR)tv e .

o 7ry

2 -ik(y +y)±

H(2)(kR,)A/_

_e o I2Ty (2) \/(a_

x)2+y2)

V Y0

e4

s (3.5.2)

Substituting (3.5.2) in (3.4.3) and retaining only the term in the differentiation of H(2)

\/

- x)+

y2 ),

we have, after cancelling common

factors,

f 22

₊

_{-2ar cose}

_{- r -cos9}

O O O O O

x± ly

=

k \

jva2+2)

is the case when r = /*2±

y becomes large. Usine polar coordinates (r, e) for the source and rectangular coordinates (x, ? for tue field point we have, for large

(3.5.3)

R x2± y2+ r2

- 2xr cos e- 2yr sin

i

t'-' xcos y sin e0

RT x2+y2+r - 2xr0eos 6+ 2yr sin

r- xcos 6±ysin 00

and for the Hankel functions ____ j7T

-lk(r - xcos

O

-ysinO )

+

-2 e ° ° 4 iir

-ik(r -xcosO +ysin9 ) +

--o o o 4 e H(2) (k u "i. H

)22

)

-ik(r-acoO) +

_{(3. 5. 5)} (3. 5. 7)

Substituting the expressions (3. 3) and retaining only the

term in the derivative of H2

J(a_x) +y ) we have, after cancelling

common factOrs,

x+iy

e0o(e15flO0 e_SflOo

_{ksO0Ç j}

_j2y2

d

'Ix-iy

- (3.5.6)

With the substitution a = x+iycost and the resulting simplifiçation, (3. 5.6)

becomes .kysinû. Ç

sin(kysin9)

₂ °

\

J0(kysint) e'

SûocoStsint

_dt.

o

Formula (3. 5. 7) is a disquised form of (3.3. 5) but perhaps a more useful representation with application in the study f frequency modulation

Although we have discussed the asymptotic results only for formula (3.4.3) which is.a very special case of (3.3.8) and (3.3.9), there is not much tO be gained in this regard from consideration of these more general cases.

Chooslìig u O would nöt cIange the asymptotth forms, choóSing Z J or N would lead to forms in which the dependence on the parameter which was taJen to be large could not be factored, and choosing Z = HW _{rather that} H(2)

would lead to essentially the same results presented here.

6. Double Integral Representations

Of particulai interest, from the point of vieW of subsequent appIications, is the expression dérivable from thé fömuia obtained by adding (3.3.8) and

(3.3.9), i.e.,

x+iy

Z(kR)cosv -

Z(kR')cosve' =

_{\ J}

(ax)+y ) F(x,y,a) da

where, adding (3.3.10) ànd (3.3 11),

F(x, y. a)

jL _{[Z(kR)cos - Z(kR') cos u}

y=o

xa

With the aid of (3.3. 19) and (3.3.20), (3.6.2) can be written as

z o 2 o

x-iy

(3.6.1) zi-1 i . zi-1

(

o _{\ I} O O /(a_x)2+y2

ra_x ±iy

_o o

''

2

\V-x)

+y o o

With the expressions (3.3. 13) and (3. 3.. 14) which give

el®

and e1® explicitly

in terms of x, y,x, and y

itis

(3.6.2) (3.6.3)

ik Zi(k\ja_+Y)

(

a-x-ly

11+1 2

2'

(car-x ) +

)

o o

CÔS7/® and cos ii z (kR) cos 1/

-F(x ,y ,a)

₀₀

F(a,y,J3)

/ x-x+i(y+y)

(y+y)J

Thus. exhibited explicitly it is clear that interchanging. x with x and y with

y in cös V and öds v® is equlvãlent tó

ltilying by (-i".

Since this interchange leaves R and R' - unaltered we find that interchanging (x, y) with

(x, y) in formula (3. 6; 1) yields, biihging the factor (-1)V td the righ,

y=0

-L.

where, using (3.6.3), ik 1

+ - Z

(k 2

v+l\

T)_{óös ?ì}

22

¡3-. a) ± y +

xx_

i(y+y)

I 2 2

y(*-0).+(Y+0)

_J

2i

+ jy o o

J(k

x -iy

. (3.6.6) _o o

where We have retiarned the integration variable to avoid confusion Ln what follows. $ubstitutirig (3. 6. 6) in (3. 6.. 2) we Obtain

-3)2+y2

_0/

F(a,y43) ay

:\

_/

.. 2

\ íf3-a--iy

y

-a)+

2 (3.6.4) (3.6.5) x;-y,13) d13 dL3 (3.6.7) y=o ii-i

Upon perfôrming the indicated differentiation and evaluation we obtain

tF(a,y43)

= k Z

We have written (3. 6 9) in this purposefully complicated form so as to stress the importance of the choice of the sigh of the square roôt. Actually, to consistent with our previous convention, there is no choice left. We required

that \J(a-x)2+ y2 and (and hence \J(13=_{x)2+ y2 )}

be real and pOsitive when a (and ¡3) were real. This implie that we must choose the sign of the square root so that When a and ¡3 are real,

\/

a

13)2

is also real and positive. Restricting the paths of integration to be straight lines connecting the end points, it is clear from (3. 6. i) that the only real

value a can assume is a

= x and from (3.. 6. 6) that the only real value ¡3 can

assume Is ¡3 x hence o Z _1(1ç13-ka) F(a, y, ¡3) = k(v -i) k(zì tFi z (k-k13) i)

vi

_a-j3 = ¡3-a

(v-i)

/

ßa

/ /(13_a)2 \V(J3)21

Z1(ka-kÍ3)

k(v+i) _a-13.

z,(1q3-ka

+k(v±l)

..--.

. -¡3-a (3. 6. 9) X >X 0 X <X o X <X (36. io) o

Substltttting (3. 6 11) In (3. 6.7) yields

F(x,

_y,

a)

=

_-

_j

Á

x±y2

)

X - jy o o

[(v-l)Z_1(kI3_kc)

+ (v+Ï)Z,1(kI3-ka)

-1)Z,.1(ka-.kI3) + (v+i)Z1(ka-kJ3)

xtiy

_O

J4

'X- iy x0_ Iy0

V-22 )

and finally substituting (3. 6 12) in (3.6. i) we obt.in the general representation

Z(kR)cosv

_{- Z(kR9cosi®}

fv1)Z1aj3ka)+

¡3-c v-Fl J

x.>x

o X <X o X .>X -o

-(v-l)Z.

1(ka-1q3)+(v+l)z 1(ka-k13) ,

*<x

L

- .

Choosing Z = H(2) _{in (3.6. 13) yields the remarkable and useful}

representation

H(kR')

+i ('x +iy

(2) da. Od JO

\J=.x)+Y2)

2) .1JT)

(3, 6. 1.2) (3.6.13) X-ly JX-13T (3.6.14)

where the positive sign must be employed when X > x and the negative sign

when x <

o The paths of, thtegration and branch cuts -are shown in Figure 3.6. 1.

a)x >x

b)x <x

o o

iy0) (xo+ iyo)

FIGtJBE 3.6: 1: BRANCH CUTSIN THE 13-PLAÑE POR FORMULA (3.6.14) The same figure can be used to represept the a-plane by interchanging

with 13,

x with x, and y with y.

As suggested by the representatioñ of the two sources,

J(2)() H(2)(kR

we may now derive a general double integral, representation of the analytic Wvê function, G(x, y, x, y.), which añishes on a segment of the line y = O

and, in additlön, is sy metric in (x y) and (x, y).

With (3. 1.1) we write- .

-G(X,YT, x, y0)

which,. we havé shown, is valid in a region óontaining x = X, y

y even.

if this is a source as as G is analytic everywhere ejse. Making ue of the symmetry property,

G(x,y,x,y.)

G(x,'y,x,y)

.

we interchange x with X. and y with in 6. 15) obtaining

where we have renamed the va. able of integrt.tion. to avoid confusion in what follows. Differentiating (3.6. 17) with respect to y, evaluating at 9. and

replacing x by a, we obtain

y, x, y0)

-yG(aYxoiYo)= \

Changing the dummy variables appropriately and substituting (3.6. 18) in (3. 6. 15), we obtain + 13 ('x± iy dJ3 J

)+y2)

_{G(, i', a, y)}

+2)

_{(x _13)2+Y2 G(, lì, a,} d13 11=0 (3.6.16) (3.6.. 17) d13.; (3.6.18) 11=0 ,y=0... 11=0 j2=0 +iy o o - jy. o -o = iy o -o c,_iy 'X0-iy0 (3.6.19)

- Again making use of the symmetry próperty, this can be Written

G(xy,x:,y)

-

da

dJ3

22)

J(%_

Where, as (3.6. 14) bears Witness1 this interôhange must be performed with considerable care in particular cases.

This representation and the corresponding fòrnis Uiparabolic and

elliptic coordinates prove to be of considerable value In solving the problem of tUffi action by a strip.

Until. now we have avoided the term "Green's function", although suggested it in the notation, because Green's functions are most often not analytic at the significant boundary (in this òase the segment of the line .y' O

Where G = 0) whereas our function must b analytic. In section 3.2 we Showed

that this iS not an essential difficulty, and heúce we shall in the future refer to (3. 6.20) as. the double integral representation for the Green's fuction for a line segment..

Aftèr having so thoroughly discussed the. representation of known wave functiOns, it would seem proper to devote some consideration to thé use of (3. 1. 1) or (3. 6.20) as an integral equation for unknoWn wave functions. Unfortuflately, the Scattering problems In rectangülar coordinates for which

': )

11=0

Ii=o

(3. 1. 1) is appropriate are either so elementary in nature or sO difficult that such consideration is fruitless. That is, wave ftthctions satisfyig a Dirichlet conditiob On the êntire Une y = O aie easily foùnd by the well known pethod of images (Rèf. 19), whereas wave functions satisfying Dirichiet conditions on a part of the line y O, as indicäted in section 3. 2, will best be treated,

Chapter 4

DIFFRACTION BY A HALF PLANE

In this chapter we consider the canonical two dimensional thffraction problem, diffraction by a half plane. The solution is expressed in the integral form compárable to (3. 6 19) in parabolic coordinates, the coordinates most

appropriate för this approach tö the pröblem. The usefulness öf parabolic coördinates in conneótion with the half-plane problemhas long been recognized, having been used by Lamb (Ref. 13) early in. this century.

First the parabolic coOrdinates arè intròduced, results comparable to those of Chapters 2 and 3 are presented, and finally the solution is given.

The half plane problem, perhaps the mOst frequently solved problem in diffraction theory, is discussed, not as a model to test a method of solu-tioú, but beäause. we assert (and prove in Chapter 5) that implIcit in the

solution of the half plane problem is the sOlution of the problem of diffraction by a strip and indeed it may, when interpreted correctly, yield the exact solution of a much wider class of two dimensional diffraction problems

1. Important Relations in Parabolic Coordinates Weset 2

x+iy = (+ir))

(4.1.1)

22

or

x=-T1

(& 1.2)

'y = 2rj

and similarly

FIGURE 4.1.1: PARABOLIC COORDINATES

In the figure the. range of and r covering the entire xy - plane is taken

as -w

<

<w,

<w. We could alsò describe the plane with O

<w.

w <y <w The appropriate chòicê really depends on the problem being considered but for the present we shall restrict the discussion tO the tipper half plane, , n .O.

The ditance R becomes

J(x_)2±(yy.)2

=

\J+)

-

(±iy)]

[(x_iy)

-2

(±.)21

.)2 ()2

a

(4. 1.3)

The Helmholtz equation is now

a2 2 2

+ T

+ (2k)

( +

ar7

T?) =

J(,u)

1(u) du (4.1.7)

Where we have used, and Will continue to use, thé short hand

Of course it must be verified that (4. 1. 6) and .(4 1.7) are indeed solutions of the He holtz equatiop.. While it. is trie that they are Solutions if 1(u) is analytic in a. simply connéôtèd. region containing the path of integration. this unfortunately does not follow directly from the already proven results. in rectangular coordinates. That is, under the conditions for whiôh (2. 4. 1) and (2. 4. 2) were shöwn. tó be solutions, it. is possible by a simple trans-formation to Show tht (4.1.6) and (4.1.7) are solutiOns fOr a wide class of furetions f(u) but not aIl,analytic functions since û O is a singular point of

(4....5) andthe solutions which corÍespônd to (2.4. 1) and (2. 4.2) are

ø(,n)

J(,u) f(u) d

(4.1.6)

J(,mu)

21: [()2

the transformation. Hówever, since the proof that (4.1.6) arid (4. i. 7) aré solutions with a a.á1ytio fuflction f(u) is completely analogous to that of section 2.2, it will úot be reproduced here. Of course thiS óan, also be established by direct substitution In the Hëlmhóltz equation but these tedious. details will alio be omitted.

2. Parâbolic Form of the Representation Theorems and Applïcatioùs

With (4; 1.6) and (4. 1. 7) we can nOw establiSh l'epresentation theOrems

comparable to (3. 1.1) and (3. L 2). -

-If Ø(, r) is an ahalytic Solution Of the HeltnhöItz eqüation in a simplr conñected regimi còntalning a Segthént of theme r? = O, and Ø = O on thi segment then, in this region, O has the integral represehtation

(

Similarly, if çli (, r) is an analytic solution of the Helmholtz equation in a simply connected region containing a segment of the line = O, and = O

on thi5 segment then, in this region, 1' has the representation

J

ç1ì(,r) =

_\

J(r1,u)

_- 1i(zi,-iu)du. (4.2.2)

-+iT1

The proOf of theekr*ns oÎ th'erepresentation theorem is exactly the same as presentéd in 3. 1. It çonsists of demonstrating that the integral

representatión preserves the value of the function and its normal derivative on the line segment and since the function is given to be à solution and the integrai has beçn shówn to be a sOlution of the Helmholtz equation, the Cauchy-Kowalewsky theorem assures that they are identical.

We could, of ourse, employ (4 2. 1) and (4.2.2) to find integral representations of products of parabolic cyflnder functiOns which vanish when :g = O or r = O, corresponding to. the representation of "rectangular"

clinder functiOns of section 3 3, büt we shall confine our remarks to tpics

more intimately connected with the half plane problem..

The expression for the difference of two sou,rces, H(2)(kR)

-

H2(kfl')

which Is of interest in this regard, can be derived using these representations. The two forms o the representation theorem, produce two different expreSsions which can also be obtained by a transformation of our previous result. Replacing

..2

. . 2

x iy with the parabolic equivalent (

-

irs) and substituting a = u in (3.4.2)

we obtain tWo expressions, because of the sign ambIguity in u (i.e. a u2= (-u)2). as follows: +

J(,îi, û)

H2)( u) udu H(2)(kR)

_{- HkR.') = -4lkn0}

V

[±l)2u2] [(-

in)2-= in H(2)kR)

-

H(2)(k.R,)

=

4ikofl0 (2)

J(,'ri,u)H1 (0,n0u) udu

\/ [(

+i

)2_u2]

-in)2-+ in

(4.2.3)

where the same abbreviation is used for H

as for J

. Under this

trans-i o

formation, straight Ïines are mapped into hyperbolas and, the branch cuts of the integrands lie along the curves

jm (±l)2u

j

i)22]

=0.

(4.2. 5)

If we write u = u1 ± iu2 then (4. 2. 5) implies that

2 2 2 -2

U1-U2±77-)

O

and the unTes iescrlbed by this equation are plotted ii Figure 4.2. 1 with the portion chosen as a branch cut indicated. A10 shown are the values

1 2

2i

2 2L

th

f arg

i) -u

J L0

lrj) - u

of the Riemann surface of log [+

ir7)2- u2 j [_ j1)2_

2]

(4.2.6)

Since it is: dèsirable to always be able to treat the path of integratio as the straight line cqnnecting the end points, we alter the cuts somewhat as shown in Figure 4. 2.. 2 We still keep track of which sheet of the Riemann surfaóe we are on by calling the thsheet that sheet whère the ai'guient of

[(+ ly)2- u2J [(Ç-

112]a1on e tted cuVes of Figure 4.2. i is

2nir and then confine our attention to the 0th sheet.

a) Branch Cuts for Formula (4.2.3) b) Branch Cuts for FdrmuJ.a (4.2.4) FIGURE 4 2 2 ALTERED BRANCH CUTS fl u-PLANE

In the same way, the double integral expression (3.6. 14) leads to two expressions 1 this parabolic form as follows:

H2(kR)

- H(kR)

-2k

=-2k

+iTì and

H(kR)

H(2)(kRT) - iT7

dv J(,

,u)

j(0,

_'y - irj 0 .0

.

[2

-(y -u

dv J(,,u)J(

00

, ,v) i

-22

±(v -u-)

-,0+ )uv f < (4.2.7) 2 21

-uJ)uv

In each case the branch cuts öf the integrand are chosen to lie along the negative real axis of the argument o1 the Hankel function. In the y plane this criterion leads to the cuts shown in Figure .2.3 NÒt that in formula

(4.2. 7), Re u > O, - w

<J

mu <w, and in formula (4.2.8), -w <Re u< w,

J

rn u.> O, so while we may use the same figures for both formulas, they correspönd to different u values as indicated. In the cases illustrated in

Figures 4.2.. 3 b and d, the point u for formula (4.2. 7) corresponds to the point -u for formula (4.2.8). In all cases, the value of the phase of the

o if n <n0

argument of the Hankel function on the branch cut and its extension (the positive real axis Of the argument) äre indicated.

a) Formula (4.2.7):

>, mu>O

b) Formula (4.2.7):

_>,4rnu<o

Formúla(4.2.8):. r°<y, Rei>O

Fòrmula(4.2.8):

r0<r, Reu<O

o'

'o

ss_u -ir _+1T +( o' c) Formula (4.2.7): cl) Fbrmula(4.2.7):

Formula (4.2.8): r)>?7, Ieu>O 'órmu1a (4.2.8): Teu<O FIGTJRE 4.2.3: BRANCH CUTS IN THE v-PLANE

The samebrarich cut criterion in the u-plañe leads to completely analogous results'. (lñdeed, the same figures can be employed with some éhanges in notatioii Y I o -ir ±lr I (O

cu

- in

3. Green's Function for the Half Plane

Corresponding to (3. 6.20) we have 'the double integral representation for the Green's fttnction which vanishes on a segment of the half lIne = 0,

82G(u,v, v,p)

J(,,u)J(,,v)

--j-' 71=0

This holds as 1on as G(, r,

_.,

n) is analytic (save för sources) in a simply connected region containing the line segment. Clearly this. representation will be. most appropriate for the. case when the segment on which G vanishes consists of the entire half line n = 0.

In Order to have a physically significant problem and still have

G(,

_{' '}n) análytic at n O we muSt. limit the problem so that the points in thé neïghborhoód of one side of the line lie outside physical Space. This

of course an be accomplished in various ways. Perhaps the simplest is

to impose another boundary condition when 0, i.e., on the complement of the half line n = 0, thus restricting physical space to the upper halt plane.

If wé require that G also vãnih when '= O we essental1y will reproduçe the problems for the entire linç treated In the previous chapter.

However, if we require that the normal derivative vanish at = O;

8G

i. e. - = 0, we have a much more complicated problem. That the

':'

representation (4 3.. 1) is ideally suited to this problem is easily seen since it is only necessary to restrict G to-be even in in order for the integral expression'to satisfy both the Hélmholtz equation .and the boundaryconditIons. Exactly the same procedure as discussed in section 3. 2 shows that this mixed problem is entirely equivalent to the problems where either the function or its normal derivative vän.sh of the half plane, i. e., the classic half plane problems. .

- Baker and .Copson (Ref. 2) give a very thorough, well referenced discussion of Sommerfeld's famous solution of this problem for plane wave incidence. Our concern is with line söurces and the exact solution for this problem, finding the field, 0. of a line source in the presence of a perfectly

soft screen (see Figure 4. 3. -1),was given by Çarslaw (Ref. 7) in a form comparable.to Sothrnerfeld's result. Macdonald (Ref. 15) simplified the result considerably and it is his form that we shall employ. . Rewriting his. result in parabolic coordinates, Macdonald found that the total field,

due to a line source, - i H(kR), could be expressed as

where i

()2

Q 1og 2 2 o 2 (4.3.2) (4.3.3)

FIUBE 4. 3.'l: A LIIE SOURCE

Th THE PRESENCE OFA SOFT. HALF PLANE

FIGURE 4.3.2: A LINE SOURCE

(kR)

₌

Ç \

cos.nw

e_05W

d

where the contour iS shown in FIgure 4.3. ..

(4.3.5)

()2 ()2

Ql = log (4.3.4)

R and R' are given by (4. 1.3) and (4. 1.4) respectIvely and the time dependence is suppressed.

With, the help of Sommerfeld's integral representatloti of the Hankel function (Ref. 29), this result has a ready interpretation aS an "incomplete" Hankel function in the following sense. Sommerfeld represents the Hankel function as a complex integral where the contour goes from wi to + wi In such a. way that convergénce is aSsure'. By choosing only a part of this path we have a "incomplete" Hankel function in the same sense that Fresnel

integrals are incomplete fictorlal functions.