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^E ET PR/!

,^. ^^ Jv. NEW YORK UNIVERSITY

b:

f#

>

"J I IN IJ^ '*'

Institute of Mothematical Sciences

^^ ^-^ ,j> Division of Electromagnetic Research

RESEARCH REPORT No. EM-81

Asymptotic Solution of Some Diffraction Problems

J. B. KELLER, R. M. LEWIS and B. D. SECKLER

>EW YORK UNIVERSITY

liN^jlii-Ui t OF MATHEMATICAL SCIENCES

LIBRARY

23 Wsvwly Place, New York 3, N. Y.

CONTRACT NO. AF19(122)-42

JUNE 1955

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM- 81

ASYMPTOTIC SOLUTION OF SOME DIFFRACTION PROBLEMS

Joseph B. Keller, Robert M. Lewis and Bernard D. Seckler

^oserm B.

2XJ^

osepf B. Keller

Robert M. Lewis

Bernard D. Seckler

Morris Kline

Project Director

The research reported in this document has been made possible through

support and sponsorship extended by the Air Force Cambridge Research

Center, under Contract No. AF 19(122)-42. It is published for techni-

cal information only and does not necessarily represent recommendations

or conclusions of the sponsoring agency.

New York, 1955

/

Abstract

Various diffraction problems are solved asymptotically in k * y

(X Â» wavelength) for k large (i.e., X small). These problems include

diffraction of a plane wave by a parabolic cylinder, a paraboloid of

revolution, a cylinder and a sphere; diffraction of a spherical wave

by a paraboloid of revolution, a hyperboloid of revolution, and a

plane interface; diffraction of a cylindrical wave by a parabolic cy-

linder, a hyperbolic cylinder and a plane interface, etc. The boundary

conditions considered are the vanishing of the functionÂ» of its normal

derivative and the impedance boundary condition. Formulas are obtained

for reflection of any wave from any two dimensional surface, and certain

formulas are deduced for three dimensional problems. The method employed

is that devised by R. K. Luneburg and extended by M. Kline,

TABLE OF CCNTEMTS

Page

1. Introduction 1

2, Formulation of the Method 2

3Â» Cylindrical Waves 3

General Considerations

Example 1. Diffraction of a Plane Wave by a Wedge or Half -Plane k

Exanple 2. Hankel Functions 5

Example 3. Diffraction of a Plane Wave by a Parabolic Cylinder( u=0) 6

Example hÂ» Diffraction of a Plane Wave by a Parabolic Cylinder 9

du â€ž

â€” - =

dv

Example 5Â» The Field of a line Source at One Focus of a Hyperbolic 13

Cylinder (u => O)

Exau^jle 6Â» The Field of a line Source at One Focus of a Hyperbolic 18

Cylinder (|^ = O)

Example ?â€¢ The Field of a Line Source over a Plane Interface 19

k* General Cylindrical Waves 29

5Â» Spherical Waves 31

General Considerations

Example 8. Diffraction of a Plane Wave by a Cone of Arbitrary 32

Cross-Section

Example 9Â» Spherical Bessel Functions 33

Example 10. Diffraction of a Plane Wave by a Paraboloid of 33

Revolution (u = O)

Example 11. Diffraction of a Plane Wave by a Paraboloid of 37

Revolution (grr ^ O)

Example 12. The Field of a Point Source at one Focus of a 37

Ffyperboloid of Revolution (u = O)

Example 13. The Field of a Point Source at one Focus of a 39

I^erboloid of Revolution (^ " O)

Example II4. The Field of a Point Source Over a Plane Interface UO

6Â» General Plane Waves III*

General Considerations

Example 15. A Line Soiirce at the Focus of a Parabolic Cylinder h^

(u - 0)

Example 16. A line Sotirce at the Focus of a Parabolic Cylinder U8

Example 17 Â» A Point Source at the Focus of a Paraboloid of U9

Revolution (u = O)

Example 18. A Point Source at the Focus of a Paraboloid of SZ

Revolution (^â€” = O)

ov

7. Arbitrary Waves 52

8. Two-dimensional Waves 58

General Considerations

Example 19. Diffraction of a Plane Wave by a Circular Cylinder /

(u - 0) V 63

Example 20. Diffraction of a Plane Wave by a Circular Cylinder

(1^-0) 66

9. Axially Symmetric Waves 68

General Considerations

Example ?1. Diffraction of a Plane Wave by a Sphere (u Â» 0) 71

Example 22. Diffraction of a Plane Wave by a Sphere (|^ Â« 0) 72

10. The Impedance Boundary Condition 73

General Considerations

Fxample 23. Hif fraction of a Plane Wave by a Parabolic Cylinder

(|^ - ikZu) 7U

Fxample 2U, Diffraction of a Plane Wave by a Circular Cylinder

(|^ - ikZu) 77

References 78

- 1 -

1, Introduction

Several authors^ J"r J have recently discussed a method by which the

asymptotic expansion with respect to frequency may be determined for periodic

solutions of electromagnetic, acoustic, and other linear wave problems. However,

virtually no applications of the method have been made. In the present paper we

explain the method, and apply it to construct the asymptotic solutions of a var-

iety of diffraction problems. Many of these problems have not been solved pre-

viously. In some cases where the problems have been solved previously we verify

that the asymptotic expansions of the known solutions coincide vdth our asymptotic

solutions and compare the exact solutions graphically with the first few terms of

the asymptotic solutions.

The asymptotic expansion under consideration here is an expansion of a

periodic solution with respect to frequency as the frequency tends to infinity.

Thus the expansion is useful at high frequencies, or for short wavelengths. In

electromagnetic theory it is therefore useful in optics, in radar problems, etc.

In fact the leading term in the expansion is Just the so-called geometrical-optics

solution.

Mathematically the problem is that of finding the asymptotic expansion of

the solution of a boimdary-value problem for an elliptic equation. The equation

and the boundary data involve a parameter k which is proportional to the frequency

of the solution and inversely proportional to the wavelength, and the order of the

equation is reduced when k" is replaced by zero (i.e., when k becomes infinite).

The method employed determines directly the as3nnptotic expansion with respect to

k near k -oo, and thus we can avoid the customary detour of first finding an exact

solution and then expanding it asymptotically in k. Consequently, the present me-

thod can be used in cases where the exact solution has not been found, and this is

its main value. But even where the exact solution is known and can be expanded,

the present method usually yields the expansion more easily. It must be noted,

however, that for most problems it has not yet been rigorously proved that the ex-

pansion yielded by this method is indeed asymptotic to the exact solution, although

formally this appears to be the case.

The numerous problems treated in this paper by the above method are selected

as exaii5>les of the types of problems for which this method is applicable* On the

other hand, it is also apparent that imless the problems conrespond to the simplest

geometries, the calculations beccane quite lengthy. In fact the problems considered

in this paper were selected partly because the related ray systems are simple, and

- 2 -

therefore the calculations are relatively easy. Further, it is not yet possible,

in general, to treat problems involving diffracted rays'- -I by this method. In

some problems where such rays do occur the present method can yield only incomplete

results, as in examples 1 and 8. However in examples 7 and lU a special method was

devised to treat the diffracted rays, and it is to be expected that more such me-

thods will be devised for other problems,

2. Formulation of the Method

We consider a solution u (scalar or vector) of the reduced wave equation

(1) (V^ + k^) u - 0.

We assume that u has an asymptotic expansion (for k -> oo) of the form

(2) u 'v e 27 â€” ,

We further assume that the asymptotic expansions of the derivatives of u are ob-

tained by differentiation of the expansion of u. Then inserting (2) into (1) and

equating to zero the coefficient of each power of k, we obtain the equations

(3) (VY)2 - 1,

(U) 2Vv^- ^ * \ '^^ â– -^^VP " â– 0' ^' â€¢â€¢â€¢^ ^-l" 0.

In (U) the coefficient v , is identically zero; it has been introduced to permit

the equation for v to be written together with those for the other v â€¢

Equation (3) is the eiconal equation of geometrical optics, and determines

the phase function T. Equations (U) are a recursive system which can be used for

dv

determining the v successively. Since 7v â€¢ 7T â– -^â€” , where s denotes arc length

along a ray (i,e,,a curve orthogonal to the wavefronts Y â– constant), equations (U)

are in fact linear ordinary differential equations along the rays. The solution

of (U), along a ray, is

9

- i / V^Yds T / - 7 / V^Tds' _

(5) v^(s) - vâ€ž(s^) e ^\ . I ^ e ^ ^^ V^v^^W df .

o

Equations ($) can be written in the simpler form (see [l] ).

- 3 -

o

In (6), G(s) denotes the Gaussian curvattire, or, in two dimensions, the ordinary

curvature, of the wavefront T Â«= constant at the point s on a ray.

For n Â» the integral in (6) is absent since v , â– 0, so v is given by

the first term. This solution shows that v varies along a ray inversely as the

square root of the cross-sectional area of a narrow tube of rays, because G(s) is

2

inversely proportional to this area. Therefore v multiplied by this area is con-

stant along each ray, which expresses the conservation of energy within such a tube.

Equation (6) also shows that if v is a vector, the orientation of v is constant

along each ray.

The preceding results are not completely new, and have been included here

primarily for the sake of completeness and convenience. M. Kline'- ^ has obtained

analogous results for Maxwell's equations, and the analogues of equations (3) and

(U) for more general linear equations. His method, irtiich is more complicated than

the present one, is that of R.K. Luneburg^ J who used it to obtain partial results

for Maxwell's equations. Other authors have also obtained partial results for spe-

cial equations - usually the analogues of (3) and of the first of equations (U)

and (6), i.a.vOf the equations for which n =Â» 0. F.G. Friedlander'- -', H. Bremmer *- â– ' ,

E.T. Copson'- -' and J. Riblet (unpublished) also treated Maxwell's equations: S.C.

ri2i hi

Lowell*- -* considered waves in shallow water; J.3. Keller"- â– â€¢ treated the equations

of acoustics; G.D. Birkhof f L^-' , L. Brillonin '-^J , G. Wentzelt^-^, P.A.M. -DiracM

and J.3. Keller'- -^ treated the Schroedinger equation of quantum mechanics. Par-

ticular problems have been solved with the aid of these asymptotic expansions by

G. SchenstedL^^-J, by K.O. Friedrichs and J.3. Keller '^â– '^-' , and by J.B. Keller L"^J* M

More general asymptotic expansions of solutions of equation (l) have been considered

by F.G. Friedlander and J.B. KelilerL-^^.

In the following sections we will construct the asymptotic expansions of

the solutions of several boundary- value problems by using equations (3) and (6).

We will also derive certain general formulas vdiich may be of use in other problems.

3Â« Cylindrical Waves

The rays, which are the orthogonal trajectories of the surfaces f Â» constant,

are straight lines, as can be shown from (3). Let us first consider the two-dimen-

sional case in which all the rays are the lines (Â© = constant )' emerging from an

- h -

origin and the wavefronts are the concentric circles (r " constant). Then

Y Â» + r + constant, and we may without loss of generality choose the constant to

be zero since equation (1) is linear and homogeneous. The curvature of the wave-

front (r Â» constant) is G(r) â– r . Thus from (6), using r instead of s, we have

(7) vjr,e) - g^(e) r-^/2^ [^^(e) = vY^W v^ f ^(e),ej] .

1/2

The factor g (6) is just r ' v (r ) evaluated on the ray (Â© =Â« constant) at r - r (Â©).

If we now use (6) and (?) we find by induction that v (r,Â©) can be written

in the form

(8) V (r,e) . f: f . (9) r-^^/2)-j ^

j=0 ^"

Inserting (8) into (6) yields the following recursive formulas for f . (9), with

ri > 1;

(9) f jn(e) - 1 I3 jTj- |) fj.1,^1 - ^j".l,n.l]> J ^ 0' - > 1*

(10)

.1/2 /Q\ ^ fr, fa\ d\ - V T."J

f (e) - r-^'^(e) V & (9), el - F r"-' f , , n > l

on^ o ^ n^-o^ * -J ^^ o jn* -

In (9) and (10) the upper sign is to be used if T - + r and the lower sign if

Y = -r. The function f (9) is Just g (9), and v [r (9), 9] is the value of v

at the point r (9) on the ray (9 = constant). Combining (2) and (8) we thus

have, for a cylindrical wave,

+ikr

- 00 - n .

(11) u^ 2~- Yl -^ t- ^in^Â®^ ^ â€¢

^ J^O (ik)" Fo ^"^

If the V [r (9), 9] are given then (9) and (lO) yield all the i.A^) success-

ively, and involve only differentiation. Thus if the expansion of u is given on a

curve r = r(9) then the v [r (9), 9] are given and (11) yields the expansion off the

curve. This way of obtaining the asymptotic expansion of a cylindrical wave is the

main result of this section, and we will now consider several applications of it.

Example 1 . Diffraction of a Plane Wave by a Wedge or Half -Plane

In the two-dimensional problem of diffraction of a plane wave by a wedge

(perfect conductcr or dielectric) or a half-plane, the asymptotic expansion of the

- 5 -

solution consists of several plane waves and of a diffracted cylindrical wave emana-

ting from the edge. The results (7)-(10) can be applied to this cylindrical wave*

We first observe that in the solution of this problem k and r can occur only in the

combination kr, since the problem contains no characteristic dimension.) In order

that the expansion also involve only the combination kr, we first multiply (11) by

"1/2

k ' , which we may do because the equation (1) is linear and homogeneoiis. Now

/â– I /O ^ ys

since the coefficient of v is k"^ ' *, and since v can contain only this same

n n

power of r, we must have f.Â»Oifj/nÂ« Then from (9) using the solution f Â« + r

J^ d

for the phase in order that the wave be outgoing, and using the notation D Â» -53, we

have

n

'12' ^nn(Â«' â€¢ ;h T, DJ- Â¥ * "5 8o(Â«' â€¢

2 nl j-1

n

Tl

J-

Inserting (12) into (ll) we obtain

00

(13) u(^,e)'^^ 2 ii * YZ -

^^ nÂ»l (2ikr

L- f; [(,. |,^ . ogj .â€ž(Â«, .

Equation (13) yields an asymptotic expansion which formally satisfies (l)

for any choice of g (6)Â« The present method cannot determine g (Â©) for the wedge

or half-plane problems^ However in other problems, some of which are treated in this

article, the present method can determine g (Â©)â€¢ If for the wedge or half -plans we

take S-.W from the first term in the asymptotic expansion of the exact solution,

which is known ifu"0org-Â«0on the wedge, then (13) will yield all the sub-

sequent terms in the expansion.

Example 2 , Hankel Functions

A cylindrical wave solution of (l) is given by the product of a Hankel

function H (kr) and sin mÂ© or cos mÂ©. Thus we have

m

(lU) u Â« H"'"(kr) cos mÂ©.

IB

Since this solution is outgoing and involves k and r only in the combination kr, its

expansion is given by (13) with g (Â©) â– A cos mÂ©. The constant A must be adjusted

so that (13) agrees with (Hi). Inserting g-,(Q) â– A cos mÂ© into (13) anl noting that

2 2

D cos mÂ© â– Â« -m cos mÂ© we obtain

- 6 -

- ikr 00 , n p. - 2 _-â–

(15) H^(kr) V A 2^ A . i: "â– IT Qj- i) - m^J .

" yS [^ n-1 (2ikr) n. j-1 "- '^ -^J

This expansion coincides exactly with the known e^qa^nsion of H if

A -/I exp[I ^ (m+ i)] (See [2o] p. 85, eq. 1).

If the phase function I is taken as -r instead of â™¦r, we obtain from (9)-

(11) the expansion of the Hankel function of the second kind, iirtiich differs from

the above merely in that i is replaced by -i, (see [20"], p. 85, eq, 2)Â»

Example 3 Â» Diffraction of a Plane Wave by a Parabolic Cylinder (u â– 0),

"ikx

We consider a plane wave e incident from the right along the axis

of a parabolic cylinder, and on the outside of it (see Fig. 1), The incident rays

are parallel to the axis, and the reflected rays are therefore radial lines which

pass through the focus if extended backward, as is well-known from the focussing

property of parabolas. Therefore the reflected field is a cylindrical wave which,

we will assume, can be described by equation (11). We will write the equation of

the parabola of focal length p as

We impose on the parabola the boundary condition

(16a) u(r,e) - on r - r^(Â«) .

Inserting the total field, incident plus reflected, into (16a) we have

-ikr (e)co3e ikT(r ,9) oÂ© v (r .Â«)

(17) e Â° * e Â° H -il-^ - .

1?0 (ik)""

Now equating coefficients of various powers of k in (1?) we obtain

(18) ^^VÂ®^ " -^0^Â®^Â°Â°^ Â®*

(19) 'o^^o'^) â– â€¢^â€¢

(20) ^n^VÂ®^ " Â°' n > 1.

- 7 -

j'lgtire 1

Reflection of a plane wave from a parabolic cylinder, or from a

paraboloid of revolution of focal length p. The plane wave is incident from

the right along the axis of the parabolic cylinder (or paraboloid) and the

origin is at the fociis. The incidait rays are parallel to the axis and the

reflected rays pass through the focus if extended backward.

- 8 -

The phase T is determined by (3) and (18), or alternatively by the dis-

tances along the rays, and the condition that the wave be outgoing. We have

(21) Y(r,0) - r-r (l-f30sÂ«) - r-2p.

Using (19) in (?) we obtain g (Â©) - â€” r ' (Â©), and thus

(22) v^(r,Â©) - -ry2(e) r"^/2 - -p^^^ (sec ^) r'^^^ .

Now we may determine the f . (6) from (9) and (10) using the bourdary conditions (20)

and the expression for g (Â©) given in (22 )o Upon calculating the first few f . we

find that the f . have the following form:

(.3) f,â€ž(Â«) - .jâ€ž p'V=)^^-"(sec |,^^*^ ,

which can then be proved by induction to hold generallyo In (23) the coefficients

a. are constants which satisfy the recursion formulas

(21^) ^jn- I (J- Â¥ ^J.l,n-1, J > 1> n > 1

n

(26) a^^- .1 .

Prom (2U) and (25) the a. can be determined successively, starting with a given

by (26).

Collecting our results, we have for the asymptotic expanaL on of the solution

(27) u-e-i^ . ei^(^-2p) Â£ (iicp)"" f a. (pr'^ sec^ h^ ^ (1/2) ^

n-0 J=0 ^^ ^

From equation (2?) with the a. computed from (2U)-(26), we find for

the first few terms in the expansion of u

- 9 -

(28) u.;e -/psec^e ^ Jl - j^^^ jl . ^ sec ^

The problem treated in this section was solved exactly by H. Lamb L-^ i

who made use of the separability of the problem in parabolic coordinates. When

his solution is expanded asymptotically in k, it yields exactly (27). The ampli-

tude and phase of the scattered wave computed from the exact solution are shown

in Figures 2 and 3. F'or comparison the same quantities computed from the first

few terras in the asymptotic expansion, i.e., from the terras through (kp) *â– , are

also shown. The agreement is seen to be good when kp >2.

Example U . Diffraction of a Plane Wave by a Parabolic Cylinder (|ii = 0)

~ oV

We consider the same situation as that of Example 3 but we replace the

boundary condition (l6a) by the requirement that the normal derivative of u

vanish, i.e., that

(29) ^-0 on r-r^(9).

Then instead of (17) we obtain

fir\\ -1, 9x ^-ikx ^ ikT/., 3T 3 V râ€” n â€ž ,-v

(30) -:Lk-e + e (ik - + -) ^ - â€” . on r = r (6).

n=0 (ik)

Equating the coefficients of the first power of k in (30) we find

(31) Y = -X Â« -r^(9) cos 8 on r = r (9),

(32) |I.|i on r.râ€ž(6).

From (31), (3) and the outgoing wave condition we find that Y is given by (21).

Then |I - |^ on r = r^(9). Therefore (32) yields

oV oV

(33) V = 1 on r Â» r (9).

o o

The coefficients of all higher powers of k yield

CNJ

I

â€¢H

(X.

CM

00

eg

o

cvi

a>

CM

oo

- 11 -

TABLE I

u =

A e ^ =

Scattering From A Parabolic Cylinder

so

2i v^ e-^i*^ f

e dz

-1

A ^ 1 +

32(kp)

7 ^

tanJ2f ^ jj^ +

3u

av

=

i^

A e '^ =

00

2 + 2i â€¢j/215'

e dz

/y^i^

-1

A '-' 1 -

32(kp)

? *

tan

'JH^*

[Lamb]

[Eq.(27)]

[Lamb]

[Eq.(27)]

u =

A e

Â±0

Scattering From A Paraboloid of Revolution

00

-l^ikp e

-2ikp

â€¢ 2 ,

xz dz

e

V2^

-1

A /^ 1 +

8(kp)

5 "â–

tanjZf ^ ^

JLamb]

[Eq.(l07)]

^ =

6v

A e '^ =

00

. 2

_ ^ , ., -2ikp / iz dz

2 + Uikp e ^ / e â€”

/)/5i5'

-1

A ~ 1 -

8(kp)

tan iZf -w - 2^ +

?*

[Lamb]

[Eq.dO?)]

-12-

5

â– p

CO

n

ctf

(4

o

t

â€¢ri

iH

>k

O

O

â€¢H

rH

O

43

CO

U

nJ

a

c^

a

0)

â€¢

1

E

o

â€¢

â€¢H

,^. ^^ Jv. NEW YORK UNIVERSITY

b:

f#

>

"J I IN IJ^ '*'

Institute of Mothematical Sciences

^^ ^-^ ,j> Division of Electromagnetic Research

RESEARCH REPORT No. EM-81

Asymptotic Solution of Some Diffraction Problems

J. B. KELLER, R. M. LEWIS and B. D. SECKLER

>EW YORK UNIVERSITY

liN^jlii-Ui t OF MATHEMATICAL SCIENCES

LIBRARY

23 Wsvwly Place, New York 3, N. Y.

CONTRACT NO. AF19(122)-42

JUNE 1955

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM- 81

ASYMPTOTIC SOLUTION OF SOME DIFFRACTION PROBLEMS

Joseph B. Keller, Robert M. Lewis and Bernard D. Seckler

^oserm B.

2XJ^

osepf B. Keller

Robert M. Lewis

Bernard D. Seckler

Morris Kline

Project Director

The research reported in this document has been made possible through

support and sponsorship extended by the Air Force Cambridge Research

Center, under Contract No. AF 19(122)-42. It is published for techni-

cal information only and does not necessarily represent recommendations

or conclusions of the sponsoring agency.

New York, 1955

/

Abstract

Various diffraction problems are solved asymptotically in k * y

(X Â» wavelength) for k large (i.e., X small). These problems include

diffraction of a plane wave by a parabolic cylinder, a paraboloid of

revolution, a cylinder and a sphere; diffraction of a spherical wave

by a paraboloid of revolution, a hyperboloid of revolution, and a

plane interface; diffraction of a cylindrical wave by a parabolic cy-

linder, a hyperbolic cylinder and a plane interface, etc. The boundary

conditions considered are the vanishing of the functionÂ» of its normal

derivative and the impedance boundary condition. Formulas are obtained

for reflection of any wave from any two dimensional surface, and certain

formulas are deduced for three dimensional problems. The method employed

is that devised by R. K. Luneburg and extended by M. Kline,

TABLE OF CCNTEMTS

Page

1. Introduction 1

2, Formulation of the Method 2

3Â» Cylindrical Waves 3

General Considerations

Example 1. Diffraction of a Plane Wave by a Wedge or Half -Plane k

Exanple 2. Hankel Functions 5

Example 3. Diffraction of a Plane Wave by a Parabolic Cylinder( u=0) 6

Example hÂ» Diffraction of a Plane Wave by a Parabolic Cylinder 9

du â€ž

â€” - =

dv

Example 5Â» The Field of a line Source at One Focus of a Hyperbolic 13

Cylinder (u => O)

Exau^jle 6Â» The Field of a line Source at One Focus of a Hyperbolic 18

Cylinder (|^ = O)

Example ?â€¢ The Field of a Line Source over a Plane Interface 19

k* General Cylindrical Waves 29

5Â» Spherical Waves 31

General Considerations

Example 8. Diffraction of a Plane Wave by a Cone of Arbitrary 32

Cross-Section

Example 9Â» Spherical Bessel Functions 33

Example 10. Diffraction of a Plane Wave by a Paraboloid of 33

Revolution (u = O)

Example 11. Diffraction of a Plane Wave by a Paraboloid of 37

Revolution (grr ^ O)

Example 12. The Field of a Point Source at one Focus of a 37

Ffyperboloid of Revolution (u = O)

Example 13. The Field of a Point Source at one Focus of a 39

I^erboloid of Revolution (^ " O)

Example II4. The Field of a Point Source Over a Plane Interface UO

6Â» General Plane Waves III*

General Considerations

Example 15. A Line Soiirce at the Focus of a Parabolic Cylinder h^

(u - 0)

Example 16. A line Sotirce at the Focus of a Parabolic Cylinder U8

Example 17 Â» A Point Source at the Focus of a Paraboloid of U9

Revolution (u = O)

Example 18. A Point Source at the Focus of a Paraboloid of SZ

Revolution (^â€” = O)

ov

7. Arbitrary Waves 52

8. Two-dimensional Waves 58

General Considerations

Example 19. Diffraction of a Plane Wave by a Circular Cylinder /

(u - 0) V 63

Example 20. Diffraction of a Plane Wave by a Circular Cylinder

(1^-0) 66

9. Axially Symmetric Waves 68

General Considerations

Example ?1. Diffraction of a Plane Wave by a Sphere (u Â» 0) 71

Example 22. Diffraction of a Plane Wave by a Sphere (|^ Â« 0) 72

10. The Impedance Boundary Condition 73

General Considerations

Fxample 23. Hif fraction of a Plane Wave by a Parabolic Cylinder

(|^ - ikZu) 7U

Fxample 2U, Diffraction of a Plane Wave by a Circular Cylinder

(|^ - ikZu) 77

References 78

- 1 -

1, Introduction

Several authors^ J"r J have recently discussed a method by which the

asymptotic expansion with respect to frequency may be determined for periodic

solutions of electromagnetic, acoustic, and other linear wave problems. However,

virtually no applications of the method have been made. In the present paper we

explain the method, and apply it to construct the asymptotic solutions of a var-

iety of diffraction problems. Many of these problems have not been solved pre-

viously. In some cases where the problems have been solved previously we verify

that the asymptotic expansions of the known solutions coincide vdth our asymptotic

solutions and compare the exact solutions graphically with the first few terms of

the asymptotic solutions.

The asymptotic expansion under consideration here is an expansion of a

periodic solution with respect to frequency as the frequency tends to infinity.

Thus the expansion is useful at high frequencies, or for short wavelengths. In

electromagnetic theory it is therefore useful in optics, in radar problems, etc.

In fact the leading term in the expansion is Just the so-called geometrical-optics

solution.

Mathematically the problem is that of finding the asymptotic expansion of

the solution of a boimdary-value problem for an elliptic equation. The equation

and the boundary data involve a parameter k which is proportional to the frequency

of the solution and inversely proportional to the wavelength, and the order of the

equation is reduced when k" is replaced by zero (i.e., when k becomes infinite).

The method employed determines directly the as3nnptotic expansion with respect to

k near k -oo, and thus we can avoid the customary detour of first finding an exact

solution and then expanding it asymptotically in k. Consequently, the present me-

thod can be used in cases where the exact solution has not been found, and this is

its main value. But even where the exact solution is known and can be expanded,

the present method usually yields the expansion more easily. It must be noted,

however, that for most problems it has not yet been rigorously proved that the ex-

pansion yielded by this method is indeed asymptotic to the exact solution, although

formally this appears to be the case.

The numerous problems treated in this paper by the above method are selected

as exaii5>les of the types of problems for which this method is applicable* On the

other hand, it is also apparent that imless the problems conrespond to the simplest

geometries, the calculations beccane quite lengthy. In fact the problems considered

in this paper were selected partly because the related ray systems are simple, and

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therefore the calculations are relatively easy. Further, it is not yet possible,

in general, to treat problems involving diffracted rays'- -I by this method. In

some problems where such rays do occur the present method can yield only incomplete

results, as in examples 1 and 8. However in examples 7 and lU a special method was

devised to treat the diffracted rays, and it is to be expected that more such me-

thods will be devised for other problems,

2. Formulation of the Method

We consider a solution u (scalar or vector) of the reduced wave equation

(1) (V^ + k^) u - 0.

We assume that u has an asymptotic expansion (for k -> oo) of the form

(2) u 'v e 27 â€” ,

We further assume that the asymptotic expansions of the derivatives of u are ob-

tained by differentiation of the expansion of u. Then inserting (2) into (1) and

equating to zero the coefficient of each power of k, we obtain the equations

(3) (VY)2 - 1,

(U) 2Vv^- ^ * \ '^^ â– -^^VP " â– 0' ^' â€¢â€¢â€¢^ ^-l" 0.

In (U) the coefficient v , is identically zero; it has been introduced to permit

the equation for v to be written together with those for the other v â€¢

Equation (3) is the eiconal equation of geometrical optics, and determines

the phase function T. Equations (U) are a recursive system which can be used for

dv

determining the v successively. Since 7v â€¢ 7T â– -^â€” , where s denotes arc length

along a ray (i,e,,a curve orthogonal to the wavefronts Y â– constant), equations (U)

are in fact linear ordinary differential equations along the rays. The solution

of (U), along a ray, is

9

- i / V^Yds T / - 7 / V^Tds' _

(5) v^(s) - vâ€ž(s^) e ^\ . I ^ e ^ ^^ V^v^^W df .

o

Equations ($) can be written in the simpler form (see [l] ).

- 3 -

o

In (6), G(s) denotes the Gaussian curvattire, or, in two dimensions, the ordinary

curvature, of the wavefront T Â«= constant at the point s on a ray.

For n Â» the integral in (6) is absent since v , â– 0, so v is given by

the first term. This solution shows that v varies along a ray inversely as the

square root of the cross-sectional area of a narrow tube of rays, because G(s) is

2

inversely proportional to this area. Therefore v multiplied by this area is con-

stant along each ray, which expresses the conservation of energy within such a tube.

Equation (6) also shows that if v is a vector, the orientation of v is constant

along each ray.

The preceding results are not completely new, and have been included here

primarily for the sake of completeness and convenience. M. Kline'- ^ has obtained

analogous results for Maxwell's equations, and the analogues of equations (3) and

(U) for more general linear equations. His method, irtiich is more complicated than

the present one, is that of R.K. Luneburg^ J who used it to obtain partial results

for Maxwell's equations. Other authors have also obtained partial results for spe-

cial equations - usually the analogues of (3) and of the first of equations (U)

and (6), i.a.vOf the equations for which n =Â» 0. F.G. Friedlander'- -', H. Bremmer *- â– ' ,

E.T. Copson'- -' and J. Riblet (unpublished) also treated Maxwell's equations: S.C.

ri2i hi

Lowell*- -* considered waves in shallow water; J.3. Keller"- â– â€¢ treated the equations

of acoustics; G.D. Birkhof f L^-' , L. Brillonin '-^J , G. Wentzelt^-^, P.A.M. -DiracM

and J.3. Keller'- -^ treated the Schroedinger equation of quantum mechanics. Par-

ticular problems have been solved with the aid of these asymptotic expansions by

G. SchenstedL^^-J, by K.O. Friedrichs and J.3. Keller '^â– '^-' , and by J.B. Keller L"^J* M

More general asymptotic expansions of solutions of equation (l) have been considered

by F.G. Friedlander and J.B. KelilerL-^^.

In the following sections we will construct the asymptotic expansions of

the solutions of several boundary- value problems by using equations (3) and (6).

We will also derive certain general formulas vdiich may be of use in other problems.

3Â« Cylindrical Waves

The rays, which are the orthogonal trajectories of the surfaces f Â» constant,

are straight lines, as can be shown from (3). Let us first consider the two-dimen-

sional case in which all the rays are the lines (Â© = constant )' emerging from an

- h -

origin and the wavefronts are the concentric circles (r " constant). Then

Y Â» + r + constant, and we may without loss of generality choose the constant to

be zero since equation (1) is linear and homogeneous. The curvature of the wave-

front (r Â» constant) is G(r) â– r . Thus from (6), using r instead of s, we have

(7) vjr,e) - g^(e) r-^/2^ [^^(e) = vY^W v^ f ^(e),ej] .

1/2

The factor g (6) is just r ' v (r ) evaluated on the ray (Â© =Â« constant) at r - r (Â©).

If we now use (6) and (?) we find by induction that v (r,Â©) can be written

in the form

(8) V (r,e) . f: f . (9) r-^^/2)-j ^

j=0 ^"

Inserting (8) into (6) yields the following recursive formulas for f . (9), with

ri > 1;

(9) f jn(e) - 1 I3 jTj- |) fj.1,^1 - ^j".l,n.l]> J ^ 0' - > 1*

(10)

.1/2 /Q\ ^ fr, fa\ d\ - V T."J

f (e) - r-^'^(e) V & (9), el - F r"-' f , , n > l

on^ o ^ n^-o^ * -J ^^ o jn* -

In (9) and (10) the upper sign is to be used if T - + r and the lower sign if

Y = -r. The function f (9) is Just g (9), and v [r (9), 9] is the value of v

at the point r (9) on the ray (9 = constant). Combining (2) and (8) we thus

have, for a cylindrical wave,

+ikr

- 00 - n .

(11) u^ 2~- Yl -^ t- ^in^Â®^ ^ â€¢

^ J^O (ik)" Fo ^"^

If the V [r (9), 9] are given then (9) and (lO) yield all the i.A^) success-

ively, and involve only differentiation. Thus if the expansion of u is given on a

curve r = r(9) then the v [r (9), 9] are given and (11) yields the expansion off the

curve. This way of obtaining the asymptotic expansion of a cylindrical wave is the

main result of this section, and we will now consider several applications of it.

Example 1 . Diffraction of a Plane Wave by a Wedge or Half -Plane

In the two-dimensional problem of diffraction of a plane wave by a wedge

(perfect conductcr or dielectric) or a half-plane, the asymptotic expansion of the

- 5 -

solution consists of several plane waves and of a diffracted cylindrical wave emana-

ting from the edge. The results (7)-(10) can be applied to this cylindrical wave*

We first observe that in the solution of this problem k and r can occur only in the

combination kr, since the problem contains no characteristic dimension.) In order

that the expansion also involve only the combination kr, we first multiply (11) by

"1/2

k ' , which we may do because the equation (1) is linear and homogeneoiis. Now

/â– I /O ^ ys

since the coefficient of v is k"^ ' *, and since v can contain only this same

n n

power of r, we must have f.Â»Oifj/nÂ« Then from (9) using the solution f Â« + r

J^ d

for the phase in order that the wave be outgoing, and using the notation D Â» -53, we

have

n

'12' ^nn(Â«' â€¢ ;h T, DJ- Â¥ * "5 8o(Â«' â€¢

2 nl j-1

n

Tl

J-

Inserting (12) into (ll) we obtain

00

(13) u(^,e)'^^ 2 ii * YZ -

^^ nÂ»l (2ikr

L- f; [(,. |,^ . ogj .â€ž(Â«, .

Equation (13) yields an asymptotic expansion which formally satisfies (l)

for any choice of g (6)Â« The present method cannot determine g (Â©) for the wedge

or half-plane problems^ However in other problems, some of which are treated in this

article, the present method can determine g (Â©)â€¢ If for the wedge or half -plans we

take S-.W from the first term in the asymptotic expansion of the exact solution,

which is known ifu"0org-Â«0on the wedge, then (13) will yield all the sub-

sequent terms in the expansion.

Example 2 , Hankel Functions

A cylindrical wave solution of (l) is given by the product of a Hankel

function H (kr) and sin mÂ© or cos mÂ©. Thus we have

m

(lU) u Â« H"'"(kr) cos mÂ©.

IB

Since this solution is outgoing and involves k and r only in the combination kr, its

expansion is given by (13) with g (Â©) â– A cos mÂ©. The constant A must be adjusted

so that (13) agrees with (Hi). Inserting g-,(Q) â– A cos mÂ© into (13) anl noting that

2 2

D cos mÂ© â– Â« -m cos mÂ© we obtain

- 6 -

- ikr 00 , n p. - 2 _-â–

(15) H^(kr) V A 2^ A . i: "â– IT Qj- i) - m^J .

" yS [^ n-1 (2ikr) n. j-1 "- '^ -^J

This expansion coincides exactly with the known e^qa^nsion of H if

A -/I exp[I ^ (m+ i)] (See [2o] p. 85, eq. 1).

If the phase function I is taken as -r instead of â™¦r, we obtain from (9)-

(11) the expansion of the Hankel function of the second kind, iirtiich differs from

the above merely in that i is replaced by -i, (see [20"], p. 85, eq, 2)Â»

Example 3 Â» Diffraction of a Plane Wave by a Parabolic Cylinder (u â– 0),

"ikx

We consider a plane wave e incident from the right along the axis

of a parabolic cylinder, and on the outside of it (see Fig. 1), The incident rays

are parallel to the axis, and the reflected rays are therefore radial lines which

pass through the focus if extended backward, as is well-known from the focussing

property of parabolas. Therefore the reflected field is a cylindrical wave which,

we will assume, can be described by equation (11). We will write the equation of

the parabola of focal length p as

We impose on the parabola the boundary condition

(16a) u(r,e) - on r - r^(Â«) .

Inserting the total field, incident plus reflected, into (16a) we have

-ikr (e)co3e ikT(r ,9) oÂ© v (r .Â«)

(17) e Â° * e Â° H -il-^ - .

1?0 (ik)""

Now equating coefficients of various powers of k in (1?) we obtain

(18) ^^VÂ®^ " -^0^Â®^Â°Â°^ Â®*

(19) 'o^^o'^) â– â€¢^â€¢

(20) ^n^VÂ®^ " Â°' n > 1.

- 7 -

j'lgtire 1

Reflection of a plane wave from a parabolic cylinder, or from a

paraboloid of revolution of focal length p. The plane wave is incident from

the right along the axis of the parabolic cylinder (or paraboloid) and the

origin is at the fociis. The incidait rays are parallel to the axis and the

reflected rays pass through the focus if extended backward.

- 8 -

The phase T is determined by (3) and (18), or alternatively by the dis-

tances along the rays, and the condition that the wave be outgoing. We have

(21) Y(r,0) - r-r (l-f30sÂ«) - r-2p.

Using (19) in (?) we obtain g (Â©) - â€” r ' (Â©), and thus

(22) v^(r,Â©) - -ry2(e) r"^/2 - -p^^^ (sec ^) r'^^^ .

Now we may determine the f . (6) from (9) and (10) using the bourdary conditions (20)

and the expression for g (Â©) given in (22 )o Upon calculating the first few f . we

find that the f . have the following form:

(.3) f,â€ž(Â«) - .jâ€ž p'V=)^^-"(sec |,^^*^ ,

which can then be proved by induction to hold generallyo In (23) the coefficients

a. are constants which satisfy the recursion formulas

(21^) ^jn- I (J- Â¥ ^J.l,n-1, J > 1> n > 1

n

(26) a^^- .1 .

Prom (2U) and (25) the a. can be determined successively, starting with a given

by (26).

Collecting our results, we have for the asymptotic expanaL on of the solution

(27) u-e-i^ . ei^(^-2p) Â£ (iicp)"" f a. (pr'^ sec^ h^ ^ (1/2) ^

n-0 J=0 ^^ ^

From equation (2?) with the a. computed from (2U)-(26), we find for

the first few terms in the expansion of u

- 9 -

(28) u.;e -/psec^e ^ Jl - j^^^ jl . ^ sec ^

The problem treated in this section was solved exactly by H. Lamb L-^ i

who made use of the separability of the problem in parabolic coordinates. When

his solution is expanded asymptotically in k, it yields exactly (27). The ampli-

tude and phase of the scattered wave computed from the exact solution are shown

in Figures 2 and 3. F'or comparison the same quantities computed from the first

few terras in the asymptotic expansion, i.e., from the terras through (kp) *â– , are

also shown. The agreement is seen to be good when kp >2.

Example U . Diffraction of a Plane Wave by a Parabolic Cylinder (|ii = 0)

~ oV

We consider the same situation as that of Example 3 but we replace the

boundary condition (l6a) by the requirement that the normal derivative of u

vanish, i.e., that

(29) ^-0 on r-r^(9).

Then instead of (17) we obtain

fir\\ -1, 9x ^-ikx ^ ikT/., 3T 3 V râ€” n â€ž ,-v

(30) -:Lk-e + e (ik - + -) ^ - â€” . on r = r (6).

n=0 (ik)

Equating the coefficients of the first power of k in (30) we find

(31) Y = -X Â« -r^(9) cos 8 on r = r (9),

(32) |I.|i on r.râ€ž(6).

From (31), (3) and the outgoing wave condition we find that Y is given by (21).

Then |I - |^ on r = r^(9). Therefore (32) yields

oV oV

(33) V = 1 on r Â» r (9).

o o

The coefficients of all higher powers of k yield

CNJ

I

â€¢H

(X.

CM

00

eg

o

cvi

a>

CM

oo

- 11 -

TABLE I

u =

A e ^ =

Scattering From A Parabolic Cylinder

so

2i v^ e-^i*^ f

e dz

-1

A ^ 1 +

32(kp)

7 ^

tanJ2f ^ jj^ +

3u

av

=

i^

A e '^ =

00

2 + 2i â€¢j/215'

e dz

/y^i^

-1

A '-' 1 -

32(kp)

? *

tan

'JH^*

[Lamb]

[Eq.(27)]

[Lamb]

[Eq.(27)]

u =

A e

Â±0

Scattering From A Paraboloid of Revolution

00

-l^ikp e

-2ikp

â€¢ 2 ,

xz dz

e

V2^

-1

A /^ 1 +

8(kp)

5 "â–

tanjZf ^ ^

JLamb]

[Eq.(l07)]

^ =

6v

A e '^ =

00

. 2

_ ^ , ., -2ikp / iz dz

2 + Uikp e ^ / e â€”

/)/5i5'

-1

A ~ 1 -

8(kp)

tan iZf -w - 2^ +

?*

[Lamb]

[Eq.dO?)]

-12-

5

â– p

CO

n

ctf

(4

o

t

â€¢ri

iH

>k

O

O

â€¢H

rH

O

43

CO

U

nJ

a

c^

a

0)

â€¢

1

E

o

â€¢

â€¢H

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