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in order that this term, which represents the reflected and diffracted waves, be
of the same order as the incident wave. Of course, due to the linearity and homo-
geneity of (1), this factor does not affect the derivation of equations (7) - (10),



- 16 -



To determine Y and the v we first use the boundary condition u ■ C on
the hyperbola. From this condition, with u given by (38), we obtain

(UO) T - R^ - r + 2a, on r - r^(©).

(Ul) V - -r""'^^/^V , < n .

^ no "^n' -

By considering distances along the reflected rays, and using the boundary value
(UO) for T on the hyperbola and the outgoing condition for the reflected wave,
we find that Y is g3.Aren by
fU2) Y = r + 2a.

Now we use (Ul) and (7) to obtain v (r,©):

(U3) v^(r,9) - -(wr)-^/^.

In {1x3) we have introduced the new variable w which is related to © by

(UU) w » P - Q cos©j P - 1 + 2ap, Q » 2aq,

From (Ul) and (U3) it appears to be convenient to consider the f . as functions of
w, i.e., as f . (w), rather than as functions of ©. Then when we use {lH) , the re-
cursion formulas (9) and (10) become the following:

(U5) f^„(w) = (2j)-l[(j - ^)'fj.i,„.i ^(P-)fj'.l,n-l

+ (Q^-P^+2Pw-w^)f"_^^ J, j > 1, n > 1,



(U6) f^ (w) = -p (2a)-"w-^/^(l-w-^)" - i: (tT^^ ^n(^^* " ^ ^-
on n tieI "



-1/2
Starting with f which is given by (U2) as f = -w ' , we may compute

OO 00

the f successively, using (U5) and (U6). Upon computing a few, we find a certain
regularity in the f . which leads us to assume that f . has the form

Jll J**



- 17 -



The c are constants. The fonn (U7) can be proved to be correct by induction,
jtn

using (U8) and (U6), if the c., satisfy the recursion fonnulas

jtn



(ue) cjtn-t^J)'"



,(o-|)^-(|*3*t)^



■ ^M.Ul.n-l'^fOt- |)(t»3)Cj.^^^_„.j



HQ'-f')(yt. |)(M- |)Cj.,^,.i,„.,



,l O), However, in place of (U9) we find that c^.^ is
jn jy/Ti — oi*(t.L)-(«'-f')( J-i'

J«0 s»o ^ ' |_ I






, < t < 2n, n > 1



As before, the asymptotic expansion of u is given by (50) with the appropriate

c.. which are now determined from (U7) and {$$) starting with c » + v — .
,ltn ooo n

The first few terms in the expansion of u are, from (50),



- 19 -



(56) u^^e l_ p^(ikR) ^ ' '+ e V ^j]^



n-o



, 1



[1 + 8P + w"^(UP+Q^-P^-5)+ 3w"^(Q^-P^)



y *BIk?1



UPw"^+ 3w"^Q^-P^)y +.



This problem has not been solved previously, so far as we know. It
applies to the scattering by a rigid hyperbolic cylinder of sound from a line
source if u denotes pressure and to the scattering by a perfectly conducting
hyperbolic cylinder of an electromagnetic wave from a line source if u denotes
the z-component of magnetic field, which must be the only non-vanishing magnetic
field component.
Example 7 * The Field of a Line Source over a Plane Interface

Let us consider spac>3 to be filled with two different media separated

by the plane interface y « 0. In the upper region y > 0, u is assumed to satisfy

equation (1) while in the lower region y < 0, u is assumed to satisfy (l) with

k replaced by k'= ^ik. Here n is a positive constant which may be interpreted

c
physically as the ratio — of propagation speed in the upper medium to that in

the lower medium. At the interface y = 0, u is assumed to satisfy the two

boundary conditions

(57) u(x,0*) - au(x,0")

/roN au(x,0"*') du(x,0")

In these conditions, a and b are two given constants. These conditions are appro-
priate to acoustic problems and certain electromagnetic problems.

In addition to the preceding conditions, the field u is assiimed to be
produced by a line source which is parallel to the interface and normal to the
xy-plane. Thus in the xy-plane the source is a point which we assume to be



- 20 -




Figure 5



The field at a line (or point) soiirce over a plane interface. The
source is located in the upper half -space y > at x = 0, y =» h (or at x= z «= 0,
y «= h) . The incident rays are lines radially directed from the source, and a
denotes the angle which a ray makes with the y-axis, which is the angle of in-
cidence of the ray at the interface y = 0. The reflected rays are lines which
appear to radiate from the image source at x = 0, y = -h (or at x = z = 0, y = -h) •
The reflected rays enter the second medium in the lower half-space making the angle
p, the angle of refraction, with the y-axis. If the propagation speed in the lower
medium exceeds that in the upper medium there is a critical angle of incidence a
for which p = n/2. The refracted ray is then along the interface, and from any
point on it a diffracted ray may split off into the upper medium making the angle
a with the y-axis. The surface normal to these diffracted rays is a diffracted
wavefront, which appears as a line segment in the figure.



-21 -



located atx»0,y«h>Oin the upper half -plane (see Figure $) . We are thus

dealing with a two-dimensional problem in which the solution u consists of a

cylindrical wave coming from the source plus outgoing waves. The cylindrical

wave is given exactly by (lU) or asymptotically by (lii), with m = in both

cases, according to the definition of a line source. We wish to determine the

asymptotic expansion of the solution u.

As a first step in the construction of the a^mptotic expansion, we

assume that the expansion has the form

^LoN ikr ^ ,., x-n-(l/2) ikT ^ / ., ^-n-(l/2) ^-

n=o n=o

00

r

n=o

This assumption is based on the expectation that in the upper region u will cop-
tain a reflected wave in addition to the incident cylindrical wave, while in the
lower region u will consist entirely of a transmitted or refracted wave. This
assumption will be justified below for ti > 1, i.e., for the case where no total
reflection occurs. However, when total reflection does occur ( n < 1), the pre-
ceding assumption must be modified by including a diffracted wave in the upper
region and an evanescent wave in the lower region. These modifications will be
considered after the case n> 1 has been treated.

Upon inserting the assumed asymptotic expansion (59), (60) into the
boundary conditions (57), (58) and equating coefficients of corresponding powers
of k ve obtain
(61)



(60) u r. e^^'^' r (ik')-"-^^/^\, y < 0.



Y(x,C) = tiT«(x,0) = r(x,0) = (h^ + x^)^/^,
(62) o^t""^^^^^ ^ v^(x,0) . a^-^^-(l/2)v^(x,0),

f - tJvI-pJ * ^^"^ v^(x,0). 1^ v^_^(x,0),

\>^-''^^^/^^^{x,Q)^^ix,0)* |p w^_^(x,0)], n = 0,1,-'-



(63) hr— (3/2)



- 22 -

The quantities p ,, v , and w , are understood to oe zero in (63) and are intro-
duced merely to unify the equations for n » and n ?< 0,

From (3) and (61), and from the condition that the second wave in (59)
must be outgoing, we find



(6U) 7(x,y) - r' -



p . (y . hj^jVS.



In (6i4), r' denotes the distance from (x,y) to the image source at (0,-h)j and
the positive square root is intended. Thus the reflected rays satisfy the law
of reflection and appear to come from the image source, and hence the reflected
wave is also cylindrical. Therefore the v are of the form (8) with the fj^j(&')
determined by (9) and (IC).

To deteinine Y' we again use (3), (61), and the outgoing wave condition.
We first note that (3) and (61) imply that the transmitted rays satisfy the law
of refraction. In terms of the angle of incidence a and the angle of refraction
p this condition is

(65) sin p ■ |ji~ sin a.

By making use of these angles, and by considering the variation in T' along a re-
fracted ray, we obtain parametric equations for the wavefront (!"' • constant) in
the form

(66) y = -(T' - hp," sec a) cos p,

(67) X ■ (T' - hp," sec a) sin ,6 + h tan ao

These equations determine T«(x,y). It is apparent that the refracted wave is not
cylindrical if [i jl 1 and therefore the w^ are given by (6), rather than by the
simpler formulas for cylindrical waves.

Now to compute the v^ and w at y - we solve (62) and (63), obtaining



w^(x,0) -^rh |bn -^ ^J



(68)

11 I

,-n-(l/2)



-r



[i- lK..-^4 .-'^^^'(-.-V '^y\



- 23 -



(69) v„(x,0) . a^-"-(l/^Vp„ - "-(^/^).

These bovmdary values together with equation (6) for the w and (8) for the v

suffice for the successive calculation of the v and w . From (7) and (68), we

n n X / »

have at once

(70) vjx,y)-^(r.)-^/^ Z-44?l|.



TTz



a cosa



To find w we use (69) and (6). The second term of (6) is zero, so the first
term alone yields w . When G(s) is obtained from equations (66) and (67) which
determine the wavefront, we find



In order to calculate v, we insert the above expressions for v and w

1 o o

into equation (68) and obtain for v, at y »

Sw_ 3v.



(72) v^(x,0) - (l*Z)-^|rh-^(bn-^/2 -^ - ^)-(8Ta'^)-^/2



1 ,2 .1/2

^ ^^
nr



VJhen this boundary value of v, and equation (70) for v are used in (9) and (10),
we find

In a similar way but with more complicated calculation, w, and the
s^icceeding terras can be computed | however we will not determine any more of
them here. The lowest-order terms v and w were previously found by K.O.
Friedrichs and J.B. Keller I- ^, It is apparent that the assumed form of
asymptotic expansion (59) and (60) can be completely determined by the preceding
methods, and that it satisfies the conditions of the problem.

Let us now turn to the case ^L < 1 in which total reflection occurs for



- 2k -



rays incident at angles greater than the critical angle a - sin" V* In this
case the trangmitted rays, corresponding to rays incident at angles between
and a^, cover the entire lower half-plane. In particular the critically re-
fracted ray, for which p •= «■> ^^^^ ^ ^^ interface y - 0, Thus if any wave
is produced in the lower half-plane by the totally reflected rays it is present
in the same region as that occupied by the transmitted wave, and it wculd have
to be included in the expansion of u. Without this additional wave we find that
the boundary conditions at the interface cannot be satisfied on that portion of
the interface where total reflection occurs. However, even if we add this wave
in the lower region the boundary conditions still cannot be satisfied. It is
necessary to add another wave in the upper region as well, and then the boundary
conditions can be satisfied. Therefore we assume that u has the asymptotic form

(7U) u . e^^^ Z P„(ikr)-"-(^/2). e^^^ £ (ik) - -(^/2)v^ . e^^^ ^ (ik) — (^2)3^^
n"o n»o n"0

y > 0,

^ ,.. ,s-n-(l/2) ik')6 ^
n«o n»o



,«c\ ik'T' ^ ,.. ,s-n-(l/2) ik«)6 ^ ., . x-n-(l/2)^

(75) u - e y" (ik') ^ ' 'w + e V' (ik) ^ ' ''t^, y < 0.

*— _ n *— n



We will call the waves in the upper region the incident, reflected and
diffracted waves respectively, and in the lower region the transmitted or refracted
wave and the evanescent wave. The reasons for this choice of names will become
clear presently.

The interface may be divided into the region of regular reflection, for
which |x| < h tan a , and the region of total reflection |x| > h tan a . In the
region of regular reflection we will assuae that only the incident, reflected,
and refracted waves are present and that they satisfy the boundary conditions. In
the region of total reflection we will assume that all the waves are present and
that the incident, reflected, and evanescent waves together satisfy the boundary
conditions while the transirltted and diffracted waves together also satisfy the



- 25 -



boundary conditions. In this way the boundary conditions will be satisfied
everywhere on the interface, provided that the diffracted and evanescent waves
are really absent in the region of regralar reflection. This will be found to
be the case when the solution is constructed.

On the basis of the foregoing method of satisfying the bouxxdary con-
ditions, we see that the transmitted and regularly reflected parts of the field
are determined as before. Similarly the evanescent wave amd the totally reflected
part of the reflected field also satisfy equations (6l)-(63). The angle of re-
fraction p is complex for these waves, however, and therefore the evanescent wave
cannot be determined in the same way as was the transmitted wave. Since the
evanescent wave is not required for the determination of the term with n = in
the totally reflected wave, this term can be found in exactly the same way as
before. Once the evanescent wave is found, the other terms in the totally re-
flected wave can be found in the same way as can those in the regularly reflected
wave. We will not consider these waves further, except to note that since p is
coratjlex the evanescent wave is really evanescent (i.e., exponentially decaying)
and can therefore be omitted in the region of regular reflection, where it is
of an exponentially lower order than all the other waves.

We now turn to the determination of the diffracted wave, which together
with the transmitted wave, has been assumed to satisfy the boundary conditions in
the region of total reflection. The transmitted wave is in principle completely
determined by the recursion formula (6) and the fact that, together with the in-
cident and reflected waves, it satisfies the boundary conditions in the region
of regular reflection. Therefore the remaining conditions in the region of total
reflection serve to determine the diffracted wave in terms of the transmitted wave.

We now insert the diffracted wave, given by the last term in (7U) and
the transmitted wave, given by the first term in (75), into the boundary conditions
(5?) and (58) in the region of total reflection, obtaining



- 26 -



(76) e"«» Z (iK)-"- h tana ,

(79) 8^(x,e) » an-"-^^/2^w^(x,0).



In deriving (8o) we have used the fact that -5 — ■ at y ■ since the wave fronts

dy

(f ■ constant) are orthogonal to the plane y - 0.

From (76), (3) and the outgoing wave condition, we have

(81) (2(x,y) - (h+y)secaQ + n |x|-(h+y)tanaQ , y > 0, |x| > (h+y)tanaQ.

The wavefronts given by (81) are parallel straight lines and the rays are consequently
the parallel lines which make the angle a with the y-axis (see Figure 5), Thus
the diffracted wave is a general plane wave of the type to be considered in Section
VI, i.e., the wavefronts are parallel planes but the amplitude on each plane is
not constant. Since (79) determines s at y ■ in terms of w , which is already
completely determined, the fonnulas of Section 6 can be used to cauLculate s off
the plane y - 0. For the first term 8 the result is, since w ■ at y ■ 0,

(82) 8^ - 0.

The next term s. is then constant along each ray, and hence



- 27 -



(83) s-(x,y) « ajji"-^' w^(x-ytano ,0), x > 0, y > 0, x-ytano > h tana ,
(8U) s-(x,y) « ap,'-^' w (x+ytana ,0), x < 0, y > 0, x+ytane < -h tana .

As we have not yet computed w, explicitly, we note that s, (x,0) can also be
found from (80) with n - 0. Thus, noting that ^ • cos a , we have instead
of (83)

(85) s^(x,y) - ibti'-^/^seca^ ~(x-ytana^,0), x > 0, y > 0, x-ytana^ > h tana^,

^0

An equation similar to (8U) applies for x < 0, If we compute -^ — from (71) we
obtain for s. from (85)

2b(sin (iJ r "i ^/p

(86) s^(x,y) 2 X- (y+h)tan a "^/^ , x- (y + h)tan a > .

a cos a l- ^ °

o



The term e ^(ik)" ' s, is the leading term in the diffracted field,
since s ■ 0. This leading term is of order ^ with respect to the incident
field. Its wave fronts are parallel planes, and its rays are parallel straight
lines which may be interpreted as having originated from the incident ray with
angle of incidence a • This critically incident ray gives rise to a refracted



ray lying in the interface, and this refracted ray in turn splits into a family
of diffracted rays all of which leave the interface at the critical angle a •

These rays are given by a modified form of Format's priniple '^ ■' . From (86)

-3/2
we see that the amplitude of s, is proportional to (x-(y-fh)tana 1 , where

the expression in brackets is just the length of that part of the ray from

the source to x, y which lies in the interface. Thus s, becomes infinite on

that ray for which this length is zero, which is just the critical totally

reflected ray. Therefore the present asymptotic expansion fails on that ray.

-3/2
For fixed y and large x, s, behaves like x , which is a more rapid decay

than that of the incident cylindrical wave. The term s, was also obtained



- 28 -

by K. 0. Friedrichs and J. B. Keller I- J. However, they did not determine
V- , which is of the same order as s,.

In siunmary, we have found that when ti > 1, no total reflection occurs
and the asymptotic expansion consists of one incident and one reflected cylindri-
cal wave in the upper medium, and of a single transmitted wave in the lower medium.
The waves in the upper medium are given by (59), and the wave 'in the lower medium
by (60). The incident wave is completely known, the first two terms of the re-
flected wave have been computed [v is given in (70), v, in (73) and T in (6U)3»
and the first term of the transmitted wave has been computed [w is given in (71)
and Y' in (66), (67)]. Additional terms can be computed making use of (68), (69)
and the formulas of Section 2,

In the case n < 1, total reflection occurs for rays incident at angles,
greater than the critical angle a ■ sin" n. In this case the asymptotic expan-
sion in the upper medium contains a diffracted wave in addition to the incident
and reflected cylindrical waves. This diffracted wave is a general plane wave
and is not present in that part of space covered by regularly reflected rays, but
only in the region covered by totally reflected rays. It may be thought of as
being produced by the transmitted wave, which travels along the interface and
'leaks' back into the upper medium. This wave is gi-ven by the third series in
(7U). Only its first two terms have been computed, the first of which (s ) is
zero and the second of which can be found from equations (86) for 8, and (61)
for ,

In the lower medium an evanescent wave occurs in addition to the ordinary
transmitted wave. This wave has not been determined although its boundary value
at y « has been found. The reflected and transmitted waves are given by the
same formulas as in the case (i > 1.

Finally it must be pointed out that although (79) completely determines
the boundary values s (x,0) of the terras in the diffracted wave, equation (80)
must also be satisfied. Since all the terras in (80) are already determined, this
equation must be an identity.



- 29 -



il» General Cylindrical Waves

By a general or three-dimensional cylindrical wave we mean a wave with
concentric circular cylinders as wavefronts but with an amplitude depending upon
all three space variables. If the axis of the cylinders is the z-axis then the
the coefficients v will depend upon z as vrell as upon r and 0, whereas for the
usual or two-dimensional cylindrical waves, such as those in Section 3> the v

are independent of z. For general cylindrical waves equation (5) yields

go(r,©,z)
(87) v^(r,9,z) - ^1^ .



From equations (6) and (8?) we find by induction that v has the form
(86) V (r,ft,z) - y- 7" a. . r"^^/^^'^ (logr)^.



i=-n j=o

Thus the asymptotic expansion of a general cylindrical wave is

j=n

±ikr 00 i=n

(69) u^^ r(ik)"'f a.. r-^(lcgr)J.

/^ *— h'— ijn
yr n=o i=-n "^

Insertion of (88) into (6) yields a rather complicated recursion formula for

the a. . . Therefore instead of giving this formula, we list below the expressions

for those a. . with n = 1 and 2 which are not zero. For n = 1 v/e find
ion



*101



®001






^-1,01" - ? Vo •



Thus



{9^) V3^(r,9,z) . a^ir-3/2 , a^^^r'^/^ . -.l,o/^^'



- 30 -

Similarly for n ^ 2



^202 ■ i'^h^l^ ^01,



^102 " 2 (it * V ^001,



^012



-?|[n"^e] ^-i,oi*^z^i /•



^ ^o (,. 1,2 A 1 ^o 2

^002 - - E ^TTT- ( (^ " 7^ * °6 J \01* ^^ ^17? ^z ^i,01



i



(J ^ ^^^1,01 * ^z ^1 ^°^ ^o * ^Y\^o*^*^ '



-1,02 r ^z ^001 »

1 _,2
^-2,02 "'Hz ^-1,01 •



Thus



(91) V2(r,©,z) - 8202^"^^^ * ^2""'^^^ * ^012^""^^^ ^°S ^

General cylindrical waves may occur in the diffraction of a plane wave
by a wedge whose properties depend upon z (a stratified wedpe), in the diffraction
of a general plane wave by a parabolic cylinder, etc. However, we will not con-
sider here any examples involving such waves. It should be noted from (88) that

the V with n > 1 contadn positive powers of r and log r, and therefore become in-
n —

finite as r does. This shows that the asymptotic expansion (89) for a general
cylindrical wave is not uniform in r. These observations will be related to corres-
ponding results for the asymptotic expansions of arbitr^ly shaped waves in Section
7.



- 31 -



5. Spherical Waves

V/e will now consider waves whose wave fronts are concentric spheres j
the rays are then lines emanating from a point. The Gaussian curvature of the
wavefront r = constant ic G(r) « r and the phase Y is given by ? = ± r + constant,
Again we may set the constant equal to zero without loss of generality. Now, using
r instead of s, we have from (6)

(92) v^(r,&,0) = -2—

The factor g (©,f!) « r (6,9')v [r (6,S2!),6,j2]] is constant on the ray 6 * const.,
^ = const.

Using (92) in (6) we find, by induction, that v (r,6,f!) can be written



in the form

n

(93) V (r,&,^) - r f.„(e,i2)r

j^o '^



-J-1



Now inserting (93) into (6) yields the following recursive formulas for f . (©,^);

(9U) ^jn = * ?J IJ^^-^) ^ ^^ Vl,n-1' J^^' "i^»

(95) fon = rj9,i2)v^[rjQ,?),©,^] -^ r^^f n > 1.

j=l

2
In equations (9U) B is the Beltrami operator B * . ' Ug sin© -^ + ■ . — «•

and the upper sign is to be used if T = +r while the lower sign is to be used
if Y = -r. The function f is defined by f ' g (9,^), and v (r ,©,;?) is the
value of V at the point v on the ray © = const., ^ = const. Using (93) in (2)
we obtain as the asymptotic expansion of a spherical wave

tikr 00 n

n=-o j=o 'J



(96) u ^



- 32 -



If the V [r^(©,{2),©,^3 ^® given, then the f. (&,^) can be computed
successively from (9U) and (95), and then the expansion of u is given by (96).
For example, if the expansion of u is given on a surface r - r (O,^) then all
the v are given on that surface, and therefore the expansion everywhere is
determined. We will now consider certain applications of these results.


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Online LibraryJoseph Bishop KellerAsymptotic solution of some diffraction problems → online text (page 2 of 5)