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time periodic wave. He obtained a series solution of the problem, the successive
terms of which contain the discontinuities in the successive time derivatives of
the reflected pulse. Consequently as Luneburg l- -l and Kline'- •• have shown, the
terms in his solution aire related to corresponding terms in the asymptotic ex-
pansion with respect to k of the solution of the time periodic problem. There-

- 52-

fore our recursion formulas are similar to his. In fact the variables ^ and
>» J which simplify the calculations considerably, were introduced by him.

Example 18, A Point Source at the Focus of a Paraboloid of Revolution (^ - O)
We now reconsider the preceding problem but with the bo-andary condi-
tion -^ « on the paraboloid. Proceeding as before from (152) we again obtain
(153)» (l5U) and (157), but (155) and (l58) occur with the minus signs replaced
by plus signs. Instead of (156) we find

(171) v^ = -(|I)-^ -^ ' P"^[(>l-P)D>^- >?dJv^_^, at ? = 0.

n '3v'

The second expression on the right in (171) results when the coordinates ^ and
» are introduced. The expression (162) for v (4, )j ) must be modified by the
addition on the right side of the right member of (171). Then we again obtain
(l6Ii) for the v (C*"*^) where the ay satisfy exactly the same recursion
formulas as in the previous example, except that

(^■^2) a^^^ - + 1 .


The asymptotic expansion of u(^, >» ) is given by (169) with the appropriate
a t . The first few terms in this expansion are
ikR ik(2p+x)

ikR ik(2p+x) 1 r £ Evl

(173) u(4,^ )^^ V * —T- ^ - ife r ^ * 5]

7. Arbitrary Waves

In Section 2 we showed how to obtain the asymptotic expansion of a
solution of (1) in the form (2). The construction of such, an asymptotic ex-
pansion first requires the determination of a phase function T satisfying
the eiconal equation (3) and then the determination of the associated system

-53 -

of rays. These rays are the orthogonal trajectories of the wavefronts (i»e«»
the surfaces T * constant) and they are straight lines. Alternatively we may
begin with a normal congruence of rays and then construct the associated phase
function T, Once the rays are knovm, the coefficients v can be computed
successively using (6), merely by integration along the rays. In the subsequent
sections this construction was carried out for cylindrical, general cylindrical,
general spherical and general plane waves, in which the wavefronts are fainilies
of concentric circular cylinders for the first two cases, and for the other two
cases, concentric spheres and parallel planes respectively. Now we will show
how to carry out the construction for an arbitrary wave, i.e., one in irtiich one
of the wavefronts is an arbitrary surface.

Let us begin by selecting an arbitrary surface Q as the wavefront T « 0.
We introduce the surface coordinates x^, x, on Q by means of the two fsunilies
of lines of curvature of Q, On each line of the first family^ Xp has a constant
value and x^ denotes arclength along the line from a fixed line of the second
family. Similarly on each line of the second family^ x_ has a constant value
and x_ denotes arclength along the line from a fixed line of the first family.
Now we introduce the normals to Q and the distance s from Q along a normal,
measured positively on one side and negatively on the other. Then s, x^, x^
are three orthogonal coordinates \diich uniquely locate a point, although a
single point may have more than one set of coordinates if it lies on more
than one normal to Q,

We can now construct a solution Y of (3) which is equal to zero on Q.
There are exactly two such solutions, namely T ■ ± s, and we will consider
the solution T » + s. The wavefronts T = constant are the siirface Q and
the surfaces parallel to it, while the rays are the normals to Q, on each
of which Xp and x- are constant.


in order to apply (6) for the detennination of the v , we must

calculate the Laplacian operator in cur orthogonal coordinate system. To

this end we introduce the principal radii of curvature p^Cx^jX,) and po(x2,x,)

of the surface Q at the point Xp, x, , In terms of these radii the metric

coefficients in this coordinate system are

p^{xr,,x^)+s p^(x2,x^)+s

(17U) h^ - 1, h^(s,X2,X3) - p^(^^^^^) , h3(s,X2,X3) - p^^^^^^^j .

Then we have

(175) y2 . iL ^ _!_ 9. . J^ 1-

>2+s'^ 3? P3CP3+S) BXg dXg p^Tpp^ axg 8X2

^p.+s'' ^TT pITp~fiT 3x, 3x- pTTpr+F) 3x_ 3x. '

3x^ ^2 ^^2^

.3 - 3 ^y^y

3 ^3

Furthermore the Gaussian curvature of the surface T - s - constant is
(176) G(3,X2,X3) - (p^,3J(p^.3) •

Thus (6) becomes

(177) v^(s,X2,X3)- v^(s^[x2,X3],X2,X3)




♦s) ' (p,+s) ' ys


In (177), v (s [x-,x ^jX^jX-) denotes the value of v at the point s^ on the
ray x- ■ constant, x_ =• constant. For example, if s - 0, then ■''n(^o**2**3^
denotes the value of v at the point x^, x- on Q,

- 55-

When n = the integral in (177) is absent, so the first term alone
yields v . Its dependence upon s is particularly siiiiple, and we note that as
s becomes large, v decreases like s" provided that both po and p^ are finite.
If either Pp or p- is infinite while the other is finite, v decreases like
|s|~ ' for large s, and if both are infinite v is independent of s. In
addition we note that v becomes infinite at the two points s ■ ~Pp* s » -p,
on each ray, or at one or no points if either or both radii are infinite.

The locus of points at which v becomes infinite is the caustic
surface of the ray system, and it has two branches which touch on rays for
which Pp ■ p.» From the fact that v becomes infinite on the caustic, we
conclude that the asymptotic expansion (2) is not valid on the caustic.
This is indeed the case, for it has been shown by I, Kay and J. B. Keller"- -•
that on a caustic u is asymptotic to a positive fractional power of k. The
expansion of u on and near the caustic can be found from the expansion (2)
by the method of [l6] .

We have thus seen that the behavior of v on the ray through the
point Xp,x_ on Q depends solely on the radii of curvature of Q at x^jX-, If
the point is elliptic, there are two caustic points on the rsy at the two
centers of principal curvature of Q, -vdiich lie on the same side of Q, and
V decreases as in a spherical wave for large s. If the point is umbilical,

the two caustic points coincide, while if it is hyperbolic they lie on opposite

sides of Q, If the point is cylindrical there is only one caustic point on the

ray and v varies as in a cylindrical wave for all s. If the point is planar

there are no caustic points on the ray and v is constant along it, just as in

a plane wave. We will now see that the nature of the point Xp, x- on Q also

determines the behavior of the other v •



By examining the expression for V in (175) and the form of the second

term in (177), we see that for large s this term is 0(s~ p.+v ^ll provided

- fc -


p- and p- are finite. This follows because in this case V reduces the order

of V ^ with regard to s by two, while the factors in the integrand of (177)
and the integration increase the order by two, and the factors outside the
integral reduce the order by one. Thus all the v tend to zero as s becomes
infinite. In the same way we see that the second terra in (177) is 0(sv ^)

if p„ is infinite and


n-1 s 3 n-1

3^r- - p^TpT?! ^ ax^ *

or if P- is infinite and

-^ - pj(^-riT 3x3 3x3

or if p_ and p^ are both infinite and

g2 p^ip^Tiy 3x2 3x2 "*■ g^2 ~ pj(p~ri7 3x3 3x3 *

Thus in any one of these cases v decreases more s]owly than v , as s increases,
^ n ' n-1

and in general v will in fact increase, becoming infinite with s. Furthermore

the greater n is, the more rapidly will v become infinite.

We thus find that on a raj- through a planar or cylindrical point of Q

the V for n > 1 will in general become infinite as s does, grovdng like
n —

s"" (or perhaps with some factors s replaced by log s) for a cylindrical
point and like s for a planar point. From this result we conclude that in
general on rays through planar or cylindrical points of Q the asymptotic ex-
pansion (2) is non-uniform. These results have been exemplified by equations
(89) and (126) for general cylindrical aid general plane waves. For ordinary
cylindrical or plane waves, on the other hand, although one or two radii are
infinite, the other quantities appearing in the conditions above are zero.


In such cases the second term in (177) is 0(s" v ,) and the v all

n-1 n

tend to zero as s becomes infinite.

By means of the preceding results we can treat the problem of re-
flection of any incident field fi*om a surface S of arbitrary shape. Let us
suppose that on S either u « or -r- = 0. Vfe begin by expand?jig the gi'^ren
incident field asymptotically and determining the associated ray system.
Then we determine the reflected ray system produced by reflection of the
incident rays from S, Next ve write the field at any point P as a sum of
terms, one term for each ray through the point P. Each of these terms is
an asymptotic expansion of the type treated in Section 2. The phase Y for
each term is determined fror- the phase appropriate to the incident ray at
the reflection point and the length of the reflected ray to the point P.
The value of each v on S is obtained from the boundary condition in which
the asymptotic expansion of the incident wave appears. The value of v off
S is computed from (177) as indicated above.

The method just described is the one which has been emplojred in
the preceding sections. It has been applied to the reflection (and trans-
mission) of a spherical wave at an arbitrary surface by J, B, Keller and

H, B. Keller'- J and to the reflection (and transmission) of an arbitrary

wave at an arbitrary surface by J, B, Keller and S, Preiser I- K However,
in both of these cases only v , the leading tern in the reflected (and trans-
mitted) field, was determined.

- 58 -

In the next two sections we will apply the considerations of this section
to two general classes of waves. In Section 8 we consider two dimensional or
cylindrical waves, i.e., waves independent of one Cartesian coordinate. The
result enables us to treat reflection of any two dimensional or cylindrical
wave from a two dimensional or cylindrical reflector. As an example we treat
reflection of a plane wave from a circular cylinder. In Section 9 we consider
three dimensional waves possessing an axis of symmetry. In this case the wave-
fronts are surfaces of revolution. This result enables us to treat reflection
from any surface of revolution. Of course the incident wave must also be sym-
metric about the same axis as the body. The result is exemplified by the treat-
ment of reflection of a plane wave from a sphere.

8. Two dimensional waves

Let us consider a two dimensional wave, i.e., one which is independent
of the Cartesian coordinate z. Then the wavefronts 1 » constant are cylinders
perpendicular to the xy plane. The intersection of a wavefront with this plane
is a curve which we will also call a wavefront. The family of wavefronts (in
the plane) is a family of parallel curves, the common normals to which are the
rays. These rays will in general possess an envelope which is called the caustic
of the family of wavefronts (or of each wavefront). We introduce the distance
s measured along a ray from the caustic as one coordinate and the angle p between
a ray and the positive x-axis as the other coordinate (see Figure 9a). Although
these coordinates are not orthogonal they seem to be more appropriate than the
coordinates of the preceding section.

We suppose that the equation of the caustic is given in terms of the para-
meter p by the equations x ■ 4(p)j y ■ ^(P)» Then for any point x, y we have
the transformation equations



Caustic coordinates. A point (x,y) is represented by the caustic coordinates (s,p).
s is the distance from the point to the caustic, measured along the ray through the
point. The angle p is the angle between the ray and the positive x-ax±s.



Figo 9b

A ray incident on the reflector at angle © yields a reflected ray with angle of re-
flection Q, The reflected ray makes the angle p with the positive x-axis. The
normal to the reflector at the point of reflection makes the angle y with fhe nega-
tive X-axis. The point of reflection is at distance s from the caustic, measured
along the reflected ray.

- 60 -

(178) X - C(P) + 8 cos p, y - v^(p) + s sin p .

By making use of (178) we find for the Laplacian operator in the new coordinates


(160) a - C + Vj , b ■ I cos p + v^ sin p, c « ^* cos p + >^ sin p •

In (IBO) the dot denotes differentiation with respect to p. The quantities a, b
and c have the following interpretation. If s(p) denotes the distance along the
ray p from the caustic to a wave front, then

— 2 2 — —

(181) a - (s ) - b , b - - 8 , - - s - b ,

Since ¥(p) also denotes the radius of curvature of the wavefront, we see that
"s(p) is constant if and only if the wavefront is a circle. Thus for a family
of circular wavefronts a«=b = c»0»

In terms of the present coordinates (6) takes the form


(182) ,„(.,?) . V.,,P)L^]'/' - ^ J r^/Vv„.,(r,p)dr .


One may show by induction, using (179) and (182), that the v are given by


(183) v^(s,p) - YL fj„(p)s-^/^-J .

The coefficient f . are given recursively by

^Jn 'hV^' l^'^J-l,n-l* Vl,n.l* ^^^ J"^) V2,n-1* ^^J" |)V2,n-l

.a(j-|)(j.|)fj.3^ A J/0

(^8U) ^on • V«o>P)^^' - £ ^in %^ -^'

t - V (s ,p)s^/^
oo o o'^ o

- 61 -

In (18U) we use the convention f , ■ if j < or j > 3n.

These results, (183) and (IBU), given the coefficients in the expansion of
any two dimensional wave. We may use these resiolts to treat reflection of an
arbitrary cylindrical wave from an arbitrary cylindrical reflector. As in the
previous problems, we may choose s « s (p) to be the equation of the reflector
surface (see Figure 9b), Then the v (s ,p) are determined by the incident field
and the boundary conditions on the reflector, so all the v can be computed.

In the application to reflection the equation of the reflector is usually
given in the form y ■ g(x) and the phase of the incident wave in the form
T ■ Y(x,y), It is therefore convenient to express a, b, and c in terms of these
given functions . To this end it is convenient to introduce the angle of incidence
given by

T^[x,g(x)] g' - T [x,g(x)]

(185) cos 6

X*- '° -• ° y'

It is also useful to define the angle y between the normal to the reflector and
the X-axis. Then the angle p between the reflected ray and the x-axis is

(186) p - n - © - Y .

Now if p denotes the radius of curvature of the reflector we find that s , the
distance along the reflected ray from the reflector to the caustic is

(187) s^ - I cos .

The radius of curvature p and the derivative of y are easily seen to be given by

(188) p- ji + (g')^^/^ k-r^

(189) Y' - - g" |l + (g')^"^ .

Since the equation of a reflected wavefront is T + s - s^ - constant, we have
from (181)

- 62 -



Fig. 10

Reflection of a plane wave incident from the left on a circular cylinder
or on a sphere of radius b . In the first case, the rectangvilar coordinates
(x,7) of a point are to be°related to (s,p) and in the latter case, the cy*-
lindrical coordinates r, z are to be related to s,p»

- 63 -

Id if D-*(g'>^[Yg' -M

(190) b - Tp-s^p - ^ ^ (T-s^) . ^.^ J ^^ ^ - TI

dx (^

a . b^, c - - -A_.b«.


It shoiild be noted that if the parameter x is used instead of p in (l8U) then
the p derivatives must be converted to x-derivatives.

As an example of (190), consider an incident plane wave for irtiich T ■ x.
Then (190) becomes

^OT S TToTT - *> * «-u, « - - 2©T

(191) b-^J J;t-xI. a-b^, c - ^b'.

In case the equation is given pararaetrically as x(t), y(t), then (191) applies
with the x-derivatives replaced by t deidvatives. If t denotes arclength then
©. is just the curvature of the reflector.

Example 19 t Diffraction of a plane wave by a circular cylinder (u = 0)

Consider the plane wave e incident from the left on a circular cylinder
whose equation is (see Figure 10)


- b cos Q( ■ - b sin 6/2
'^ o '^'

b sin (^ ■ b cos p/2.

We will calculate the asymptotic expansion of the reflected field. The field
associated with the diffracted rays will not be computed since its order is
higher than that of the complete asymptotic expansion of the reflected field©

Upon applying the boundary condition u » to the incident plus reflected
fields we find
(193) T(Sq,P) - X - - b^sin p/2

(19U) ^o^^o'P^ " '^* ^n^^o'^^ " ° f or n > 1 ,

- 6U -

From (193) and (18?), one finds that the phase of the reflected field is given by

(195) Y(6,p) - T(s^,p) + 8-s^ - s - I b^sin p/2.

To use the recursion relations (18U), we must express a, b and c in terms
of p. From (191) and (192) we find

(196) a • I^ ^o °°^^ P/^' ^ ' -I ^o^^ ^ " - E ^0^=°^ P/^

c - I b^cos ?l - I t^sin p/2.

Using these coefficients in (l8U) we find, and verify by induction, that the
f have the form

(197) fj„ . 2-(^*J*3/2> bJ-"*l^^ t »jtn(=^ P/2)^'"'"* ^ •

Here the numerical coefficients a., satisfy the following recursion relations


for J ^

(198) a^^^ - j"^J(2j+Ut+2n-3)(6j-Ut-2n-l) aj_i^t,n-l

+ (2j-Ut-2n+5)(2j-Ut-2n+3) ^ ^.^^^ ^^^.^ ^j,,^

* [2U(J-l)(J-2t-n)-6]a._2^^^^.1 * 12(l-j)(2j-Ut-2n.3)aj_2,t-l,n-l
. 9(2J-5)(1.2J)aj_3^^^„., * 9(2J.5)(2J-l)aj,3^^_,^„_,l

For j = we have

(199) a^tn " " ^ ^jtn ' ^oo " -2'

The complete asymptotic expansion of the incident and reflected fields is
thus given by

- 65 -

(200) u '^e


. i/5sin

2 ' 2s


ik(s- K- isin §) oo 3n n

' ^ r E r

n=o j«=o t»o

«dtn^l6ib^ksin |)-"(^)J(sin £)>^*.

For the first few terms, we obtain

ikx / bo ■ i ikCs-Ksinf)] ^ r 8

u^e -/^sxnfe Ji._gj^^_^




( -Vl - 3) - (^)' ( - ^ - 9 sin |)
sin ^ sin ^

+ (2|)^(15 sin^ I - 15)

These results can be used to compute the amplitude of the reflected far field,

For a point in the far field (see Figure ID), p ^^^ a, s -^R, and T -^ R - 2b^ sin ^ .

From Eq. (200 ) we see that the reflected field has the form

ik(R-2b^sin |)




A -^ f

^01 ^02 /

oo ik

^ . a

y Sin ^


1 +




• • • •

c^V^ sm f k/^in^


^XV 1 +



■ + • • • •


o . a
•5- sin J

y^ Sin f (b„.)2 2(^ Sin |)(b„.)^

From (198) and (199) we finally obtain

- 66 -

y -J- sin ^ ^

The asymptotic form of (201), for the far field (s~R »b ) coincides with
the result of I. Imai {2l{\ obtained by expansion of the explicit solution. Since
he does not give the term in k~ his result cannot be used to compute (a| given by
(205) and shown in Figs. 11 and 12.

Example 20 1 Diffraction of a plane wave by a circular cylinder {-rr - 0)

Let us reconsider the previous problem, replacing the boundary condition
u - by ~ - 0. Then instead of (193) and (19U) we find


(206) %(8o'P^ " ■"■' ^(s^»P) " * = -b^sin |

(^^) \(%*P^ - ^3?) a? 7-71 -^p

2 + COS I 3v^ i^^o*^^

-7— Tj- • -^^ii .">!•

2 sin *

The phase of the reflected field is again given by (195). From (I8U) and

(196) we again find that the f . are given by (197) with a for 3^0 satisfying

jn jx>n

(198). However for j = we have instead

(208) a^,„ - - r^ ^a^^^ . I6(2t*n-l)aj.^^^^„_, . l6(U-2t-n.2j)aj.,^^_,^„.,

^000- 2.

The complete asymptotic expansion of the incident and reflected field la
again given by (200) with the appropriate a., and for the first few terras,
we obtain

-67 .

h A













Fig. 11

The back scattering aii5)litvide A divided by the geometrical optics value ■>/b /2 ' as
a function of kb » A denotes the absolute value of the amplitude of the field reflected
frcan a circiilar cylinder of radius b when a plane wave is incident. It is given by
(210) with o = Ji. °

- 68 -

., /b ~ ik(E- -j^ Bin |)| ^

(209) u ^ e^''^ / ^ sin I e ^1 * j^^;^

8 *_2.

sin ^ sin

. ^ (3- -4^).(^)^(9 »in I - ^).(^)'(«-15 .in^ |)

2« sii^ r ^=' - =in

+ .

For the amplitude of the reflected far field we find instead of (205)

/oinN 1A| 1 1 f 3,1x17 6,6U2 ^ 3,OU9 ^ .

(16b k) sin J sm ^ sin ^


o . a
^sin J

The asymptotic form of (209) for the far field coincides with the result

of I, Imai [2U] obtained by expansion of the explicit solution. As in the

previous case, he does not ?ive the term in k and therefore his result cannot

be used to compute |A| given by (210) and shown in Figs. 11 and 12.

9. Axially symmetric waves

To find the asymptotic expans'on of an axially symmetric field we introduce
the coordinates s, p and (fl , Here ^ is the ordinary rotational angle of cylindrical
coordinates and s, p are defined as above In the planes ^ = const. If the caustic
surface is given by the equations z - 4(p)j r " ^(p), then the cylindrical coor-
dinates z and r are related to s and p by

(211) s cos p = z - ^ } s sin p « r - vj •

These relations are similar to the previous relations between x, y and s, p.
Therefore when the Laplacian in cylindrical coordinates is converted into s, p,
^ coordinates, and (jf derivatives are omitted, one obtains (179) with the addi-
tional terms

(^.r)) I sin p i li_ ♦ , cos P . ^

^ ' "^ s sin p+>| ' s(,s sin p+ vj ) / 3s e(s sin P+ "^ ) 3^ '

-6? ^










C -H

^< O H


flJ iH ft

13 -P P.

and a







U (U



o cd (0

^ o o

»T} ft

ed b
al o



^H o o

(D en -H

-P Jh

-P o; -P

05 ^ 0)

O Eh e



XJ • W)

iH a


0) m
•H © ^


«H rH -P





»H H

O 0)


•d -P O •

I-) CO V( CM

ft O - ^


m o


,P O Vi !3


- 70 -

Since the wavefronts of axiallj symmetric fields are surfaces of revolution,
the principal curvatures at any point are given by s and esc p(s sin p+ >^ ) . Thus
Eq, (6) assumes the form

(s (s sin p+ y^) ll/2


Fop n • 0, 1, 2 we find

(21U) v^(s,p) - < s(s sin p+>/ )

-1/2 f3n _j n

r V "Ti 'tn^' sinP*^)
j=o '' t=«l


Here f ■ v (s ,3) Vs (s sin P+ ^ ) > and the f . , g. , satisfy the following
recursion formxilas with a, b, c given by (I8O) (or by (I8I) or (190))

(215) for J / 0; fj^ - ^j(j- ^)'f^_i,„.i-fV.i,,.i> c(j- |)fj.2,„.i


3n-3 (-l)^-J-^C^-j)(sinp)^-^-V
/^*1 h^^^^

for j - 3, add

^a ^

" - ^,n-l

for j

n-1 I eg. T+Ube , Stasinflg. ^
2, add r J H.n-1 ^,n-l ^Hn-l

hi 1 8^^ 16 r, ^*1

- 71 -

for j


. 2,

^tn ■ 2t i


16 v^

5a csin£v3/^"^*' n _1 (3p-^t)sinp

- ^4.^ (P^-sp . ilzfs^^ g^^^_^


2pcosp (p-t)(p-t-t-l)bsin p

p-t+1 ' p-t-t-2


3n-3 (-l)^(sinp)^"^f . ^ ^
t = 1, add 5~ 3 iili=i

. ^ sinp(p-k)

p,n-l p-t+1 ^P,n-1

j-o 8 >|


(216) for J = 0, f - - Z. f C - r gan ( V^^M )"'^ •

on *— , pn •'— T "=qn o
p=l ^ q«l ^

For n > 2, V is given by (21U) plus additional terms in log s, and

f. , e^ still satisfy (215) and (216), If the logarithmic terms cancel
jn' Hn

then (21U) gives v completely.

Example 21 ; HLf fraction of a plane wave by a sphere (u » O)

Suppose that a plane wave e is incident from the left on a sphere given



(see Figure 10), The caustic surface of the reflected wavef rents is given para-
metric ally by

z « -b sin p/2 ,

r « b^coa p/2

(see (192))


z • 4(0)


72 -

-^ sin I ^ 2 + cos p

b^cos^l .


Substitution of the incident plus reflected fields into the boundary-
condition u » leads to the results (193) - (195). If (213) - (217)) are
now used, the first few terms in the expansion of the fields are found to be

(219) u A/ e



b sin B
o ^

Us(s sin p+b cos £)

ik (s - |b^sin |)j

1 2 4

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