Joseph Bishop Keller.

# Asymptotic solution of some diffraction problems online

. (page 3 of 5)
Online LibraryJoseph Bishop KellerAsymptotic solution of some diffraction problems → online text (page 3 of 5)
Font size Example 8 . Diffraction of a Plane Wave by a Cone of Arbitrary Cross-Section

In the diffraction of a plane wave by a cone, the asymptotic expansion

contains a spherical wave centered at the vertex, in addition to other waves.

The preceding considerations apply to this spherical wave . Since the problem

contains no characteristic dimension, k and r can occur only in the combination

kr. To achieve this dependence in the asymptotic expansion (2) we first multiply

(2) by k" , which we may do since (2) is linear and homcgeneous. Then since the

coefficient of v is k~ ' , all f. must be zero except f , as we see from (93 )Â»
n in nn

Thus we have, choosing T Â«Â» +r in order to obtain an outgoing wave,

(97) ^nn^"*'^*^^ " ^nn^^**^^""""""^ â€˘

The f can be found successively from f - gA^}9) by means of (9U).
From this equation we obtain

(98) f (e,5') - ^ ^ TT |j(J-l)*B|y gâ€ž(Â©,?l), n>l.

'^^^''^^'^i {ti li^^-'^^4^o(^Â»^^>

Collecting our results, we have
ikr

(99) u ^ Â®

"TT

1 +

f ^ â€ž -Ifr rj(j-i)-Bl

h?l (2ikr) ni j-1 ^ J I

go(e,?).

Here as in the wedge problem of Example 1, the present method does not yield

g (0Â»5')>but once g is found in some other way, it yields the complete expansion

of the spherical wave u.

- 33 -

Example 9 Â« Spherical Bessel Functions

Spherical wave solutions of (1) are given by the product of (kr)~ '

with a spherical Bessel function H /, ,â€žx(kr) and a spherical harmonic Y (0,0)

m+(,Â±/-.(Â».i)]} .

This expression for the spherical Bessel function agrees exactly with the

nâ€”T P 1 /P

known representation if A â–  i (-) ' (see , p. 78, eq, 3). If we

had chosen T = -r we would have obtained the expansion of the incoming spherical

Bessel function \+(^i/2)^^^^ * ^^^^ expansions differs from the above merely in
having -i everywhere in place of i. (See [2o] * P* 78, eq. U).

Example 10 . Diffraction of a Plane Wave by a Paraboloid of Revolution (u - )

^ikx
Let us consider a plane wave e incident from the right along the

axis of a paraboloid of revolution, and on the outside of it (see Fig. 1) Â» This

problem is similar to Example 3Â» except that the reflected wave is now spherical

instead of cylindrical. If we let r denote distance from the focus, we may write

the equation of the paraboloid in the form (16), Again assiuning u â–  on the

- 3U -

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-35-

boundary we obtain (17) and from it (18) - (20). The phase T is again given
by (21), but instead of (22) we find from (92) that

(102) v^(r,Â©) - -pr'-'-sec^ | .

We see that v is independent of flf and therefore so are all the other v , as
o "

was to be expected from the synunetry of the problem.

Now using (102) and the boundary conditions (20) in the recursion
formulas (9U) and (95) for the f . (9), we find - and prove by induction - that
f . (Â©) is given by

(103) fj^(Â©) - a^^v^-^'^^isec^ |)J*1 .
The a. are given by the recursion formulas

(lOU)

%- ?^j-l,n-l'

J > 1Â» n> 1.

(105)
(106)

n

^on " - ^ ^jn '

00

Â°>1.

Inserting equations (103) for the f . (Â©) into (96), we obtain as the
asymptotic expansion of u

/â– ta'7\ -ilex ik(r-2p) ^ ... .-n ^ / -1 2 Â©nJ+1

(107) u 'vj e + e '^ H (^^p) 11 ^in^P^ ^^^ ?^

nÂ«o j-o ''

By computing the first few a. from (lOU) - (106) we obtain for the first few
terms of (107)

2 e

,,^ftv -ikx ik(r-2p) -1 2 9 17 1 ,, ^ ^Â®Â° ?, 1

(108) u 'v' e - e ' ^'r psec ^ 1- ^jjâ€” (1 ) â™¦ â€˘â€˘â€˘

This problem has been solved exactly by H, Lamb'- J and the asymptotic
expansion of his solution is precisely (107). The amplitude and phase of the
scattered wave given by his exact solution are compared with those given by (107)
in Figures 6 and 7Â«

- 36 -

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(75') uv e^'' ' r (ik')""w + e^*" '^ 7" (ik-)-"t , y < .

*â€” _ n ^- n '

Proceeding as in the previous example, we find that v , w , Y, Y'

and V- are given by the formi'las obtained above and that Of and s are also
1 "^ o

given by the same formulas as Y and v . The value of p is complex for the
totally reflected rays, so the evanescent wave decays exponentially. The
value of w (x,0) for this wave is also given by (65'), No other terms in
the totally reflected and evanescent waves will be considered.

Now we consider the diffracted wave. Proceeding as in the previous
example we insert the expressions for the diffracted and totally reflected
wavea into the boundary conditions. Upon equating coefficients of powers of

-U3 -

k we obtain (78) and thus 9r(x,y) is given by (81), Thus the diffracted wave-
fronts are cones with the y-axis as axis. In the x-y plane the section of
the diffracted wavefront consists of two straight line segments, one of which
is shown in Figure \$,

From the other powers of k we find at y = 0, noting that T'(x,0) - 0,
-n

(79.) ^n " ^ ''n^

3s - 9w
n+1 . , -n n

(80.) cosa^s^^^ * -^ . * bix " ^ .

Sinca w - at y = 0,(79') shows that s =OatyÂ«0. But then s vanishes
all along each diffracted ray, so we have

(82') s^(x,y) .-0.

Then from (6) ve find for s^

(85Â») s^(x,y) - s^(x-y tana^,0)x" ' (x-y tana^) ' .

Finally after computing s-(x,0) from (80') we have

(86') s-(x,y) Â« ^ x"^/^t-(y+h)tana1"^/^, (^-(y+h)tanaj] > 0.

a cos ct Â«- J '_ J
o

The leading term in the diffracted wave, e ^(ik)~ s^,was discussed

ri7i

in connection with Exsimple 7> and was also discussed by E. Ger juoy â€˘- -â€˘ and by
K. 0. Friedrichs and J. B. Keller'- -â€˘, who also obtained it. We will therefore
only remark that none of these authors had obtained v- , which is of the same
order in k as s- . For large x, s, behaves like x which is a more rapid decay
than that of either the incident or the reflected wave . However as x increases

these two waves tend to cancel each other, and their suia also ultimately behaves

_2
like X â€˘ Therefore the diffracted wave may be comparable to this sum at large

horizontal distances from the source.

6, General Plane Vavee

By a genereil plane wave we mean a wave in which the rays are parsLllel
straight lines and the wavefronts therefore parallel plaines, but in which the
amplitude is not necessarily constant on each plane. For such waves the curva-
ture G " 0. We will assume that the rays are parallel to the x-axis and use x
instead of s to denote arclen^h along them. For such waves equation (3) yields
T â–  Â±x + const, and as before the constant may be taken as zero without loss of
generality. Then V Y â–  0, so the solution of (U) is instead of (6)

X

(1?3) v^(x,y,z) - v^(x^,y,z) - j- / ^^-i^^) n > .

o

From (123) we see that v is independent of x. Thus we have

(12U) v^(x,y,2) - g^Cyjz).

Now from (123), (12U) we find by induction that vâ€ž has the form

n

(125) V (x,y,z) - J3 f (y,z)xJ.

J"0

Inserting (125) into (123) we obtain the following recursion formulas for the
f^ (y.z) with n > 1'.

(126)

fjâ€ž - - ? [f"'A ^j-l,n-.l^(J*l)^J.l,n-l]> 1 < J 5 n.2

1 .-1

(127) ^Jn â–  - ? J A f j_i,n-l > ^ " "-^* ">

n

(126) f - V (x^,y,z) - f" f, x^ , n > 1 .
^ ' on no'-"' ^, jn o ' â€”

In these equations ^ denotes the two-dimensional Laplacian ^ â–  â€” Â«â€˘ â™¦ â€” Â«â€˘ ,

dy dz

^oo " ^oÂ»^"^ "^n^^o*^*'^^ ^Â® ^^ value of v^ on the surface x - x (v,z). Using
the equation (125) for v in (2) we obtain as the asymptotic expansion of a

-U5 -

general plane wave

(129) u ~ e*^^ ÂŁ (ik)-" r f.â€ž(y,z)xJ.

n=o 5=0 *^

We will now consider some applications of these results,

Fxample 15 â€˘ A Line Source at the Focus of a Parabolic Cylinder (u = 0)

Consider a line source at the focus of a parabolic cylinder (see
Figure 8). The line source creates a cylindrical wave given by (II4) or asympto-
tically by (15), with m â– = 0. From the focussing property of the parabola we con-
clude that this wave will be reflected as a plane wave traveling along the axis
of the parabola. Thus this wave will have an asymptotic expansion of the form

(129) with the f . (y) satisfying (126), (127) and (128).

To derive these facts in detail let the equation of the parabola be

2

(130) X Â» x^(y) Â« g^- p.

The line source is at the focus of the parabola (0,0), and, we denote distance
from it by r. We assume that the solution u has the expansion

ikr 00 ., _ 00 /-, /n\

(131) u~2_ 21 P (ikr)-^ + e^^^ ^ (ik)-"-(l/2)

/ 1 1 ^â€” n *â€” n

/ikr nÂ«o nÂ«o

On the parabola we assume that u = 0, and applying this condition to (131) yields

(132) T - r, on X = x^(y)

(133) V . -p r-"-^^/^\ onx-xjy).

n n ' o'

The solution of (3) for T which is outgoing and satisfies (132) can
be found, by ray considerations, to be
(13U) T = X + 2p.

Thus the reflected wave is a general plane wave and v is of the form (12U),
Maicing use of the value of v^ on x - x^(y) given by (133) we have

- ii6 -

Figure 8

Reflection from a parabolic cylinder of the cylindrical wave
emanating from a line source at the focal line. The focal line is
at (0,0), the incident rays are radially directed from it and the
reflected rays are parallel to the x-axis. The figure also repre-
sents the reflection from a paraboloid of revolution of a spherical
wave emanating from a point source at the focus.

-U7-

(135) V (x,y) - -p^ ( -^ )^/^ .

Up +y

We note that v is independent of 2, and hence so are all the other â–Ľ , as
was to be expected.

In determining the v (x,y) for n > 1 it is convenient to introduce two
new independent variables ^ and >j instead of x and y. These variables are de-
fined by

2

(136) 4 - X + p - ^ ,

2

(137) >^ - p + ^ .

In terms of these variables, we find

(138) r^ - (C+>|)^ - Up^ .

Equation (123) for the v now becomes, when (133) is used,

(m) vâ€ž(5, r^ ) . .p_^^-i:-(v2)3. i.jrr .^,,^2. j^^j,^)^ ^^1^ 1 ^^. 1 ^^^^_^^^ ^

From (135) we find that v^ is given by

(lUO) ^o^^*''^ Â° '"^'^^^'

Now using (ihO) in (139) we find that v^ is given by

(lui) v(^,>^)-f i:%,n.i^"'n

Z"0 mÂ»o '

The constants a, are determined by the recursion formulas
xjnn

(11x2) a =0, m y n,

^^ ' omn ' ' *

(1U3) a Â« -P ,

^ ^â– ''' onn n '

(^^) ^im,n.l- - ^ ^i.l,m,n*(^â€” ?)^^^"- l^^^ ^'^ Vl,-l,n

"^ " / -(Z+m-(l/2)] -n+m

-1Â»8 -

We finally have from (131), (13U) and (llj.) the following asyaptotic expansion

for u:

m-n ^

Ikr CO â€ž lk(2pÂ«) 00 i;n .j^ / .1 X*mÂ» j

^ V^lo^ n=o " Vi^p n-o X-o

mÂ»o

The first few terms in (i)i5) yield

ikr 00 ^ gikC^+x)

(ia6) u(C,>|) - ^ H Pn^i*^^^" -^

yTIo^ n-o y^cp

[, , ^ [jpn-^,'/=-35n-' V'^ Hn-^'V/^^ - }

Exainple 16 Â» A Line Source at the Focus of a Parabolic Cylinder (â€˘â€” - - )

If we replace the boundary condition u Â» of the previous problem by

the condition -5- â–  0, we obtain a different problem which can be treated in essen-
dV

tially the same way. The first difference between the two problems appears in
the expression for v on the boundary, which is given by the following equations

(1U7) v^ - + pQ^'^^^ â–  '?'^^^' """ "" = ""o^y^j

(1,,8) v^ . -(|^)-'[|?(Pn.ir-'/'-n-l) ^ Pn-"'/' l]' ^ ^ = ^o^^^ ^

vâ€ž - P-^EVP)B, -..,][v,r-^/\.,]^ P,.-&^^^/^2,on C - 0.

The result (liil) is again obtained for v (ÂŁ:, ^ ) and a. is again given by (lUU)
for X ^ 0, while for / Â» we have instead of (1U2), (1U3) the following:

- U9 -

(1U9) a^ - 1 - P ;
o,n+l,n+l n'

The expansion of u is given by (lii\$) with the appropriate a.s . The first

few terms in this expansion yield

ikr 00 ik(2p+x)

(151) u(.f,r,)^2 YL Pâ€ž(ilcr)-" + 5 .

iV^ n=o ^ >/ik^

Example 17 * A Point Source at the Focus of a Paraboloid of Revolution ( u Â» )
We will now determine the field produced by a point source located at
the focus of a paraboloid of revolution with u * on the paraboloid. As in the
two preceding examples, we expect the reflected field to be a general plane wave,
and this is indeed the case. The wave produced by a point source is spherical and

independent of 9 and ^, so we may describe it by (100 ) or (101) with ra Â« 0. We

â€”1 ikR
â€˘will use (101) with A = k,so the spherical wave becomes Re , which is the

customary form. Then we assume that

ikR .._ 00

(152) u /v Ijj- + e^^^ XI (ik)"\ â€˘

n=o

Here R denotes distance from the focus located at (p,0,0) on the x-axis, which
is the axis of the paraboloid. We write the equation of the paraboloid as

- 50 -

2
(153) x^ . x^(y,z) - ^ - P .

2 2 1/2
Here r â– = (x + y ) ' is radial distance from the x-axis.

Upon inserting (152) into the boxindary condition u Â« and equating

coefficients of powers of k, we obtain on the paraboloid

(I51i) T - R,

(155) v^ - -n'\

(156) V - 0, n > 1 .

As in the preceding problems, we find that Y is given by

(157) T = X + 2p.

From (155) and (l?li) v is found to be

(158) \" ( -^ )^^^ .

Up +r

From (158) and (125) it is clear that all the v will be independent of the

angular coordinate 6 in the y, z plane, as was to be expected.

To determine the v for n > 1 it is again convenient to replace x

n â€” '^

and y by the two new independent variables 5, >i defined by

(159) 4 - X . p - ^

2

*i = p ^ 5p â€˘

(160)

We then have

(161) R^ - U*>^f - UpC

E^iuation (123) for the v now becomes, when use is made of (l50).

n

(162) v^(4, ^ ) - - ^ ^ [(>^ -p)(D^-2D^D ) Md| * y D^] Vl^

From (158) we see that v is given by

- 51-

(163) ^o^^**^^ Â» , i^ -^ .

Now beginning with this value for v , and determining the v successively
from (16?) we find

JL^o m=o ^

The constants aÂ« are given by the formulas
ÂŁmn

(165) -000 - V

(166) a - 0. n > 1,
^ ' onn > â€” '

(167) a - = -(m+l)a +ma ^ -a. , 0

1 3 5

Online LibraryJoseph Bishop KellerAsymptotic solution of some diffraction problems → online text (page 3 of 5)