- Email: [email protected]
Scattering of acoustic, electromagnetic and elastic SH waves by two-dimensional obstacles
ANNALS
OF PHYSICS
Scattering
91,
l-39
(1975)
of Acoustic, Electromagnetic and Elastic 9-l Waves by Two-Dimensional Obstacles D. L. JAIN AND R. P. KANWAL Department
of Mathematics, University Park,
Pennsylvania Pennsylvania
State 16802
University,
Received September 30, 1974
We present a complete and a simplified discussion of the scattering of acoustic, electromagnetic and earthquake SH waves by various shapes such as infinite cylinders, strips, slits, cracks, cavities, and semiinfinite planes. Formulas are derived for the velocity potentials, electromagnetic fields, displacement fields, far-field amplitudes, scattering cross sections, stresses, and dynamic stress intensity factors.
1. INTRODUCTION Scattering of waves by two-dimensional configurations is of considerable interest in physical sciences. These configurations include infinite cylinders, strips, slits, cracks, cavities, and semiinfinite planes. Attention devoted to them is not entirely based on the physical existence of scatterers such as an infinite cylinder. It is based also on the fact that the solution for an infinite cylinder gives an insight into the solution for a finite cylinder. Our aim is to discuss three kinds of waves-acoustic, electromagnetic, and elastic SH wa.ves. We present the analysis for the scattering of acoustic waves by several perfectly soft and rigid bodies. They are an infinite strip, a slit, a circular cylinder, an elliptic cylinder, a parabolic cylinder and a semiinfinite plane. For the scattering of electromagnetic waves also we consider the same obstacles which are assumed to be perfectly conducting. In the case of elastic SH waves we consider an infinite rigid strip, a circular cylinder, an elliptic cylinder, a parabolic cylinder and a semiinfinite plane and corresponding cracks and cavities of these shapes. These rigid inclusions, cracks, and cavities are embedded in an infinite expanse of a homogeneous and an isotropic elastic medium. Solutions of these problems are known up to various degrees of generality. A survey article on the acoustic and electromagnetic part has been presented by Bouwlsamp [3]. Recently, formulas for the quantities of physical interest in the 1 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
2
JAIN AND KANWAL
case of acoustic and electromagnetic waves have been compiled in [4], where the references to the original papers can be found. The corresponding analysis for the elastic SHwaves is also available to some extent [l, 2, 5-7, 12-15, 17, 19,21-23, 251. However, there are many aspects of the solutions that can be simplified, improved and generalized. For instance, in the case of scattering of elastic SH waves by a rigid elliptic cylinder, or an elliptic cylindrical cavity the solutions are presented in terms of Mathieu functions and it is very cumbersome to derive approximate and asymptotic solutions from this analysis. Moreover, by the method of wave-function expansion it is extremely difficult to deduce quantities of physical interest near the scatterer such as the scattered field and the stress concentrations and, accordingly, very little is known about them. We have succeeded in deriving approximate values of the displacement as well as stresses due to the scattered field in the entire medium. From these results we obtain the dynamic stress intensity factors for the elliptic cylindrical cavity. Similarly, when the scatterer is a parabolic cylinder, which is an appropriate model for a hill [20], the solutions are known for a plane wave. We present the solution for a line source and obtain the case of a plane wave as a limit. Our approach is based on the integral equation formulation and a perturbation technique [I 1, 181 which has already been demonstrated to be very effective for three-dimensional scattering problems [9-l 11. For a parabolic cylinder we present the solution by combining integral equation methods and series expansions in Weber functions. Furthermore, it suffices to present the solutions for elliptic cylinder, a parabolic cylinder, and the corresponding cavities. The solutions for other configurations are derived by taking suitable limits.
2. MATHEMATICAL
FORMULATION
Let (x, y, z) be the Cartesian coordinates such that the z-axis is drawn in the horizontal direction and is parallel to the axis of symmetry of the scatterer. Let r be the section of this scatterer by the vertical plane z = 0 (see Fig. 1). A two-dimensional scattering problem in acoustics consists of finding the scattered field U, exterior to the scatterer when an incident plane wave (throughout our analysis we omit the time factor e-iwt, where w is the frequency), q(x, y) = Aeimx.6 = A exp i[m(x cos y0 + y sin &I,
(2.1)
impinges on it. Here A is a constant, m z ka is a dimensionless parameter, k = w/c, c is the speed of sound and a is the characteristic length of the scatterer, ii is the unit vector normal to the wave front which lies in the xy-plane and makes an angle qa with x-axis. The total field is u = Ui + U, and all these three fields
SCATTERING
OF
WAVES
I x’ FIG. 1. Section of the scattererby the vertical plane z = 0.
satisfy the Helmholtz equation. We shall be concerned with solving this equation for the scattered field U, , that is, (V2 + m”) u,(x) = 0,
x E R,
V-2)
where x = (x, y), R is the region exterior to the scatterer and we have nondimensionalize the equation with the help of the length u. The boundary condition for a perfectly soft body is x E r,
u,(x) = -4(x),
(2.3)
and that for a perfectly rigid one is
f%(x) _
----3
aui(x)
bV
where v is normal to I’ at x. Moreover, condition,
li+i d/p (4$
x
E r,
8V
u, satisfies the Sommerfeld
p = (x” + y2)1/2.
- imu,) = 0,
(2.4)
radiation (2.5)
The integral representation formula for the exterior Dirichlet problem consisting of Eq. (2.2) and (2.3) is [6-8, Ill, 44
= -G/4)
s,
~(xlwf~'(~
I x
-
Xl
I)IX@ ds, ,
(2.6)
where HJ1) is the Hankel function of the first kind, x1 = (xl , yl) is the source point, I(x) = au(x)/& is the single layer density and dsI is the element of length
4
JAIN AND KANWAL
at the point x1 along r. The corresponding integral representation for the exterior Neumann problem (2.2) and (2.4) is u,(~) = $1, J(x,) [*
f@‘(m I x - XI l)]xlEr 4 9
where J(x) = U(X) is the double layer density. The problem of scattering of electromagnetic waves by two-dimensional perfectly conducting bodies can be related to the foregoing scattering problems in acoustics, Indeed, the time-independent parts Eti’ and Hu) of an incident E-polarized monochromatic plane wave propagating along the direction ii perpendicular to the axis of symmetry of the scatterer are (see Fig. 2a) ,(i’(x)
= &z,(i’(x) z
== [email protected];2
= ($)“” (sin q&
- cos q&
(2.8)
1
PX.;.
i
Y vX’ (a) E-polarized x
(i) E
FIG.
(b) H-polarized
2. Electromagnetic
incident waves.
(2.9)
SCATTERING
OF
5
WAVES
Here &, Cy, C, are the unit vectors along the coordinates axes, iI = cos yoC, + sin ~&Z, , m = ka = wa/c, c = (
E’“‘(x) = &El”‘(x),
= _ ;
($)“’
(sz %$
- &, c).
(2.10)
The quantity EAs) satisfies the Helmholtz equation in the region exterior to the scatterer. The boundary condition that the tangential components of the total field E = Eti) + EtS) vanish on the scatterer leads to the relation that Els)(x) = -Eti)(x) on I’. Furthermore, I?:“) satisfies the Sommerfeld radiation condition. Thus, we have the exterior Dirichlet problem equivalent to (2.2), (2.3), and (2.5). For an incident H-polarized electromagnetic wave propagating in the direction fi perpendicular to z-axis, the values of the time-independent parts of Eti) and HCi) are (see Fig. 2b)
@(x) = ; (+)1’2 (vfff’
H’i’(x)
=
@ ffi”(x) z
=
X 5,)
& &mdi z
The scattered field is also H-polarized
so that
E’“‘(x) = ; ($)li2 (vf@‘(x) =
H”‘(x)
;
(2.12)
(q2
(8% 2A!$
X cz) -
c, ?gl),
(2.13)
= --c,H,‘“‘(x),
(2.14)
and His) satisfies the Helmholtz equation in the region lying outside the scatterer. The boundary condition is that the tangential components of the total electric field vanish on the scatterer, i.e., E(x) - 4 = E(x) . (6, x q = 0,
XEF
where 2 and 9 are the unit vectors in the directions of the tangent and the normal to r at x. In view of (2.11) and (2.13), the above relation becomes [v(Hy(x)
+ H:)(x))
x C,] * (C, x q = 0,
x E r.
(2.15)
6
JAIN
AND
KANWAL
Finally, using the identity (A x B) * (C x D) = (A * C)(B . D) - (A . D)(B . C), and the relation (& * G) = 0; (2.15) becomes aJ$ZS(x)/av = -aHii)(x)/&, x E r. Also, Hi”) satisfies the Sommerfeld readiation condition. Hence, we have the exterior Neumann problem equivalent to (2.2), (2.4), and (2.5). We have, thus, proved that the solution for the problem of scattering of E (or H)-polarized electromagnetic wave by a two-dimensional perfectly conducting body, when the wave is propagating in a direction perpendicular to the axis of symmetry, is equivalent to that for the problem of scattering of an acoustic wave by a soft (or rigid) body. For scattering of elastic waves we deal with steady-state Navier-Cauchy equations of elastodynamics valid for a homogeneous and isotropic medium, (A + ,u) grad div U + E~.PLJ + p,02a2U = 0,
(2.16)
which has also been nondimensionalized with the help of the characteristic length a. Here h and p are Lame’s constants, p,, is the density of the medium and U is the time-independent displacement field. Let the displacement field U@) due to the incident plane elastic SH wave be lP(x,
v) = Qi(x,
v) = A&, exp i[m(x cos P)~ + y sin vO)],
(2.17)
where A is a constant, m = ka, k = w/c and c = (,LL/~,,)~/~is the speed of S wave. The field U@) clearly satisfies the Eq. (2.16). The scattered wave is also an SH wave whose displacement field is UYX, Y) = &4x,
Y),
(2.18)
which also satisfies (2.16) in the region lying outside the scatterer. Hence, uS(x, y) satisfies the Helmholtz equation (2.2) with m defined as above. In elastodynamics also we have two kinds of scattering problems. The first kind relates to the rigid inclusions. In this case the boundary condition is that the total displacement vanishes on the surface of the inclusion. This means that u = ui + u, = 0 so that the boundary condition in this case is (2.3). Moreover, u, satisfies the radiation condition. Accordingly, we have the exterior Dirichlet problem consisting of Eqs. (2.2), (2.3), and (2.5). The second kind of scattering problems in elasticity relate to the cracks and cavities embedded in the medium. In this case the boundary condition is that the total stress vanishes on the surface of the crack or the cavity, that is, T(X) * G = 0,
x E r,
(2.19)
where z = T(Z)+ T(S) is the total stress tensor while G) and ~5~) are the stress
7
SCATTERING OF WAVES
tensors for the incident and scattered fields. The nonzero components of &) and ds) are Tfj(X) = p-, ad4 ay
(2.20)
T:(x)= p-.au,(x) ay
(2.21)
and
7(,;(x)= p,,,au,(x) Hence, (2.19) becomes
m4(x> + G41} - 3 = 0,
x Er,
or au,(x) _ ---al,’ av
au,(x)
x E r,
(2.22)
which is the same as (2.4). Also, u,(x) satisfies the radiation condition and we are presented with the exterior Neumann problem consisting of (2.2), (2.4), and (2.5). Thus, the above-mentioned scattering problems in the fields of acoustics, electromagnetism and elastodynamics are equivalent mathematically. In discussing the scattering of acoustic and electromagnetic waves it is the far-field behavior of the field and the scattering cross section which are of fundamental importance. In the case of scattering of elastic waves, we not only need these quantities but also the near-field as well. For instance, in the case of cavities we require the values of the dynamic stress intensity factors. This factor is defined as the ratio of the stress at a point on the cavity to the stress due to the incident field at the same point in the absence of the scatterer. It gives the measure of the stress concentrations at that point. Accordingly, we present the complete solution for the scattering of elastic waves. The solutions for the corresponding problems in acoustics and electromagnetism follow immediately. It is precisely for this reason that we have used the same notation for various quantities such as m, k, and c. Therefore, we shall merely mention the various configurations only in the rest of the sections and it is undlarstood that we are discussing SH waves. When some results for acoustic scattering need an emphasis, we shall mention them.
3. RIGID
ELLIPTIC
CYLINDER
The elliptic cylindrical coordinate system is defined as x = b cash f cos 7,
y = b sinh f sin 7,
z = z,
0 < 6 < co, 0 < 77< 271, (3.1)
8
JAIN AND KANWAL
where b is the distance between the two focii. The element of length ds is ds2 = dx2 + dy2 + dz2 = b2(cosh2 6 - cos2 v)(dt2 + dq2) + dz2.
(3.2)
Thus the scale factor h, , h, , h, for this coordinate system are h, = h, = b(cosh2 .$ - cos2 ~)l/~,
h, = h, = b(cosh25 - cos2$j2,
h, = h, = 1.
(3.3) Next, we nondimensionalize
all lengths with the help of b so that,
x = cash 5 cos q,
y = sinh .$ sin 7.
(3.4)
The fixed elliptic cylinder, the scatterer, is given by 5 = to and its principal axes have lengths 2 cash to and 2 sinh [, . Then from (3.2) we find that the element of length ds along a section z = a constant of this elliptic cylinder and the normal derivative a/an are 1 a ds = (cosh2 .$ - co? r#i2 dq, & = (3.5) (cosh2 e - cos2 ~)r/~ z. The plane SH wave moving along the direction ii impinges on the cylinder so that (see Fig. 3) ii = cos pJ& + sin &, , (3.6) where C, and C+,are unit vectors in the x and y directions and relation (2.9) becomes Ui = Szui , where ui = A exp im(cosh .$ cos 7 cos q~,,+ sinh [ sin r] sin y,,).
(3.7)
The scattered field is U, = QL, where u,([, 7) satisfies the boundary value problem, (V2 + m”) &(5,4 = 0, (3.8) to < 5 --c a, uS([,, ,T) = ---A exp im(cosh to cos 7 cos CJ+,+ sinh f,, sin 77sin CJ+,), 0<7]<<, (3.9) V2 = h2 (5
+ +),
h2 = (cosh2 .$ - cos2 7)-l,
FIG. 3. Elastic SH incident wave.
in = bw(po/p)1/2.
SCATTERING
OF
WAVES
The integral representation formula for u,(& ?) is
where I(771
Applying
=
[ aUy]f=r..
the boundary condition
Xl
=
k-1,
Tl>.
(3.9) to (3.10), we obtain the integral
= A exp im(cosh t,, cos yecos q,, + sinh 5, sin 7 sin q+,), Next we expand I@)(rn 1x - x1 I) in powers of
nz
equation
0 < ?j < 27T. (3.11)
for m < 1,
where
#o(% 71) =p+to- 2 f cz 124
x
ft
(cash nf,, cos nv cos ny, + sinh n[,, sin nq sin no),
&(q:, Q) = - T
[(l + e-“‘“)(l
+ e-2ETcos 277+
cos
(3.13)
- cos(7j - Q)) - 2e-“O cos(vj + Q)
27jIMM7j, Q) - 11,
(3.14)
and p = log(m/4) + y - k-/2.
Furthermore,
(3.15)
we expand the density function I(q) and the right side of (3.11) as
I(rl) = f m”&L(d,
(3.16)
T&=0
and exp im(cosh to cos 7 cos q. + sinh to sin 7 sin I& = 1 + im(cosh e, cos 7 cos q. + sinh E. sin 7 sin qo) - (m2/8)[(cos 2y, + cash 25,) + (1 + cash 2.$, cos 2~~) cos 27 + sinh 2co sin 2q sin 2q~,] + O(m3). (3.17)
10
JAIN
AND
KANWAL
Substituting these expressions in integral equation (3.11) we get simple integral equations in Z,,(q), Z,(q), Z,(q), etc. These equations can be solved by setting Zj(q) = ajo + f
j = 0, 1, 2 )... .
(q, cos nq + b,, sin UT),
(3.18)
n=l
The result is + inz& cos(v, - cpO) in2 (~0s 293, - co& %,I + (p : E > sinh 2c, 1 0 + S(P + to) I *2po -__ (3.19) cos 2(r)l - (PO) 4 I Substituting this value in the formula (3.10) we obtain the value of the scattered field G5 4 u,([, 7) = -A
I(*)
+ inze-(‘-‘0)
x (cash 5, cos 7 cos v. + sinh 5, sin 77sin ?o) - $
I($$-)(cosh sinh 2f
-i
P + to 1
25 + cos 27 + cos 2~0 + y;z)
+ e-2(c-Po)(cosh 2[, cos 2711~cos 27
+ sinh 2(, sin 2~, sin 27)] + O(m3)1,
!t > 40 2
(3.20)
which satisfies the boundary condition (3.9) at t = to . Furthermore,
= -phA
[&
- ime -(E-E”f(cosh to cos 7j cos y.
+ sinh to sin rl sin po) - $ x
I
[(&)
2(p + $) sinh 2c - cash 25‘ + cos 27) + cos 29z+,+
sinh 2fo P + &I I
- 2eP2(‘-“){cosh 2fo cos 2~,, cos 27 + sinh 25, sin 29, sin 27}] + O(m3)1,
(3.21)
SCATTERING
OF
11
WAVES
= A~.~inzlz e-(E-‘O)(cosh E, sin 7 cos q,, - sinh to cos 7 sin F,,) I + f
[(*)
sin 27 + f~?(~-‘~)
x (cash 2f,, cos 2q1,, sin 2~ - sinh 25, sin 2~” cos 241 + O(mz)/, (3.22) which are valid in the region 5 > &, . Far-field amplitude and scattering cross section. The far-field behavior of the scattered field follows from the formula (3.10) by using the asymptotic value of the Hankel function f$)(m
1x - xl I) E (&)1’2
ei(mo-n/Q
x exp -inz(cosh [r cos y cos Q $ sinh El sin CJIsin 73, (3.23) where we have set x = (p cos cp,p sin y),
x1 = (cash [I cos rI , sinh ,$I sin Q).
(3.24)
Thus, u,((, 7) ‘v (-$J””
ei(mp--n/4)Al(rp),
(3.25)
where
A,(y) ==- f Jo2nexp -itn(cosh f,, cos v cos Q + sinh to sin v sin Q) I(Q) dql
’ == A 5 +
1 I p + 5.
qp
1 +
to)2
1
1
- 4 m2 c 2(P + 50) (cos 2% + cos 2y) sinh 2&l
1
+ 2eEo(cosh to cos ~~ cos q + sinh co sin v. sin q) + O(m4)\.
(3.26)
Hence, the scattering cross section is S.C.S. =
2b mqlA12
so2n I A,(d2
dg?
= -!I!?!? mZs2 11 - i m2 [cos 29,, + &
(sinh 2fo)(log(me’0/4)
+ y)] + @ml)/ , (3.27)
12
JAIN
AND
KANWAL
z,2 = 4 [log ($)
(3.28)
+ YIP + 572.
We now derive the corresponding formulas for the limiting
configurations.
A. Infinite Strip When &, --+ 0, the elliptic cylinder reduces to a rigid strip of width 2b which occupies the region -b ,( x < b, y = 0, -co < z < co. Then the values of the displacement field u, , the stress field TV, the far-field amplitude A,(v) and the scattering cross section follow from relations (3.20), (3.21), (3.22), (3.26) and (3.27) to be
= -A
(( 1 + b [) + ime-’ cos 7j cos fpO
-$[(
1 + j‘) 5 (cash 25 + cos 27 + cos 2~~) - j sinh 2f
+ eC2’ cos 2~, cos 2~1 + O(m3)/,
06f
0<77<2r, (3.29)
P(Y cc?.. >y) = [&f * 2 $1 E.=0vx -
[T% dl+o vy , 0 < t < a, 0 < 7 < 237,
7yx =0 VY+ yz 3 y) = [Tj”‘(5 2 7 v)].c,J
kk
(3.30)
dl vx 9
O<(
o
where
bk)(t?
v&,=0
=
-
(Cosh2
-- 1’
b2(5> rl)lto=o=
[i
PA _
cos2 rl)l,2
]j-
-
ime-’
‘OS
7
‘OS
v0
{2(p + f) sinh 2f - cash 2.$ + cos 2~) + cos 2?,}
- 2ev2’ cos 2~, cos 27 I pimA e-’ sin 7 cos ~~ (cosh2 5 - COG $1/z I + $
vx -
f
[$ (p + 5) + eC2’ cos 2po] sin 27) + O(m2)1,
10 sinh 5 cos 7 (cash 25 - cos 2~)l/~ ’
vy =
~‘2 cash .$ sin q (co& 2,$ - cos 241’2 ’
(3’32)
SCATTERING
OF
13
WAVES
In particular,
i I -A
t&(x:, 0) =
1 + imx cos C& - q
\ -A
I(1 +;
x2 COG qJ0+ O(m”) , 1x 1 < 1. I
cash-l ) x ) + im(x - sgn x(x’ - I)‘/“) cos q. 1
-- m2 1 + j cosl1-1 / x I) (2x2 + cos 2cp,) - $ ( x / (x2 - 1y/* 8 [( i
~+(2r2-1-21X1(.~2-l)~~~)COS2y0]+o(m~)~,
Ixj>l, (3.33)
u,(O, y) = ---A I( 1 + i sinh-l 1y I) -_
99t*
[(I + j sinh-l / y I) (2y* + cos 2qo) - f / y 1 (1 + yy*
8
- (2y2 + 1 - 2 I y I (1 + y”)““) cos 2g,o] + O(m3$ ---co
1 -Apim T~(X, 0) = I
-A,u
cos qo{l + imx cos cpo + O(m2)},
[(P + cash-l I x I) 2x - i - l)sgnx ((';i2
_
-&O,
-
1)1/2
I$
- 2(2-U” -
\O,
cos 2%] + ow$
- imx cm q. - $-
1) COS 2~,] + O(m3)/,
1 wx I x I > 1,
[ +- (2.9 - 2 + cos 2~~)
1x1
IYI (1 + yy -_ t2
2(Y2
24
IX/ (x” - 1)1/2 ) I cos2qJ,, (9 - 1)1/z - -2(x2 - ])1/2
lx/ > 1,
y) = -,~~irnA I it1
-
1x1
(3.35)
F (1 “A,2)l,e 9(x YZ 7 o&-t) =
(3.34)
sgn x j + in2 cos po 1 l p(x” - 1)1/Z (
- $ i - p
a,
--co
1
< y < co, (3.37)
[$ f4(p + sinh-l I y I) I y 1 (1 + y2)*/*
+ 1) + cos 2%)
+ 2(2Y2 + 1 - 2 I y I (1 + y2)l’2) cos 2p70] + O(m3)/, --a
< J’ < co.
(3.38)
14
JAIN
AND
KANWAL
Similarly,
(3.39)
S.C.S. = -y52;
11 - $
(3.40)
cos 2q$l + o@n‘q,
where Z12 = 4{log(m/4) + r}” + n2. The corresponding results for the scattering of acoustic waves by a slit in an infinite rigid plane screen follow from the above analysis by using Babinet’s principle. The same is true for the scattering of E-polarized electromagnetic waves by a slit in an infinite conducting plane screen. B. Rigid Circular Cylinder When b + 0, to + co, such that b cash to + a, b sinh to -+ a, the elliptic cylinder reduces to the circular cylinder of radius a. Since 4 > to, .$ also tends to infinity and b cash 5 -+ ap, b sinh 5 -+ ap, q -+ q~, where (p, y) are polar coordinates of the field point (E, 7) in the limit such that p is nondimensionalized with the help of a. Taking these limits in the formula (3.20) for the scattered field, we obtain %(P, ~1 =
-A -
ii1 + i log p) + $ P2 + +s
qg, -
po]
COS(P- To) - g + Wm3),
p z
[(q + log p) (p2 + ;)
(3.41)
1,
where q=logT+y--,
nz = aw(po/p)1/2.
(3.42)
From this relation we readily obtain the nonzero components riz and T.& of the stress tensor associated with the scattered field,
rzz
-4
]$
-
+ (p + $1
+os(FJ - 3
cm 2(p,
vo) - g
-
poll-
[2Pb
+
O(m?j,
-
1 + logp)
(3.43)
SCATTERING
OF
15
WAVES
and
= 9
m [sin(y -
‘pO) + $
sin 2(9, - ‘pO) + OC,rZ,].
(3.44)
These stress components can also be obtained from (3.21) and (3.22) by taking the limits as explained above. Similarly, the values of the far-field amplitude and the scattering cross section follow from (3.26) and (3.27) to be A,(y) = y
jl -;$mq;+44
cos(g, -
%I]
+ o(m3/,
(3.45)
and S.C.S. = g
[I -
2nr2(yy
+ r) + O(n12)],
(3.46)
2
where zz2 = [4(log(m/2) + 7)” + n”]. 4. ELLIPTIC CYLINDRICAL CAVITY
We flollow the notation of the previous section so that the time-independent part of the incident field is Ui = &(f, T>, where ui = A exp inz[cosh 5 cos 7 cos vO + sinh [ sin 71sin ~~1.
(4.0
The scattered field zr,(f, 7) satisfies the boundary value problem P2 + m”> ~,(f, ‘I) = 0, ~EGl
2 7) = ~&o
3 4 = ~&od
E, < t < 03, = 0,
0 < 7) <27-r.
(4.2) (4.3)
The nonzero components of the stress tensors G and + associated with the incident and scattered field are a& & == $1 x = ipAnA(sinh x
5 cos q cos cp,,+ cash 5 sin 17sin yO)
exp im[cosh 5 cos 7 cos qO f
sinh 6 sin r) sin ~~1, (4.4)
’ = ph s = -ipAmh(cosh 5 sin 77cos qO - sinh 5 cos 7 sin yO) Tnz 877 x exp im(cosh 5 cos 77cos q,, + sinh [ sin 77sin q& q
(4.5) (4.6)
16
JAIN
AND
KANWAL
where h = (cosh2 [ - cos2 q)- rf2. From relations (4.3) to (4.6) it follows that the boundary condition is
W&77)/x- = 0,
4 = t-0,
0 e r) = 2n-.
(4.7)
The integral representation formula for u,([, q) is
where J(q) = u(&, , 7). From the expansion (3.12) for the Hankel function, we find that
I where @,(I$, ?j, 7jJ = f
P-J
cos n(?j - 7jl) -
?Z=l
jJ e-n(c+co) cos n(7j + Q) n=1
= 2 f
e-“‘[cash n&, sin nq sin nq, + sinh n& cos nq cos qll,
Tb=l
(4.10) and Q1([, v, Q) = $[{sinh 26, - 2 cash 5 sinh 5, cos 7 cos qI - 2 sinh [ cash 5, sin 17sin Q} p + 5 - + - 2 f I
x
Substituting
[cash n&, cos nq cos q
G
Vl=l
+ sinh a$,, sin nr) sin nql] . I
(4.9) in (4.8) and letting t + t,-, , we obtain
(4.11)
SCATTERING
OF
17
WAVES
where x0(7, 71) = -
4 -
f
(4.13)
e-2nSo cos “(7 + ql),
?k=l
X1(%71)= Pl(5, 72%)lP=to = & [sinh 2,$,,(1 - cos(r) - &](p I
+ & -
l/2 - 2 f
@“‘O/n)
n=l
;< (cash nf,, cos ny cos nql + sinh n[,, sin r17 sin HQ) By virtue of Poisson’s
summation
(4.14)
formula
n=-m
?L=l
relation (4.12) yields, 450 3 4 = 9417) + (1/W
/02r
J(?h)[Xo(%
71)
+
1"2x1(%
71)
- Qb4)1 4 3
0 < 7) < 2n.
(4.15)
Finally, we use the relation
= J(q) - A exp im(cosh f,, cos 7 cos qO + sinh 5, sin v sin v,,), in (4.15) and derive the Fredholm
integral equation of the second kind
J(v) = 2A exp z+n[cosh e, cos 77cos CJ+,+ sinh f,, sin 77sin CJ+,] + (l/d
J2 J(rll)[XO(% rll) + 0
m2x1(777 71)+ W~31 61 9
0 < ?j < 2n. (4.16)
We solve this integral equation in precisely the same manner as we solved the Eq. (3.11). We spare the reader of the details. The result is J(q) = A(1 + imecO cos(q -
(m2/8)[(cosh
vo)
25, - 2(p + 8,) sinh 2(, + cos 2~~)
+ cos 27j + e2’0 cos 2(77 -
yo)] + O(nz3)}.
(4.17)
18
JAIN
AND
KANWAL
Next, we substitute this value in the integral representation obtain the value of the scattered field, u,(f, T) = iAm
I
formula
(4.8) and
e-(E-EO’(sinh [, cos 7 cos y0 -+ cash [, sin 7 sin vO)
je-B(C-PO)
+
m[sinh 2c0 cos 2q,, cos 27 + cash 2(, sin 271, sin 2771
8
- 6 sinh 25,&p
[I + O(m21!, 5 > 5, .
+
(4.18)
Stresses. The formulas for the non-zero components T& and T,“~ of the stress tensor follow from relations (4.6) and (4.18) to be T&($, v) = -i~LhAn2{e-‘5-Eo’(sinh + (im/4) e-‘(‘-@[sinh
5, cos v cos ?. + cash E, sin 7 sin v,,) 2f,, cos 2171,cos 2r) + cash 2& sin 271, sin 271
+ O(m2)1, E-+ 6, ,
+ (im/4) sin X0
r,$(t, 7) = -i~LlzANz[e-‘E-‘a’(sinh
(4.19)
.$, cos q,, sin 7 - cash 5, sin ‘p,, cos 7)
+ (im/4) e- “(‘-‘O)(sinh 2t, cos 293, sin 27 - cash 2[, sin 27+, cos 27)
+ anz2)1,
f > 5, ’
The above values of stresses satisfy the boundary Furthermore, from relation (4.1) we have &(.$, 7) = -ipLhAm[(cosh
(4.20) condition
(3.3) for [ = f,, .
6 sin 71cos q+-,- sinh [ cos 7 sin p,,)
+ (im/4)(sin 27 + cash 25 sin 277 cos 27~,, - sinh 2f cos 277sin 2q7,,) (4.21)
+ O(m2)1, The addition of (4.20) and (4.21) gives 7,&,
q) = -iphAm{[(e-(5-50)
sinh E,, + cash f) cos y0 sin q
- (e-(S-E0) cash So + sinh 5) sin CJI,,cos q] + (jm/W
1 + (e- 2(c--c0)sinh 25, + cash 2.5) cos 2~~) sin 27
- (eP2(‘-‘O) cash 2t0 + sinh 20 sin 2q1,,cos 291 + O&2)}, The only nonzero component
t > to, (4.22)
of the total stress tensor at a point (to, q), on the
SCATTERING
OF
WAVES
19
surface of the cavity, is 711z. Hence the value of the dynamic stress intensity factor N,, is T,t(L d Nn3 = lim t-+50+[ - ipAm I = h, eEosin(r) - y,,) f :’ i
[sin 2~ + eztOsin 2(7 - q+)] + O(WZ~)/, (4.23)
where h,, is the value of h evaluated at the point (5, , 7). Far-jield amplitude and scattering cross section. We apply the asymptotic formula
2 112 = -1171 ,( ~TrnP 1 ei(“~~-“J4)exp -irn(cosh x
f, cos g, cos Q + sinh .$,,sin p sin qI)
(sinh 5, cos 9 cos yl + cash 5, sin cp sin qr),
where 1; = (p cos F, p sin y), xl = (cash [I cos Q, sinh e1 sin Q), in the integral representation formula (4.8) and find that u,([, 27)= ($)l”
The value of the amplitude -4d~)
ei(me-n/4)A2(q).
A,(v) is
= T jo2n J(rlJ(sinh 5,cos g, cos y1 + x
exp -im(cosh
= - y
cash fO sin v sin ql)
to cos g, cos ?I + sinh 5, sin 9 sin v,I~)dql
[sinh 2(, - 2eE0(sinh [,, cos q~cos 9?0
+ cash 5, sin cp sin y,,) + O(M~)].
(4.25)
The value of the scattering cross section is
S.C.S.:= 2b j"" ~mlAla
:= F Let us
now
o
I A2(~)1~
dg,
[sinh’ 2.$, + 2e2’O(sinh2 f,, cos2 q. + cash’ & sin’ v,,) + O(W?)]. (4.26) attend to the limiting configurations.
20
JAIN
AND
KANWAL
A. GrijEth Crack When to + 0, the elliptic cylindrical cavity reduces to the Griffith crack which occupies the region -b < x < b, y = 0, - co -C z -=c 03. Then the formulas (4.18) to (4.20) for the displacement field and stress components reduce to
fdx,
Y> =
M5, .)7)1+rJ
= iAm
I
e-’ sin 7 sin pO + 7
e-” sin 27 sin 2y, + O(f7zz)/,
0 < 5 < ~0, 0 < 7 < 2~r, (4.27)
TXX, Y> = [TX53rl)lE,=O v, - bL(E7?l>le,=o y11 ,
o
0<7jd2n,
(4.28)
where v, and vy are defined in (3.32) and [T&([,
q)]ro=o = -iphAm
[T$@,
q)]+.
/e@ sin 7 sin q+, f $
= iphAm /e-e cos 17sin yO + q
In particular,
fiAm(1 i 0,
- x2)1/2 sin qO 1 + I
IxI>
+
=
(1 - 9)1/2
0,
0) =
~40,
Y>
/x+9(2x2-l)cosp,+O(m2)~,
Ixl
(4.31)
1x1 > 1,
-ipAm
sin q,,{ 1 + imx cos y+ + O(m2)},
-ipAm
sin q~,, 1 I(
+ im (x Similarly,
(4.30)
1%
ipAm sin q,,
Tiz(X,
e@f cos 2q sin 2~p,, + O(n?)l.
the values of these quantities at the faces of the crack are
%(X, @t> =
T;;(X, Of)
e-25 sin 2~ sin 29, + O(m2)/,
(9
‘!I? ;)I,2
(2x2 - 1) sgn x 2+x2
_
1)1/2
IX < 1,
1
)co~~~+O(m~)~,
Ix1 >l.
(4.32)
the values of these quantities in the plane x = 0, are
= iAm sin v. sgn y{(l
pAm2
ddo, Y> = 4
2y
_
+ y2)l12 -
CO2 + 1) w (1 + yy2
I y I) + O(m2>}, Y ] sin 2yo + O(m)/,
--oo
< y < a.
(4.33)
SCATTERING
Ml
Y) = -ipAm [l -
(1 yd2)l,2 ]
OF
21
WAVES
sin y, + O(m2) ,
I
---co
y < co. (4.34)
Since, ui(x, JJ) = A exp im(x cos q+, + y sin plO), we have
*rtz(x,0) = P [$$ (x3Y)]g=o = ipAm sin y,,{l + imx cos q,, + O(m2)>,
1x1
>. 1.
(4.35)
Adding the second part of (4.32) and (4.35) we obtain (2x2 - 1) sgn x cos vO + O(m2) 2(x2 - 1)1/Z
T~Jx, 0) = ipAm sin CJI,,
’ x j > 1.
(4.36)
t ;]
(4.37)
The dynamic stressesat the edgesof the crack are IV,+, = lim(j x 1 - 1)1/2[ T$:z)] =- ““;p
[1 & y
Id>1 ’
x -+ [1':
cos yO + O(m2)\.
(4.38)
The values of the far-field amplitude and scattering cross section are A,(Y)
=
iArm 4
sin CJIf m2 ,sm v, + 16 [sin 2g, cos v - 2 sin vO cos2 F
- (2 cm2 y. + 4p - 3)l + O(m2)/,
s c s = bm37r2 sin v. . . .
8 t
I
(4.39)
sin v. - +6 17t2[sin~~ + 2(cos 2~,, - 2
4@x(m/4) + 241 + Wan”)/.
(4.40)
The first term in each of the above formulas can be obtained as limits from relation (4.25) and (4.26). We have evaluated them independently to give the higher order terms since the analysis in this case is simpler. The corresponding results for the scattering of acoustic waves by a slit in an infinite solft plane screen follow by Babinet’s principle. The sameis true for the scatteri:ng of H-polarized electromagnetic waves by a slit in an infinite conducting plane screen.
22
JAIN
AND
KANWAL
B. CircuIar Cylindrical Cavity Let b-+0, to-+ co in such a way that b cash &,-+ a, b sinh [,, --t a and the elliptic cylindrical cavity becomes circular cylindrical cavity of radius a. Since 8 > to , 5 also tends to infinity so that b cash [ + ap, b sinh 5 + ap, 7 -+ y, where (p, y) are the polar coordinates as explained in Section 3 for the case of a rigid circmar cylinder. Taking these limits in formula (4.18), we obtain the scattered field u,(p, y) = iAm if cos(g, - vO>- f -
$
[(i
(2q + ;j
[2q + 2 log p - $ cos 2(9, - PO)]
+ 2p (4 - ; + 1% Pjj 4g,
-
%>
(4.41)
p > 1,
where q and m are defined in (3.42) and we have included the term of order m3 by following the previous steps independently for this simple limiting case. Same is true for the following quantities of physical interest. The stress components ~pS~(p, y) and ~,“,(p, y) are
= - y
jcos(p: - pJ + 9
‘a2 [(29 + ; -
+ $ cos3(g, -
[p + ; cos 2(p, - %)I
2p2 (4 + ; + log Pjj
WJ] + OW) f ,
cos(g, -
%I
p > 1)
(4.42)
and g012 (p 3 ‘p) = .P- %(PT F> P
89
ipAm
= - 2 -
$
+ &
P
t
sin(p, -
[((2q
+ i)
sin 3(9, -
vO) + E!- sin 2(p, 2P
+ 2p2 (4 A]
+ O(m31
I&
& + log p)) sin(y p > 1.
-
94 (4.43)
SCATTERING
= --ipAm
t
- T
OF
23
WAVES
sin(g, - yO) + y
p sin 2(~ - ?J
[sin(g, - I& + sin 3((p - qO)] + O(m3)j.
(4.44)
Therefore, Tmzh VI = --ipAm
](I + $)
- -$- [($ + k[$-
sin(p, - yo) + f
(p + f)
(24 + i) + 2 (q - k + log p + g)
+ p2] sin 3(g, - vol] + Ob3)l,
sin 2(9, - p)J sin(g, - q+)
p > 1,
(4.45)
and the value of the stress intensity factor N,, is
=
[
2 sin(g, -
- q
q$) + in2 sin 2(g, - vO)
[( 4q + 35) sin(g, - d
+ sin 3(9 - v,)] + OWI].
(4.46)
The first two terms in this expression for N,, can also be directly obtained from formula. (4.23) by taking the appropriate limits as explained above. Similarly, the values of the far-field amplitude A,(F) and the scattering cross section in this case are A2(g?) := - q
I(; - cos(p, - q+i) + $
x cos(g,- PO)-
2
[(4q -
03s2(9, - %)I + ow$
3) + (8q + 10) (4.47)
and (4.48)
24
JAIN AND KANWAL 5. LINE SOURCE PLACED
IN FRONT OF A RIGID PARABOLIC
CYLINDER
In view of Fig. 1, we take the z-axis parallel to the generators of the cylinder, such that the parabolic cylindrical coordinates are x = gg2 - q2),
Y = &I,
z = z,
--00 < 6 < co, 0 d 7 < co, (5.1)
The cylinder is defined by 7 = M, so that the coordinates on its surface are x = $([” - M2),
Y=tM
z = z,
--co<‘$
(5.2)
and the parabolic section in the xy-plane has the equation y2 = 2fw(x + (W/2)).
(5.3)
Thus, the latus rectum has the length 2M2 while the focus and the vertix are at the points O(0, 0) and V(--M2/2, 0) (see Fig. 4). The polar coordinates (p, y) of the field point ([, 7) are P = 3(5” + T2),
91= 2 tan-‘(r/&J).
(5.4)
FIG. 4. Section of the parabolic cylinder and the line source by the vertical plane z = 0.
It is convenient to nondimensionalize all lengths by the semilatus rectum A42 so that over the cylinder 77= 1, -co < 5 < co. The value of ~~(6,v) is (5.5) which is a line source of strength A parallel to the z-axis placed in front of the cylinder so that it intersects the xy-plane at the point P,, , x,, = (St, , r],,). The
SCATTERING
OF
25
WAVES
field point is, as usual, x = (6, r)). The parameter m is equal to kM2 which is (~cIJ~M~/cL)~/~for the case of elastic waves and wM~/c in acoustics. When we expand EQ) in terms of parabolic coordinates of the field point ([, 7) and the source point (f,, , q,) we have [16]l
where q<: = minimum (7, v,,); 7, = maximum (7, Q). The quantities Bm(z) and g-,-,(z) are Weber functions of the first and the second kind, respectively, g?‘,(z) = (-1)” B-+,(z)
/‘4
= 42 q n.
!&
(p2’2),
e?*‘4 f&
n 2 0. FZ> 0,
[ezzi2 erfc ($1,
while X := (-2im)lj2 and /i is its complex conjugate. The boundary value problem for ~~(4, 7) in this coordinate form (V2 + m2) f4f, d = 0, q > 1, %(f, 7) = -45,
$3
--co
(5.8)
system, takes the (5.9)
r = 1,
(5.10)
where
The integral representation outside the parabola is
formula for u,(f, 7) at a fixed point x = @, 7) lying
udf, 4 = - $ Jvrn m %Afd%
I x - x1 l)>nl=ldtl ,
17> 1,
(5.11)
where xzl = (tl , Q); and
1 There is a slight error in the formula given by Morse and Feshback which we have corrected in this relation.
26
JAIN
AND
KANWAL
Applying the boundary condition (5.10) to (5.11) we get the Fredholm equation of the first kind for the density function,
s -1
GWc%~
I x
-
Xl
integral
I)17J=n,=l &I = A[H$% I x - x0 l)ln=l) --co < c$< co. (5.13)
Next we use the expansion (5.6) of HA1) in terms of Weber functions and set
m = f an~,@5>,
(5.14)
7L=O
and use the orthogonality
condition (5.15)
Then the integral equation (5.13) yields the value of the coefficients a,, as
Thus, the value of the density function is completely known. Finally, we substitute this value of I(t) in formula (5.11) and derive the value
-q > 1, (5.16) where we have used the orthogonality condition (5.15). This obviously satisfies the boundary condition (5.10). With the help of this relation and (5.6) we can evaluate the total stress components (5.17) Plane wave. When the source PO + cc along the line OP, , 1 f. I--+ co, y. + co. In terms of its polar coordinates (p. , qo) p. + co, but v. = 2 tan-l(~o/~o) remains constant because PO + co along the line OP, so that
80 = (2PoY KS yo/2,
7jo = (2~,)l/~ sin ~~12.
(5.18)
SCATTERING
OF
27
WAVES
When we use the polar coordinates for both x and x, , then we find that for large p,, ,
Accordingly,
we take the constant factor A in such a way that
Then (5.5) reduces to the plane wave ui(p,
v)
=
e-i~m3s(~-~o)
=
eim~cos(~-d
=
eim(zcos~+tdkd,
(5.21)
where we have set q+, = n + 01.Thus, we have a plane wave which is moving in the direction which makes an angle 01with the x-axis (see Fig. 5). To obtain the corresponding asymptotic value of relation (5.6) we use the asymptotic formulas for the Weber functions [24] LBn(X&J = Qn (-2 -
(-ipom)1/2
ei~omsin2(a/2)
(
sin 5)
-2 (-ipomz)1’2 sin f)“,
I a I < r/2,
(5.22)
1a I < 77/2.
(5.23)
I 01/ < z-/2.
(5.24)
/ a 1 < 7r/2.
(5.25)
and %n-1(X7jo)
= ~+, exe
(2(--ipom)“~
ic+cos%/2
(
cos +)
2(--ipom)l’2
cos f)
--n-l
,
Then
~a,G%J%-l(GJ
c-e
eiDom(- 1)” tann cyj2 set olj2 2( - ipom)lj2 ’
Then from (5.1), (5.6), (5.20), (5.21), and (5.24) we have u&f, 7) = exp im[$(t2 - +) cos OL+ 57 sin LX] = set +
f
in q
9,(hf)
Lq$i?jQ,
?l=O
Similarly, we can derive the expression for ui(<, 7) when the angle of incidence 01 lies between z-12 and 3rr/2, but we shall discuss the case when 1 01I < rrj2.
28
JAIN AND KANWAL
FIG. 5. Geometry for the plane waves incident on a parabolic cylinder.
The corresponding limit for the scattered field u,(t, 7) is obtained in the same way. Indeed, using (5.20) and (5.24) we find from (5.16) that II > 1.
(5.26)
The result of adding (5.25) and (5.26) is
which obviously satisfy the boundary condition (5. IO) at 7 = 1. To expand the values of u in powers of m for vz < 1, we need the expansions
s2n(Z) = (- 1)” @I ! 2”n! [I - (n+ t, zq+ Wz4), g2n+l(z) = (-11% P2% !+ l)! z + (qz3) 3 B-212-l(z) = (g2
&
- Jg
+ (g”
@ ;j;
(5.28) (5.29) z2 + U(z3),
(5.30)
and a-2,-2(z)
=
(2n2y1)!
-
_-7r lj2 z ( 2 > 2% ! + tn + ;,
2
(2;y
‘I)!
+ U(z3).
(5.31)
SCATTERING
OF
29
WAVES
Then relation (5.17) yields ~(4, rl) = im(~ -
1) [-2
(&jr”
cos 5 + 5 sin cy. (-4
tan2 +)”
+ c+P)],
77 3 1. (5.32)
The values of the nonzero stress components are 7&,
(5.33)
7) = $2 -& = @&z(n - l)[sin 01+ 0(m1/2)],
and T&,
77) = ph $-
= pimh [-2
(&)l”
cos T + [ sin cr (-4
tan2 F)n + O(m1/2)],
(5.34)
which are valid in the region r] > 1. Line source in front of a rigid half plane. The parabolic cylinder given by Eqs. (5.2) reduces to a rigid half-plane x > 0, y = 0, z = z, when A4 - 0. Since there is no characteristic length we deal with the wave number k. The incident displacement field due to line source of strength A, palced at the point I’,(&, QJ which does not lie on the half plane (Q, > 0), is (see Fig. 6) u&S 7) = (i/4) A fft’(k - &n;o
Ix -
m (-i)” 7 n.
xo
I)
Sn(do)
~,(a3
9n(Q<)
~-n-l(~rl>)~
(5.35)
p,(Eo.To ) FIG.
z = 0.
6. Ox and P0 are the sections of the half-plane and the line source by the vertical plane
30
JAIN
AND
KANWAL
where (T = (-2ik)l12, and 0 is its complex conjugate. The solution u,([, q) for this case is obtained from solutions (5.16) by first writing it in its proper dimensions, namely,
and then letting M--f 0. Since from (5.28) to (5.31) 92n+m
=
S2JO)
0,
=
(-;Jny
)
9-2n-1(0)
=
(i)li2
A)
the required limit is %(5,
rl)
=
-
4
f ?I=0
~2n(4392n(d
~-27&d
g-zn-l(T).
(5.37)
Plane wme. When we take the limits of relations (5.35) and (5.37) as \ 5, \ -+ co, ~7~+ co such that p. -+ co while ro/fo remains constant and equal to tan yo/2, and ‘I2
e i(ko o-?r/4) -+ 1,
(5.38)
we obtain ui((, 7) = exp ik(x cos a + y sin cz) = exp ik[$(t2 - 7”) cos a + & sin LX]
and u,(E, 7) = - (4)“”
set E nto tan2” 2 g2,(u5) B-2n-1(u$,
(5.40)
where we have used the result (5.24) and q. = n + a, 1 a! I < z-/2. Relation (5.40) gives the scattered field when a plane wave moving in the direction perpendicular to z-axis and making an angle CYwith x-axis impinges on the rigid half-plane x > 0, y = 0, z = z. It is convenient to write the left side of (5.39) as a product of two Weber functions. This is achieved with the help of relation go(z) = e-z2/4 and we have u&f, q) = exp ik[+(t2 - +‘) cos a + (17 sin CY]
SCATTERING
31
OF WAVES
Next we use the identity (2+/z 530(z) = %,(iz)
+ .sl(-iz),
the abovle formula becomes
X go (J f cos -5 + 71sin + [ ( 2
)I,
I a! I < 7T/2.
(5.42)
Let us put the formula (5.40) in this form by appealing to the relation
as well a.sthe one obtained from it by changing 01to -01, namely, gz [CT(7j cos $ - 5 sin +j] =
(COS
T)” j.
‘! ‘i:ii
TSO[CT(c cos 2 + 7 sin T)] FEyF’2)
where we have used the identity relation (5.40) is simplified to u,(L q> = - -(27&
t go
[i
~n(uLJ) L@-n(aq),
9,,(-z)
j cdj < 77/2, (5.44)
= g,,(z). By adding (5.43) and (5.44),
a 7j sin 5 - .$ cos f )]~-l[~(~cos~+&sin~)]
+ L?20[IJ (5 cos + -I- r sin +j]
%,
[CT(7 cos 4 - 4 sin T)] 1, 1 011 < z-/2. (5.45)
Finally, we add (5.42) and (5.45) and derive,
- go [U (7 sin $ - E cos %)] 9-l
[U (7 cos 4 + E sin T)] 1, 1 a / < z-/2. (5.46)
595/91/I-3
32
JAIN
AND
KANWAL
To derive the famous Sommerfeld formula we use the identities _
go(&) = pw,
.9’-1(d)
e--ikt2/2
-6t
= 1/2 t/i
s
eia2
d BY
--m
(5.47)
where t is real. Then (5.46) becomes 4cY ?I) = ($1,2
I
exp ik[&(t2 - 77”)cos 01+ EV sin
-~~(fsin(./2)-~C0S(~i2)leis2 X
al
dp,
s-72
- exp ik[$(k” - q”) cos a: - &j sin a] -z/lc?Ssin(a/2)+rlcos(njz))
X
ei@ d/3].
s -m
(5.48)
In terms of polar coordinates .$ = (2p)li2 cos p/2, 7 = (2p)1/2 sin v/2, the above relation becomes the Sommerfeld formula (2ok)%n((m-a)/2)
e@ d/3
eikpcos(m-~)s --m [ (ZdPi2Sin((m+a)/2) _
ei@ d/l].
eikoCos(~+a)
s --m
Stresses on the faces of the half-plane. identities
(5.49)
By applying the formula (5.17) and the
g,‘(z) = - g 9,(z) + ng),-,(z) = ;i gn(z) - ~n+l(z),
(5.50)
and (27r)1’2 q)(z)
= 9Ll(iZ)
(5.51)
+ z%l( AZ),
we obtain from (5.46) the values of the stresses on the upper and low faces x > 0, y = Ok, z = z of the half-plane. They are
= p (-$)1’2
x go [
ik sin 01 1~3~~[a 1 5 1 sin T]
- (0 / [ 1 sin $)-’
I 4> 0,
a.$ cos + ,
(5.52)
SCATTERING
OF
33
WAVES
and
zz
-[T&X,
O+)]e,o
+ 2(ikp sin 01)go [65 sin 51
t < 0.
:< q)
(5.53)
The low-frequency approximation for the displacement field and the stresses for the present case can be obtained from the corresponding formulas (5.32) to (5.34) for the parabolic cylinder. For this purpose, we first write them in the following forms by setting m = KM2 and introducing the original values of the coordina.tes f, 7 with proper dimensions, u(f, 97) = ik(7j -
x g T&
___ g;;
*7) = (p f’~2jl,2 T,;
=
(&)“”
M) [-2
Pik
(-4
cos T + ( sin 01- ?
tan2 5)”
set 5
+ O(W2)],
(5.54) (5.55)
(7 - M)[sin x + O(W)], [-2
(J&y’
cos f
(-4
tan2 T)”
+ O(k1!2)].
Cc?”+ q2)1’2
4M
+ c sin 01 - -
77
set E2
(5.56)
Then by letting A4 -+ 0, we obtain the formulas for the half-plane in the limit. They are, in polar coordinates, u(p, 71) = 2ikp sin 22 [ -
(-&-)I’”
cos q + cos T sin a: +- O((kp)1/2)], (5.57)
TJP, ~1, = pik [sin 5 sin 01+ O((kp)‘/“)],
(5.58)
and Q-&P, p’) = pik [cos 5 sin 01 -
for kp<< 1.
($)1’2
cos $- + O((kp)llz)],
(5.59)
34
JAIN AND KANWAL
6. PARABOLIC CYLINDRICAL
CAVITY
Let us now consider a hollow parabolic cylinder whose surface is stress free. In terms of the notation and configuration of the previous section the incident wave ui due to a line source at x0 = (& , T,,) is given by relation (5.6). The scattered field U, satisfies the boundary value problem
-cc
< f < 03, (6.2)
where we have used the expansion for the Hankel function as given by (5.6). The integral representation formula for u,([, 77)is
where the double layer density is J(t) = ~(6, 1). To obtain the required integral equation we substitute the Weber function expansion for Hi11 in the integrand, differentiate both sides of (6.3) with respect to 77,let 7 --f I and use the boundary condition (6.2). The result is
=-&I$
~?&@0)~n@O%‘(a ~-n-db>,
--co<(
?I=0
(6.4) We solve this integral in the same manner as Eq. (5.13). Indeed we set
use the orthogonality
property (5.15) and find that (6.6)
SCATTERING
OF
35
WAVES
The value of the scattered field u.& 7) is obtained from (6.3), (6.5), and (6.6),
rl > 1, --a3 < ( < GO, (6.7) which obviously satisfies the boundary condition (6.2) for 77 = 1. The total field as well as the total stresses can now be readily calculated. Plane wave. Taking precisely the same limits as we did in the corresponding situation. in the previous section, we derive from relations (5.1) and (6.7), the values of the incident field for a plane wave
~~(5, 77) = exp inz(x cos 01+ y sin LX) = set $ C 2 (tan f)n n=o .
Bn(h.$) L@J&),
(6.8)
where CJI,)= 7r + 01,j o( j < rr/2, while the scattered field is
The result of adding (6.8) and (6.9) is
(6.10) Accordingly,
the values of stress components are
X
[
B,(Xr))
- i ~?a’(a sz-l(b) %,-,Gv
1’
-q > 1,
(6.11)
and
7 > 1. (6.12)
36
JAIN
AND
KANWAL
In the next stage we use the expansion of the Weber functions as given by (5.28) to (5.31) and those for their derivatives and obtain for m < 1, t sin+
a((, 7) = im [ 2 (-g2
+ 2’([” - q2 + 27) cos 01 a, (2n + 3/2)(n !>”
( -4 tan2 $)% +
O(m1/2)],
(6.13)
sin fi2 + 5 cos 01+ 4 set -Z tan -!? n2 2
~~~(5,rl) = pimh [2 (&,“’ x f Vl=O
(2~ + 3f2)(n!)2 (-4 tan2(,/2))” (2n + l)!
(6.14)
+ ow3],
and T,&,
q) = @mh[(l
(6.15)
- T) cos 01 + O@Z~/~)],
where we have dropped all the constant terms upto order O(m) in (6.13). To derive the limiting results for a crack of the shape of a Half-plane crack. half plane defined as x 3 0, z = 0, we use the expansion (5.35) for ui which is given in its dimensional form. Furthermore, we write the scattered field (6.7) also in its dimensional form,
Taking the limit of relation (6.16) when M-+ a,(&
7)
=
-
$
f
(-
1)2n+l
9 zn+l(do>
0, we get
=%+d~~)
g-zn-2(wJ
~-zn-z(4,
(6.17)
?k=O
where we have used the relations
%2,(O)= 0,
J%n+1(0>=
(-1)”
(2n + l)! 2”n! ’
9Y2n--2(0) =
-zy
.
Relation (6.17) gives the scattered field when a line source of strength A is placed at (co, qo). The other quantities of interest can now be readily calculated. To solve this problem we take the (5.35) and (6.17) when 1cl, j + CO, ~~-j co such that
Plane wave incident on a half-plane crack.
limits
of relations
SCATTERING
OF
37
WAVES
r),/& = tan q,/2. We further use the limit (5.38). Then the incident field ui reduces to relation (5.42) as explained there while the scattered field becomes,
a-7 7)==(2;)‘,2 190[i(5rl sin2 - f cos $,)I S1 [U (q cos - 90[u (4 cos 212 + 7 sin E2)] Lie1 [Q (7j cos $- -
f
+
t sin
$)]
4 sin :)]I, (6.18)
where we have used the same steps and notation (5.45). Adding (5.42) and (6.18) we have
+ go [U ([ cos + - q sin f)]
a-l
as we did in deriving
[U (t sin f
relation
+ 7 cos :)]I. (6.19)
Next we use relations (5.47) and derive from (6.19) the formula sponds to Sommerfeld formula (5.49). It is
s
-(2p~)1i2sin(m+(~/2))
+
eik!msb+a)
which
1
e@ dfl .
--3o
corre-
(6.20)
Stresses on the faces of the crack. From relation (6.19) we can derive the stresses ‘on the faces of the crack by using formulas (5.17), which gives
b&G w)lr>O = [+Tdt, 0)1,<,,m = 0. -m
stress component
b&G o+)lz>o = b&5 dlO<,<~ = 4 [ y; Tl=O
= (a)“’
pik 1~0s &-,
(6.21)
is r,, so that
-+I] O
[CT1( / sin 11
+ 2(u 1( I)-’ sin + B0 ~7$sin ; [
1 t>o (6.22)
a.$ cos $ ,
38
JAIN
AND
KANWAL
and
[- . “I [
= 2pik cos c&9,, & sm -2- .9,, ut cos -&] -
[T&X, O+)lZ,,, , (6.23)
where we have used the identities (5.50) and (5.51). Low frequency approximation for the displacement and stress fields, inside the material can be obtained from relations (6.13) to (6.15) by first writing them in the following forms by setting m = kM2 and putting the original values of the coordinates 5, r) with proper dimensions, u([, v) = ik [2 (&)l”
f sin 01+ 1/2(f2 - y2 + 2$4)
cos 01
+ $ (M set 5 tan T 2OD(2n + 3/2)(n !)” (-4 tan2 f)” %=() m + I>!
+ O(k112)], (6.24)
T&,
77) =
x 2 (2n + 3/2)(n !)” (-4 n=() (2n + I>!
tan2 T)”
+ O(kl!z)],
(6.25)
and pik
TJ*(!f, 4 = (P + qy2
[(M - 7) cos a + O(k112)].
Now we let M+ 0 and introduce polar coordinates (2~)l/~ sin v/2. Then relations (6.24) to (6.26) reduce to u(p, y) = 2ikp [(--$-)I’” T&P,
91 =
pik
[(--$-)I’”
(6.26)
5 = (2p)li2 cos v/2,
cos 5 sin a: + k cos y cos a! + O((kp)l/2)], sin $ + cos 01cos T + O((kp)1!2)],
(6.27) (6.28)
and T&P, 91>= -pik
[cos cy.sin $- + O((kp)‘l”)].
(6.29)
REFERENCES
1. M. L. BARON AND J. T. MATHEWS, Diffraction of a pressure wave by a cylindrical cavity in an elastic medium, J. Appl. Mech. 28 (1961), 347-354. 2. M. L. BARON AND J. T. MATHEWS, Diffraction of a shear wave by a cylindrical cavity in an elastic medium, J. Appl. Mech. 29 (1962), 205-207.
SCATTERING
OF WAVES
39
3. C. J. BOUWKAMP, Diffraction theory, Rept. Phys., 17 (1954), pp. 35-100. 4. J. J. -BOWMAN, T. B. A. SENIOR, AND P. L. E. USLENGHI, “Electromagnetic and Acoustic Scattering by Simple Shapes,” North Holland Publishing Co., Amsterdam, 1969. 5. K. HARUMI, Scattering of plane waves by a rigid ribbon in a solid, J. Appl. Phys. 32 (1961), 1488-1497; 33 (1962), 3588-3593. 6. D. L. JAIN AND R. P. KANWAL, Diffraction of elastic waves by two coplanar and parallel rigid strips, Ipzr. J. Eng. Sci. 10 (1972), 925-937. 7. D. L. JAIN AND R. P. KANWAL, Diffraction of elastic waves by two coplanar Griffith cracks in an infinite elastic medium, 1~. J. Solids Structures 8 (1972), 961-975. 8. D. L. JAIN AND R. P. KANWAL, Acoustic diffraction of a plane wave by two coplanar parallel perfectly soft or rigid strips, Can. J. Whys. 50 (1972), 428439. 9. D. L. JAIN AND R. P. KANWAL, Diffraction of a plane shear elastic wave by a circular rigid disk and a Penney shaped crack, Quart. Appl. Math. 30 (1972), 283-297. 10. D. L. JAIN AND R. P. KANWAL, An integral equation perturbation technique in applied mathematics, II, Applicable Analysis, to appear. 11. R. P. KANWAL, “Linear Integral Equations,” Academic Press, New York, 1971. 12. A. K. MAL, A note on the low frequency diffraction of elastic waves by a Griffith crack, hr. J. Eng. Sci. 10 (1972), 609-612. 13. A. W. MAUE, Die beugung elastischer Wellen an der Halbebene, ZAMM 33 (1953), I-10. 14. L. J. MENTE AND F. G. FRENCH, Response of elastic cylinders to plane shear waves, J. Eng. Mech. Div., Proc. ASCE 90 (J964), 103-118. 15. J. W. MILES, Motion of a rigid cylinder due to a plane elastic wave, J. Acousf. Sot. Amer. 32 (1960),
1656-1659.
16. P. M. MORSE AND H. FESHBACH, “Methods of Theoretical Physics,” McGraw-Hill, New York, 1953. 17. C. C. Mow AND L. J. MENTE, Dynamic stresses and displacements around cylindrical discontinuities due to a plane harmonic shear wave, J. Appl. Mech. 30 (1963), 598-604. 18. B. NOBLE, Integral equation perturbation methods in low frequency diffraction, in “Electromagnetic Waves” (R. E. Langer, ed.), pp. 323-360, University of Wisconsin Press, WI, 1962. 19. Y. H. PAO AND C. C. Mow, “Diffraction of Elastic Waves and Dynamic Stress Concentrations.,” Crane Russak Publishers, New York, 1973. 20. S. 0. RICE, Diffraction of plane radio waves by a parabolic cylinder-calculation of shadows behind hills, Bell System Tech. J. 23 (1954), 417-504. 21. S. A. THAU AND Y. H. PAO, Diffraction of horizontal shear waves by a parabolic cylinder and dynamic stress concentrations, J. Appl. Mech. 33 (1966), 785-782. 22. S. A. THAU AND Y. H. PAO, Stress intensification near a semi-infinite rigid smooth strip due to diffraction of elastic waves, 1. Appl. Mech. 34 (1967), 119-126. 23. R. Ml. WHITE, Elastic wave scattering at a cylindrical discontinuity in a solid, J. Acoust. Sac. Am. 30 (1958), 771-785. 24. E. T. WHITTAKER AND G. N. WATSON, “A Course of Modern Analysis,” Cambridge University Press, London, 1927. 25. T. YOSHIHARA, A. R. ROBINSON,AND J. L. MERRIT, Interaction of plane elastic waves with an elastic cylindrical shell, Structure Research Series, No. 261, University of Illinois, Urbana, Jan. 1963.
OF PHYSICS
Scattering
91,
l-39
(1975)
of Acoustic, Electromagnetic and Elastic 9-l Waves by Two-Dimensional Obstacles D. L. JAIN AND R. P. KANWAL Department
of Mathematics, University Park,
Pennsylvania Pennsylvania
State 16802
University,
Received September 30, 1974
We present a complete and a simplified discussion of the scattering of acoustic, electromagnetic and earthquake SH waves by various shapes such as infinite cylinders, strips, slits, cracks, cavities, and semiinfinite planes. Formulas are derived for the velocity potentials, electromagnetic fields, displacement fields, far-field amplitudes, scattering cross sections, stresses, and dynamic stress intensity factors.
1. INTRODUCTION Scattering of waves by two-dimensional configurations is of considerable interest in physical sciences. These configurations include infinite cylinders, strips, slits, cracks, cavities, and semiinfinite planes. Attention devoted to them is not entirely based on the physical existence of scatterers such as an infinite cylinder. It is based also on the fact that the solution for an infinite cylinder gives an insight into the solution for a finite cylinder. Our aim is to discuss three kinds of waves-acoustic, electromagnetic, and elastic SH wa.ves. We present the analysis for the scattering of acoustic waves by several perfectly soft and rigid bodies. They are an infinite strip, a slit, a circular cylinder, an elliptic cylinder, a parabolic cylinder and a semiinfinite plane. For the scattering of electromagnetic waves also we consider the same obstacles which are assumed to be perfectly conducting. In the case of elastic SH waves we consider an infinite rigid strip, a circular cylinder, an elliptic cylinder, a parabolic cylinder and a semiinfinite plane and corresponding cracks and cavities of these shapes. These rigid inclusions, cracks, and cavities are embedded in an infinite expanse of a homogeneous and an isotropic elastic medium. Solutions of these problems are known up to various degrees of generality. A survey article on the acoustic and electromagnetic part has been presented by Bouwlsamp [3]. Recently, formulas for the quantities of physical interest in the 1 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
2
JAIN AND KANWAL
case of acoustic and electromagnetic waves have been compiled in [4], where the references to the original papers can be found. The corresponding analysis for the elastic SHwaves is also available to some extent [l, 2, 5-7, 12-15, 17, 19,21-23, 251. However, there are many aspects of the solutions that can be simplified, improved and generalized. For instance, in the case of scattering of elastic SH waves by a rigid elliptic cylinder, or an elliptic cylindrical cavity the solutions are presented in terms of Mathieu functions and it is very cumbersome to derive approximate and asymptotic solutions from this analysis. Moreover, by the method of wave-function expansion it is extremely difficult to deduce quantities of physical interest near the scatterer such as the scattered field and the stress concentrations and, accordingly, very little is known about them. We have succeeded in deriving approximate values of the displacement as well as stresses due to the scattered field in the entire medium. From these results we obtain the dynamic stress intensity factors for the elliptic cylindrical cavity. Similarly, when the scatterer is a parabolic cylinder, which is an appropriate model for a hill [20], the solutions are known for a plane wave. We present the solution for a line source and obtain the case of a plane wave as a limit. Our approach is based on the integral equation formulation and a perturbation technique [I 1, 181 which has already been demonstrated to be very effective for three-dimensional scattering problems [9-l 11. For a parabolic cylinder we present the solution by combining integral equation methods and series expansions in Weber functions. Furthermore, it suffices to present the solutions for elliptic cylinder, a parabolic cylinder, and the corresponding cavities. The solutions for other configurations are derived by taking suitable limits.
2. MATHEMATICAL
FORMULATION
Let (x, y, z) be the Cartesian coordinates such that the z-axis is drawn in the horizontal direction and is parallel to the axis of symmetry of the scatterer. Let r be the section of this scatterer by the vertical plane z = 0 (see Fig. 1). A two-dimensional scattering problem in acoustics consists of finding the scattered field U, exterior to the scatterer when an incident plane wave (throughout our analysis we omit the time factor e-iwt, where w is the frequency), q(x, y) = Aeimx.6 = A exp i[m(x cos y0 + y sin &I,
(2.1)
impinges on it. Here A is a constant, m z ka is a dimensionless parameter, k = w/c, c is the speed of sound and a is the characteristic length of the scatterer, ii is the unit vector normal to the wave front which lies in the xy-plane and makes an angle qa with x-axis. The total field is u = Ui + U, and all these three fields
SCATTERING
OF
WAVES
I x’ FIG. 1. Section of the scattererby the vertical plane z = 0.
satisfy the Helmholtz equation. We shall be concerned with solving this equation for the scattered field U, , that is, (V2 + m”) u,(x) = 0,
x E R,
V-2)
where x = (x, y), R is the region exterior to the scatterer and we have nondimensionalize the equation with the help of the length u. The boundary condition for a perfectly soft body is x E r,
u,(x) = -4(x),
(2.3)
and that for a perfectly rigid one is
f%(x) _
----3
aui(x)
bV
where v is normal to I’ at x. Moreover, condition,
li+i d/p (4$
x
E r,
8V
u, satisfies the Sommerfeld
p = (x” + y2)1/2.
- imu,) = 0,
(2.4)
radiation (2.5)
The integral representation formula for the exterior Dirichlet problem consisting of Eq. (2.2) and (2.3) is [6-8, Ill, 44
= -G/4)
s,
~(xlwf~'(~
I x
-
Xl
I)IX@ ds, ,
(2.6)
where HJ1) is the Hankel function of the first kind, x1 = (xl , yl) is the source point, I(x) = au(x)/& is the single layer density and dsI is the element of length
4
JAIN AND KANWAL
at the point x1 along r. The corresponding integral representation for the exterior Neumann problem (2.2) and (2.4) is u,(~) = $1, J(x,) [*
f@‘(m I x - XI l)]xlEr 4 9
where J(x) = U(X) is the double layer density. The problem of scattering of electromagnetic waves by two-dimensional perfectly conducting bodies can be related to the foregoing scattering problems in acoustics, Indeed, the time-independent parts Eti’ and Hu) of an incident E-polarized monochromatic plane wave propagating along the direction ii perpendicular to the axis of symmetry of the scatterer are (see Fig. 2a) ,(i’(x)
= &z,(i’(x) z
== [email protected];2
= ($)“” (sin q&
- cos q&
(2.8)
1
PX.;.
i
Y vX’ (a) E-polarized x
(i) E
FIG.
(b) H-polarized
2. Electromagnetic
incident waves.
(2.9)
SCATTERING
OF
5
WAVES
Here &, Cy, C, are the unit vectors along the coordinates axes, iI = cos yoC, + sin ~&Z, , m = ka = wa/c, c = (
E’“‘(x) = &El”‘(x),
= _ ;
($)“’
(sz %$
- &, c).
(2.10)
The quantity EAs) satisfies the Helmholtz equation in the region exterior to the scatterer. The boundary condition that the tangential components of the total field E = Eti) + EtS) vanish on the scatterer leads to the relation that Els)(x) = -Eti)(x) on I’. Furthermore, I?:“) satisfies the Sommerfeld radiation condition. Thus, we have the exterior Dirichlet problem equivalent to (2.2), (2.3), and (2.5). For an incident H-polarized electromagnetic wave propagating in the direction fi perpendicular to z-axis, the values of the time-independent parts of Eti) and HCi) are (see Fig. 2b)
@(x) = ; (+)1’2 (vfff’
H’i’(x)
=
@ ffi”(x) z
=
X 5,)
& &mdi z
The scattered field is also H-polarized
so that
E’“‘(x) = ; ($)li2 (vf@‘(x) =
H”‘(x)
;
(2.12)
(q2
(8% 2A!$
X cz) -
c, ?gl),
(2.13)
= --c,H,‘“‘(x),
(2.14)
and His) satisfies the Helmholtz equation in the region lying outside the scatterer. The boundary condition is that the tangential components of the total electric field vanish on the scatterer, i.e., E(x) - 4 = E(x) . (6, x q = 0,
XEF
where 2 and 9 are the unit vectors in the directions of the tangent and the normal to r at x. In view of (2.11) and (2.13), the above relation becomes [v(Hy(x)
+ H:)(x))
x C,] * (C, x q = 0,
x E r.
(2.15)
6
JAIN
AND
KANWAL
Finally, using the identity (A x B) * (C x D) = (A * C)(B . D) - (A . D)(B . C), and the relation (& * G) = 0; (2.15) becomes aJ$ZS(x)/av = -aHii)(x)/&, x E r. Also, Hi”) satisfies the Sommerfeld readiation condition. Hence, we have the exterior Neumann problem equivalent to (2.2), (2.4), and (2.5). We have, thus, proved that the solution for the problem of scattering of E (or H)-polarized electromagnetic wave by a two-dimensional perfectly conducting body, when the wave is propagating in a direction perpendicular to the axis of symmetry, is equivalent to that for the problem of scattering of an acoustic wave by a soft (or rigid) body. For scattering of elastic waves we deal with steady-state Navier-Cauchy equations of elastodynamics valid for a homogeneous and isotropic medium, (A + ,u) grad div U + E~.PLJ + p,02a2U = 0,
(2.16)
which has also been nondimensionalized with the help of the characteristic length a. Here h and p are Lame’s constants, p,, is the density of the medium and U is the time-independent displacement field. Let the displacement field U@) due to the incident plane elastic SH wave be lP(x,
v) = Qi(x,
v) = A&, exp i[m(x cos P)~ + y sin vO)],
(2.17)
where A is a constant, m = ka, k = w/c and c = (,LL/~,,)~/~is the speed of S wave. The field U@) clearly satisfies the Eq. (2.16). The scattered wave is also an SH wave whose displacement field is UYX, Y) = &4x,
Y),
(2.18)
which also satisfies (2.16) in the region lying outside the scatterer. Hence, uS(x, y) satisfies the Helmholtz equation (2.2) with m defined as above. In elastodynamics also we have two kinds of scattering problems. The first kind relates to the rigid inclusions. In this case the boundary condition is that the total displacement vanishes on the surface of the inclusion. This means that u = ui + u, = 0 so that the boundary condition in this case is (2.3). Moreover, u, satisfies the radiation condition. Accordingly, we have the exterior Dirichlet problem consisting of Eqs. (2.2), (2.3), and (2.5). The second kind of scattering problems in elasticity relate to the cracks and cavities embedded in the medium. In this case the boundary condition is that the total stress vanishes on the surface of the crack or the cavity, that is, T(X) * G = 0,
x E r,
(2.19)
where z = T(Z)+ T(S) is the total stress tensor while G) and ~5~) are the stress
7
SCATTERING OF WAVES
tensors for the incident and scattered fields. The nonzero components of &) and ds) are Tfj(X) = p-, ad4 ay
(2.20)
T:(x)= p-.au,(x) ay
(2.21)
and
7(,;(x)= p,,,au,(x) Hence, (2.19) becomes
m4(x> + G41} - 3 = 0,
x Er,
or au,(x) _ ---al,’ av
au,(x)
x E r,
(2.22)
which is the same as (2.4). Also, u,(x) satisfies the radiation condition and we are presented with the exterior Neumann problem consisting of (2.2), (2.4), and (2.5). Thus, the above-mentioned scattering problems in the fields of acoustics, electromagnetism and elastodynamics are equivalent mathematically. In discussing the scattering of acoustic and electromagnetic waves it is the far-field behavior of the field and the scattering cross section which are of fundamental importance. In the case of scattering of elastic waves, we not only need these quantities but also the near-field as well. For instance, in the case of cavities we require the values of the dynamic stress intensity factors. This factor is defined as the ratio of the stress at a point on the cavity to the stress due to the incident field at the same point in the absence of the scatterer. It gives the measure of the stress concentrations at that point. Accordingly, we present the complete solution for the scattering of elastic waves. The solutions for the corresponding problems in acoustics and electromagnetism follow immediately. It is precisely for this reason that we have used the same notation for various quantities such as m, k, and c. Therefore, we shall merely mention the various configurations only in the rest of the sections and it is undlarstood that we are discussing SH waves. When some results for acoustic scattering need an emphasis, we shall mention them.
3. RIGID
ELLIPTIC
CYLINDER
The elliptic cylindrical coordinate system is defined as x = b cash f cos 7,
y = b sinh f sin 7,
z = z,
0 < 6 < co, 0 < 77< 271, (3.1)
8
JAIN AND KANWAL
where b is the distance between the two focii. The element of length ds is ds2 = dx2 + dy2 + dz2 = b2(cosh2 6 - cos2 v)(dt2 + dq2) + dz2.
(3.2)
Thus the scale factor h, , h, , h, for this coordinate system are h, = h, = b(cosh2 .$ - cos2 ~)l/~,
h, = h, = b(cosh25 - cos2$j2,
h, = h, = 1.
(3.3) Next, we nondimensionalize
all lengths with the help of b so that,
x = cash 5 cos q,
y = sinh .$ sin 7.
(3.4)
The fixed elliptic cylinder, the scatterer, is given by 5 = to and its principal axes have lengths 2 cash to and 2 sinh [, . Then from (3.2) we find that the element of length ds along a section z = a constant of this elliptic cylinder and the normal derivative a/an are 1 a ds = (cosh2 .$ - co? r#i2 dq, & = (3.5) (cosh2 e - cos2 ~)r/~ z. The plane SH wave moving along the direction ii impinges on the cylinder so that (see Fig. 3) ii = cos pJ& + sin &, , (3.6) where C, and C+,are unit vectors in the x and y directions and relation (2.9) becomes Ui = Szui , where ui = A exp im(cosh .$ cos 7 cos q~,,+ sinh [ sin r] sin y,,).
(3.7)
The scattered field is U, = QL, where u,([, 7) satisfies the boundary value problem, (V2 + m”) &(5,4 = 0, (3.8) to < 5 --c a, uS([,, ,T) = ---A exp im(cosh to cos 7 cos CJ+,+ sinh f,, sin 77sin CJ+,), 0<7]<<, (3.9) V2 = h2 (5
+ +),
h2 = (cosh2 .$ - cos2 7)-l,
FIG. 3. Elastic SH incident wave.
in = bw(po/p)1/2.
SCATTERING
OF
WAVES
The integral representation formula for u,(& ?) is
where I(771
Applying
=
[ aUy]f=r..
the boundary condition
Xl
=
k-1,
Tl>.
(3.9) to (3.10), we obtain the integral
= A exp im(cosh t,, cos yecos q,, + sinh 5, sin 7 sin q+,), Next we expand I@)(rn 1x - x1 I) in powers of
nz
equation
0 < ?j < 27T. (3.11)
for m < 1,
where
#o(% 71) =p+to- 2 f cz 124
x
ft
(cash nf,, cos nv cos ny, + sinh n[,, sin nq sin no),
&(q:, Q) = - T
[(l + e-“‘“)(l
+ e-2ETcos 277+
cos
(3.13)
- cos(7j - Q)) - 2e-“O cos(vj + Q)
27jIMM7j, Q) - 11,
(3.14)
and p = log(m/4) + y - k-/2.
Furthermore,
(3.15)
we expand the density function I(q) and the right side of (3.11) as
I(rl) = f m”&L(d,
(3.16)
T&=0
and exp im(cosh to cos 7 cos q. + sinh to sin 7 sin I& = 1 + im(cosh e, cos 7 cos q. + sinh E. sin 7 sin qo) - (m2/8)[(cos 2y, + cash 25,) + (1 + cash 2.$, cos 2~~) cos 27 + sinh 2co sin 2q sin 2q~,] + O(m3). (3.17)
10
JAIN
AND
KANWAL
Substituting these expressions in integral equation (3.11) we get simple integral equations in Z,,(q), Z,(q), Z,(q), etc. These equations can be solved by setting Zj(q) = ajo + f
j = 0, 1, 2 )... .
(q, cos nq + b,, sin UT),
(3.18)
n=l
The result is + inz& cos(v, - cpO) in2 (~0s 293, - co& %,I + (p : E > sinh 2c, 1 0 + S(P + to) I *2po -__ (3.19) cos 2(r)l - (PO) 4 I Substituting this value in the formula (3.10) we obtain the value of the scattered field G5 4 u,([, 7) = -A
I(*)
+ inze-(‘-‘0)
x (cash 5, cos 7 cos v. + sinh 5, sin 77sin ?o) - $
I($$-)(cosh sinh 2f
-i
P + to 1
25 + cos 27 + cos 2~0 + y;z)
+ e-2(c-Po)(cosh 2[, cos 2711~cos 27
+ sinh 2(, sin 2~, sin 27)] + O(m3)1,
!t > 40 2
(3.20)
which satisfies the boundary condition (3.9) at t = to . Furthermore,
= -phA
[&
- ime -(E-E”f(cosh to cos 7j cos y.
+ sinh to sin rl sin po) - $ x
I
[(&)
2(p + $) sinh 2c - cash 25‘ + cos 27) + cos 29z+,+
sinh 2fo P + &I I
- 2eP2(‘-“){cosh 2fo cos 2~,, cos 27 + sinh 25, sin 29, sin 27}] + O(m3)1,
(3.21)
SCATTERING
OF
11
WAVES
= A~.~inzlz e-(E-‘O)(cosh E, sin 7 cos q,, - sinh to cos 7 sin F,,) I + f
[(*)
sin 27 + f~?(~-‘~)
x (cash 2f,, cos 2q1,, sin 2~ - sinh 25, sin 2~” cos 241 + O(mz)/, (3.22) which are valid in the region 5 > &, . Far-field amplitude and scattering cross section. The far-field behavior of the scattered field follows from the formula (3.10) by using the asymptotic value of the Hankel function f$)(m
1x - xl I) E (&)1’2
ei(mo-n/Q
x exp -inz(cosh [r cos y cos Q $ sinh El sin CJIsin 73, (3.23) where we have set x = (p cos cp,p sin y),
x1 = (cash [I cos rI , sinh ,$I sin Q).
(3.24)
Thus, u,((, 7) ‘v (-$J””
ei(mp--n/4)Al(rp),
(3.25)
where
A,(y) ==- f Jo2nexp -itn(cosh f,, cos v cos Q + sinh to sin v sin Q) I(Q) dql
’ == A 5 +
1 I p + 5.
qp
1 +
to)2
1
1
- 4 m2 c 2(P + 50) (cos 2% + cos 2y) sinh 2&l
1
+ 2eEo(cosh to cos ~~ cos q + sinh co sin v. sin q) + O(m4)\.
(3.26)
Hence, the scattering cross section is S.C.S. =
2b mqlA12
so2n I A,(d2
dg?
= -!I!?!? mZs2 11 - i m2 [cos 29,, + &
(sinh 2fo)(log(me’0/4)
+ y)] + @ml)/ , (3.27)
12
JAIN
AND
KANWAL
z,2 = 4 [log ($)
(3.28)
+ YIP + 572.
We now derive the corresponding formulas for the limiting
configurations.
A. Infinite Strip When &, --+ 0, the elliptic cylinder reduces to a rigid strip of width 2b which occupies the region -b ,( x < b, y = 0, -co < z < co. Then the values of the displacement field u, , the stress field TV, the far-field amplitude A,(v) and the scattering cross section follow from relations (3.20), (3.21), (3.22), (3.26) and (3.27) to be
= -A
(( 1 + b [) + ime-’ cos 7j cos fpO
-$[(
1 + j‘) 5 (cash 25 + cos 27 + cos 2~~) - j sinh 2f
+ eC2’ cos 2~, cos 2~1 + O(m3)/,
06f
0<77<2r, (3.29)
P(Y cc?.. >y) = [&f * 2 $1 E.=0vx -
[T% dl+o vy , 0 < t < a, 0 < 7 < 237,
7yx =0 VY+ yz 3 y) = [Tj”‘(5 2 7 v)].c,J
kk
(3.30)
dl vx 9
O<(
o
where
bk)(t?
v&,=0
=
-
(Cosh2
-- 1’
b2(5> rl)lto=o=
[i
PA _
cos2 rl)l,2
]j-
-
ime-’
‘OS
7
‘OS
v0
{2(p + f) sinh 2f - cash 2.$ + cos 2~) + cos 2?,}
- 2ev2’ cos 2~, cos 27 I pimA e-’ sin 7 cos ~~ (cosh2 5 - COG $1/z I + $
vx -
f
[$ (p + 5) + eC2’ cos 2po] sin 27) + O(m2)1,
10 sinh 5 cos 7 (cash 25 - cos 2~)l/~ ’
vy =
~‘2 cash .$ sin q (co& 2,$ - cos 241’2 ’
(3’32)
SCATTERING
OF
13
WAVES
In particular,
i I -A
t&(x:, 0) =
1 + imx cos C& - q
\ -A
I(1 +;
x2 COG qJ0+ O(m”) , 1x 1 < 1. I
cash-l ) x ) + im(x - sgn x(x’ - I)‘/“) cos q. 1
-- m2 1 + j cosl1-1 / x I) (2x2 + cos 2cp,) - $ ( x / (x2 - 1y/* 8 [( i
~+(2r2-1-21X1(.~2-l)~~~)COS2y0]+o(m~)~,
Ixj>l, (3.33)
u,(O, y) = ---A I( 1 + i sinh-l 1y I) -_
99t*
[(I + j sinh-l / y I) (2y* + cos 2qo) - f / y 1 (1 + yy*
8
- (2y2 + 1 - 2 I y I (1 + y”)““) cos 2g,o] + O(m3$ ---co
1 -Apim T~(X, 0) = I
-A,u
cos qo{l + imx cos cpo + O(m2)},
[(P + cash-l I x I) 2x - i - l)sgnx ((';i2
_
-&O,
-
1)1/2
I$
- 2(2-U” -
\O,
cos 2%] + ow$
- imx cm q. - $-
1) COS 2~,] + O(m3)/,
1 wx I x I > 1,
[ +- (2.9 - 2 + cos 2~~)
1x1
IYI (1 + yy -_ t2
2(Y2
24
IX/ (x” - 1)1/2 ) I cos2qJ,, (9 - 1)1/z - -2(x2 - ])1/2
lx/ > 1,
y) = -,~~irnA I it1
-
1x1
(3.35)
F (1 “A,2)l,e 9(x YZ 7 o&-t) =
(3.34)
sgn x j + in2 cos po 1 l p(x” - 1)1/Z (
- $ i - p
a,
--co
1
< y < co, (3.37)
[$ f4(p + sinh-l I y I) I y 1 (1 + y2)*/*
+ 1) + cos 2%)
+ 2(2Y2 + 1 - 2 I y I (1 + y2)l’2) cos 2p70] + O(m3)/, --a
< J’ < co.
(3.38)
14
JAIN
AND
KANWAL
Similarly,
(3.39)
S.C.S. = -y52;
11 - $
(3.40)
cos 2q$l + o@n‘q,
where Z12 = 4{log(m/4) + r}” + n2. The corresponding results for the scattering of acoustic waves by a slit in an infinite rigid plane screen follow from the above analysis by using Babinet’s principle. The same is true for the scattering of E-polarized electromagnetic waves by a slit in an infinite conducting plane screen. B. Rigid Circular Cylinder When b + 0, to + co, such that b cash to + a, b sinh to -+ a, the elliptic cylinder reduces to the circular cylinder of radius a. Since 4 > to, .$ also tends to infinity and b cash 5 -+ ap, b sinh 5 -+ ap, q -+ q~, where (p, y) are polar coordinates of the field point (E, 7) in the limit such that p is nondimensionalized with the help of a. Taking these limits in the formula (3.20) for the scattered field, we obtain %(P, ~1 =
-A -
ii1 + i log p) + $ P2 + +s
qg, -
po]
COS(P- To) - g + Wm3),
p z
[(q + log p) (p2 + ;)
(3.41)
1,
where q=logT+y--,
nz = aw(po/p)1/2.
(3.42)
From this relation we readily obtain the nonzero components riz and T.& of the stress tensor associated with the scattered field,
rzz
-4
]$
-
+ (p + $1
+os(FJ - 3
cm 2(p,
vo) - g
-
poll-
[2Pb
+
O(m?j,
-
1 + logp)
(3.43)
SCATTERING
OF
15
WAVES
and
= 9
m [sin(y -
‘pO) + $
sin 2(9, - ‘pO) + OC,rZ,].
(3.44)
These stress components can also be obtained from (3.21) and (3.22) by taking the limits as explained above. Similarly, the values of the far-field amplitude and the scattering cross section follow from (3.26) and (3.27) to be A,(y) = y
jl -;$mq;+44
cos(g, -
%I]
+ o(m3/,
(3.45)
and S.C.S. = g
[I -
2nr2(yy
+ r) + O(n12)],
(3.46)
2
where zz2 = [4(log(m/2) + 7)” + n”]. 4. ELLIPTIC CYLINDRICAL CAVITY
We flollow the notation of the previous section so that the time-independent part of the incident field is Ui = &(f, T>, where ui = A exp inz[cosh 5 cos 7 cos vO + sinh [ sin 71sin ~~1.
(4.0
The scattered field zr,(f, 7) satisfies the boundary value problem P2 + m”> ~,(f, ‘I) = 0, ~EGl
2 7) = ~&o
3 4 = ~&od
E, < t < 03, = 0,
0 < 7) <27-r.
(4.2) (4.3)
The nonzero components of the stress tensors G and + associated with the incident and scattered field are a& & == $1 x = ipAnA(sinh x
5 cos q cos cp,,+ cash 5 sin 17sin yO)
exp im[cosh 5 cos 7 cos qO f
sinh 6 sin r) sin ~~1, (4.4)
’ = ph s = -ipAmh(cosh 5 sin 77cos qO - sinh 5 cos 7 sin yO) Tnz 877 x exp im(cosh 5 cos 77cos q,, + sinh [ sin 77sin q& q
(4.5) (4.6)
16
JAIN
AND
KANWAL
where h = (cosh2 [ - cos2 q)- rf2. From relations (4.3) to (4.6) it follows that the boundary condition is
W&77)/x- = 0,
4 = t-0,
0 e r) = 2n-.
(4.7)
The integral representation formula for u,([, q) is
where J(q) = u(&, , 7). From the expansion (3.12) for the Hankel function, we find that
I where @,(I$, ?j, 7jJ = f
P-J
cos n(?j - 7jl) -
?Z=l
jJ e-n(c+co) cos n(7j + Q) n=1
= 2 f
e-“‘[cash n&, sin nq sin nq, + sinh n& cos nq cos qll,
Tb=l
(4.10) and Q1([, v, Q) = $[{sinh 26, - 2 cash 5 sinh 5, cos 7 cos qI - 2 sinh [ cash 5, sin 17sin Q} p + 5 - + - 2 f I
x
Substituting
[cash n&, cos nq cos q
G
Vl=l
+ sinh a$,, sin nr) sin nql] . I
(4.9) in (4.8) and letting t + t,-, , we obtain
(4.11)
SCATTERING
OF
17
WAVES
where x0(7, 71) = -
4 -
f
(4.13)
e-2nSo cos “(7 + ql),
?k=l
X1(%71)= Pl(5, 72%)lP=to = & [sinh 2,$,,(1 - cos(r) - &](p I
+ & -
l/2 - 2 f
@“‘O/n)
n=l
;< (cash nf,, cos ny cos nql + sinh n[,, sin r17 sin HQ) By virtue of Poisson’s
summation
(4.14)
formula
n=-m
?L=l
relation (4.12) yields, 450 3 4 = 9417) + (1/W
/02r
J(?h)[Xo(%
71)
+
1"2x1(%
71)
- Qb4)1 4 3
0 < 7) < 2n.
(4.15)
Finally, we use the relation
= J(q) - A exp im(cosh f,, cos 7 cos qO + sinh 5, sin v sin v,,), in (4.15) and derive the Fredholm
integral equation of the second kind
J(v) = 2A exp z+n[cosh e, cos 77cos CJ+,+ sinh f,, sin 77sin CJ+,] + (l/d
J2 J(rll)[XO(% rll) + 0
m2x1(777 71)+ W~31 61 9
0 < ?j < 2n. (4.16)
We solve this integral equation in precisely the same manner as we solved the Eq. (3.11). We spare the reader of the details. The result is J(q) = A(1 + imecO cos(q -
(m2/8)[(cosh
vo)
25, - 2(p + 8,) sinh 2(, + cos 2~~)
+ cos 27j + e2’0 cos 2(77 -
yo)] + O(nz3)}.
(4.17)
18
JAIN
AND
KANWAL
Next, we substitute this value in the integral representation obtain the value of the scattered field, u,(f, T) = iAm
I
formula
(4.8) and
e-(E-EO’(sinh [, cos 7 cos y0 -+ cash [, sin 7 sin vO)
je-B(C-PO)
+
m[sinh 2c0 cos 2q,, cos 27 + cash 2(, sin 271, sin 2771
8
- 6 sinh 25,&p
[I + O(m21!, 5 > 5, .
+
(4.18)
Stresses. The formulas for the non-zero components T& and T,“~ of the stress tensor follow from relations (4.6) and (4.18) to be T&($, v) = -i~LhAn2{e-‘5-Eo’(sinh + (im/4) e-‘(‘-@[sinh
5, cos v cos ?. + cash E, sin 7 sin v,,) 2f,, cos 2171,cos 2r) + cash 2& sin 271, sin 271
+ O(m2)1, E-+ 6, ,
+ (im/4) sin X0
r,$(t, 7) = -i~LlzANz[e-‘E-‘a’(sinh
(4.19)
.$, cos q,, sin 7 - cash 5, sin ‘p,, cos 7)
+ (im/4) e- “(‘-‘O)(sinh 2t, cos 293, sin 27 - cash 2[, sin 27+, cos 27)
+ anz2)1,
f > 5, ’
The above values of stresses satisfy the boundary Furthermore, from relation (4.1) we have &(.$, 7) = -ipLhAm[(cosh
(4.20) condition
(3.3) for [ = f,, .
6 sin 71cos q+-,- sinh [ cos 7 sin p,,)
+ (im/4)(sin 27 + cash 25 sin 277 cos 27~,, - sinh 2f cos 277sin 2q7,,) (4.21)
+ O(m2)1, The addition of (4.20) and (4.21) gives 7,&,
q) = -iphAm{[(e-(5-50)
sinh E,, + cash f) cos y0 sin q
- (e-(S-E0) cash So + sinh 5) sin CJI,,cos q] + (jm/W
1 + (e- 2(c--c0)sinh 25, + cash 2.5) cos 2~~) sin 27
- (eP2(‘-‘O) cash 2t0 + sinh 20 sin 2q1,,cos 291 + O&2)}, The only nonzero component
t > to, (4.22)
of the total stress tensor at a point (to, q), on the
SCATTERING
OF
WAVES
19
surface of the cavity, is 711z. Hence the value of the dynamic stress intensity factor N,, is T,t(L d Nn3 = lim t-+50+[ - ipAm I = h, eEosin(r) - y,,) f :’ i
[sin 2~ + eztOsin 2(7 - q+)] + O(WZ~)/, (4.23)
where h,, is the value of h evaluated at the point (5, , 7). Far-jield amplitude and scattering cross section. We apply the asymptotic formula
2 112 = -1171 ,( ~TrnP 1 ei(“~~-“J4)exp -irn(cosh x
f, cos g, cos Q + sinh .$,,sin p sin qI)
(sinh 5, cos 9 cos yl + cash 5, sin cp sin qr),
where 1; = (p cos F, p sin y), xl = (cash [I cos Q, sinh e1 sin Q), in the integral representation formula (4.8) and find that u,([, 27)= ($)l”
The value of the amplitude -4d~)
ei(me-n/4)A2(q).
A,(v) is
= T jo2n J(rlJ(sinh 5,cos g, cos y1 + x
exp -im(cosh
= - y
cash fO sin v sin ql)
to cos g, cos ?I + sinh 5, sin 9 sin v,I~)dql
[sinh 2(, - 2eE0(sinh [,, cos q~cos 9?0
+ cash 5, sin cp sin y,,) + O(M~)].
(4.25)
The value of the scattering cross section is
S.C.S.:= 2b j"" ~mlAla
:= F Let us
now
o
I A2(~)1~
dg,
[sinh’ 2.$, + 2e2’O(sinh2 f,, cos2 q. + cash’ & sin’ v,,) + O(W?)]. (4.26) attend to the limiting configurations.
20
JAIN
AND
KANWAL
A. GrijEth Crack When to + 0, the elliptic cylindrical cavity reduces to the Griffith crack which occupies the region -b < x < b, y = 0, - co -C z -=c 03. Then the formulas (4.18) to (4.20) for the displacement field and stress components reduce to
fdx,
Y> =
M5, .)7)1+rJ
= iAm
I
e-’ sin 7 sin pO + 7
e-” sin 27 sin 2y, + O(f7zz)/,
0 < 5 < ~0, 0 < 7 < 2~r, (4.27)
TXX, Y> = [TX53rl)lE,=O v, - bL(E7?l>le,=o y11 ,
o
0<7jd2n,
(4.28)
where v, and vy are defined in (3.32) and [T&([,
q)]ro=o = -iphAm
[T$@,
q)]+.
/e@ sin 7 sin q+, f $
= iphAm /e-e cos 17sin yO + q
In particular,
fiAm(1 i 0,
- x2)1/2 sin qO 1 + I
IxI>
+
=
(1 - 9)1/2
0,
0) =
~40,
Y>
/x+9(2x2-l)cosp,+O(m2)~,
Ixl
(4.31)
1x1 > 1,
-ipAm
sin q,,{ 1 + imx cos y+ + O(m2)},
-ipAm
sin q~,, 1 I(
+ im (x Similarly,
(4.30)
1%
ipAm sin q,,
Tiz(X,
e@f cos 2q sin 2~p,, + O(n?)l.
the values of these quantities at the faces of the crack are
%(X, @t> =
T;;(X, Of)
e-25 sin 2~ sin 29, + O(m2)/,
(9
‘!I? ;)I,2
(2x2 - 1) sgn x 2+x2
_
1)1/2
IX < 1,
1
)co~~~+O(m~)~,
Ix1 >l.
(4.32)
the values of these quantities in the plane x = 0, are
= iAm sin v. sgn y{(l
pAm2
ddo, Y> = 4
2y
_
+ y2)l12 -
CO2 + 1) w (1 + yy2
I y I) + O(m2>}, Y ] sin 2yo + O(m)/,
--oo
< y < a.
(4.33)
SCATTERING
Ml
Y) = -ipAm [l -
(1 yd2)l,2 ]
OF
21
WAVES
sin y, + O(m2) ,
I
---co
y < co. (4.34)
Since, ui(x, JJ) = A exp im(x cos q+, + y sin plO), we have
*rtz(x,0) = P [$$ (x3Y)]g=o = ipAm sin y,,{l + imx cos q,, + O(m2)>,
1x1
>. 1.
(4.35)
Adding the second part of (4.32) and (4.35) we obtain (2x2 - 1) sgn x cos vO + O(m2) 2(x2 - 1)1/Z
T~Jx, 0) = ipAm sin CJI,,
’ x j > 1.
(4.36)
t ;]
(4.37)
The dynamic stressesat the edgesof the crack are IV,+, = lim(j x 1 - 1)1/2[ T$:z)] =- ““;p
[1 & y
Id>1 ’
x -+ [1':
cos yO + O(m2)\.
(4.38)
The values of the far-field amplitude and scattering cross section are A,(Y)
=
iArm 4
sin CJIf m2 ,sm v, + 16 [sin 2g, cos v - 2 sin vO cos2 F
- (2 cm2 y. + 4p - 3)l + O(m2)/,
s c s = bm37r2 sin v. . . .
8 t
I
(4.39)
sin v. - +6 17t2[sin~~ + 2(cos 2~,, - 2
4@x(m/4) + 241 + Wan”)/.
(4.40)
The first term in each of the above formulas can be obtained as limits from relation (4.25) and (4.26). We have evaluated them independently to give the higher order terms since the analysis in this case is simpler. The corresponding results for the scattering of acoustic waves by a slit in an infinite solft plane screen follow by Babinet’s principle. The sameis true for the scatteri:ng of H-polarized electromagnetic waves by a slit in an infinite conducting plane screen.
22
JAIN
AND
KANWAL
B. CircuIar Cylindrical Cavity Let b-+0, to-+ co in such a way that b cash &,-+ a, b sinh [,, --t a and the elliptic cylindrical cavity becomes circular cylindrical cavity of radius a. Since 8 > to , 5 also tends to infinity so that b cash [ + ap, b sinh 5 + ap, 7 -+ y, where (p, y) are the polar coordinates as explained in Section 3 for the case of a rigid circmar cylinder. Taking these limits in formula (4.18), we obtain the scattered field u,(p, y) = iAm if cos(g, - vO>- f -
$
[(i
(2q + ;j
[2q + 2 log p - $ cos 2(9, - PO)]
+ 2p (4 - ; + 1% Pjj 4g,
-
%>
(4.41)
p > 1,
where q and m are defined in (3.42) and we have included the term of order m3 by following the previous steps independently for this simple limiting case. Same is true for the following quantities of physical interest. The stress components ~pS~(p, y) and ~,“,(p, y) are
= - y
jcos(p: - pJ + 9
‘a2 [(29 + ; -
+ $ cos3(g, -
[p + ; cos 2(p, - %)I
2p2 (4 + ; + log Pjj
WJ] + OW) f ,
cos(g, -
%I
p > 1)
(4.42)
and g012 (p 3 ‘p) = .P- %(PT F> P
89
ipAm
= - 2 -
$
+ &
P
t
sin(p, -
[((2q
+ i)
sin 3(9, -
vO) + E!- sin 2(p, 2P
+ 2p2 (4 A]
+ O(m31
I&
& + log p)) sin(y p > 1.
-
94 (4.43)
SCATTERING
= --ipAm
t
- T
OF
23
WAVES
sin(g, - yO) + y
p sin 2(~ - ?J
[sin(g, - I& + sin 3((p - qO)] + O(m3)j.
(4.44)
Therefore, Tmzh VI = --ipAm
](I + $)
- -$- [($ + k[$-
sin(p, - yo) + f
(p + f)
(24 + i) + 2 (q - k + log p + g)
+ p2] sin 3(g, - vol] + Ob3)l,
sin 2(9, - p)J sin(g, - q+)
p > 1,
(4.45)
and the value of the stress intensity factor N,, is
=
[
2 sin(g, -
- q
q$) + in2 sin 2(g, - vO)
[( 4q + 35) sin(g, - d
+ sin 3(9 - v,)] + OWI].
(4.46)
The first two terms in this expression for N,, can also be directly obtained from formula. (4.23) by taking the appropriate limits as explained above. Similarly, the values of the far-field amplitude A,(F) and the scattering cross section in this case are A2(g?) := - q
I(; - cos(p, - q+i) + $
x cos(g,- PO)-
2
[(4q -
03s2(9, - %)I + ow$
3) + (8q + 10) (4.47)
and (4.48)
24
JAIN AND KANWAL 5. LINE SOURCE PLACED
IN FRONT OF A RIGID PARABOLIC
CYLINDER
In view of Fig. 1, we take the z-axis parallel to the generators of the cylinder, such that the parabolic cylindrical coordinates are x = gg2 - q2),
Y = &I,
z = z,
--00 < 6 < co, 0 d 7 < co, (5.1)
The cylinder is defined by 7 = M, so that the coordinates on its surface are x = $([” - M2),
Y=tM
z = z,
--co<‘$
(5.2)
and the parabolic section in the xy-plane has the equation y2 = 2fw(x + (W/2)).
(5.3)
Thus, the latus rectum has the length 2M2 while the focus and the vertix are at the points O(0, 0) and V(--M2/2, 0) (see Fig. 4). The polar coordinates (p, y) of the field point ([, 7) are P = 3(5” + T2),
91= 2 tan-‘(r/&J).
(5.4)
FIG. 4. Section of the parabolic cylinder and the line source by the vertical plane z = 0.
It is convenient to nondimensionalize all lengths by the semilatus rectum A42 so that over the cylinder 77= 1, -co < 5 < co. The value of ~~(6,v) is (5.5) which is a line source of strength A parallel to the z-axis placed in front of the cylinder so that it intersects the xy-plane at the point P,, , x,, = (St, , r],,). The
SCATTERING
OF
25
WAVES
field point is, as usual, x = (6, r)). The parameter m is equal to kM2 which is (~cIJ~M~/cL)~/~for the case of elastic waves and wM~/c in acoustics. When we expand EQ) in terms of parabolic coordinates of the field point ([, 7) and the source point (f,, , q,) we have [16]l
where q<: = minimum (7, v,,); 7, = maximum (7, Q). The quantities Bm(z) and g-,-,(z) are Weber functions of the first and the second kind, respectively, g?‘,(z) = (-1)” B-+,(z)
/‘4
= 42 q n.
!&
(p2’2),
e?*‘4 f&
n 2 0. FZ> 0,
[ezzi2 erfc ($1,
while X := (-2im)lj2 and /i is its complex conjugate. The boundary value problem for ~~(4, 7) in this coordinate form (V2 + m2) f4f, d = 0, q > 1, %(f, 7) = -45,
$3
--co
(5.8)
system, takes the (5.9)
r = 1,
(5.10)
where
The integral representation outside the parabola is
formula for u,(f, 7) at a fixed point x = @, 7) lying
udf, 4 = - $ Jvrn m %Afd%
I x - x1 l)>nl=ldtl ,
17> 1,
(5.11)
where xzl = (tl , Q); and
1 There is a slight error in the formula given by Morse and Feshback which we have corrected in this relation.
26
JAIN
AND
KANWAL
Applying the boundary condition (5.10) to (5.11) we get the Fredholm equation of the first kind for the density function,
s -1
GWc%~
I x
-
Xl
integral
I)17J=n,=l &I = A[H$% I x - x0 l)ln=l) --co < c$< co. (5.13)
Next we use the expansion (5.6) of HA1) in terms of Weber functions and set
m = f an~,@5>,
(5.14)
7L=O
and use the orthogonality
condition (5.15)
Then the integral equation (5.13) yields the value of the coefficients a,, as
Thus, the value of the density function is completely known. Finally, we substitute this value of I(t) in formula (5.11) and derive the value
-q > 1, (5.16) where we have used the orthogonality condition (5.15). This obviously satisfies the boundary condition (5.10). With the help of this relation and (5.6) we can evaluate the total stress components (5.17) Plane wave. When the source PO + cc along the line OP, , 1 f. I--+ co, y. + co. In terms of its polar coordinates (p. , qo) p. + co, but v. = 2 tan-l(~o/~o) remains constant because PO + co along the line OP, so that
80 = (2PoY KS yo/2,
7jo = (2~,)l/~ sin ~~12.
(5.18)
SCATTERING
OF
27
WAVES
When we use the polar coordinates for both x and x, , then we find that for large p,, ,
Accordingly,
we take the constant factor A in such a way that
Then (5.5) reduces to the plane wave ui(p,
v)
=
e-i~m3s(~-~o)
=
eim~cos(~-d
=
eim(zcos~+tdkd,
(5.21)
where we have set q+, = n + 01.Thus, we have a plane wave which is moving in the direction which makes an angle 01with the x-axis (see Fig. 5). To obtain the corresponding asymptotic value of relation (5.6) we use the asymptotic formulas for the Weber functions [24] LBn(X&J = Qn (-2 -
(-ipom)1/2
ei~omsin2(a/2)
(
sin 5)
-2 (-ipomz)1’2 sin f)“,
I a I < r/2,
(5.22)
1a I < 77/2.
(5.23)
I 01/ < z-/2.
(5.24)
/ a 1 < 7r/2.
(5.25)
and %n-1(X7jo)
= ~+, exe
(2(--ipom)“~
ic+cos%/2
(
cos +)
2(--ipom)l’2
cos f)
--n-l
,
Then
~a,G%J%-l(GJ
c-e
eiDom(- 1)” tann cyj2 set olj2 2( - ipom)lj2 ’
Then from (5.1), (5.6), (5.20), (5.21), and (5.24) we have u&f, 7) = exp im[$(t2 - +) cos OL+ 57 sin LX] = set +
f
in q
9,(hf)
Lq$i?jQ,
?l=O
Similarly, we can derive the expression for ui(<, 7) when the angle of incidence 01 lies between z-12 and 3rr/2, but we shall discuss the case when 1 01I < rrj2.
28
JAIN AND KANWAL
FIG. 5. Geometry for the plane waves incident on a parabolic cylinder.
The corresponding limit for the scattered field u,(t, 7) is obtained in the same way. Indeed, using (5.20) and (5.24) we find from (5.16) that II > 1.
(5.26)
The result of adding (5.25) and (5.26) is
which obviously satisfy the boundary condition (5. IO) at 7 = 1. To expand the values of u in powers of m for vz < 1, we need the expansions
s2n(Z) = (- 1)” @I ! 2”n! [I - (n+ t, zq+ Wz4), g2n+l(z) = (-11% P2% !+ l)! z + (qz3) 3 B-212-l(z) = (g2
&
- Jg
+ (g”
@ ;j;
(5.28) (5.29) z2 + U(z3),
(5.30)
and a-2,-2(z)
=
(2n2y1)!
-
_-7r lj2 z ( 2 > 2% ! + tn + ;,
2
(2;y
‘I)!
+ U(z3).
(5.31)
SCATTERING
OF
29
WAVES
Then relation (5.17) yields ~(4, rl) = im(~ -
1) [-2
(&jr”
cos 5 + 5 sin cy. (-4
tan2 +)”
+ c+P)],
77 3 1. (5.32)
The values of the nonzero stress components are 7&,
(5.33)
7) = $2 -& = @&z(n - l)[sin 01+ 0(m1/2)],
and T&,
77) = ph $-
= pimh [-2
(&)l”
cos T + [ sin cr (-4
tan2 F)n + O(m1/2)],
(5.34)
which are valid in the region r] > 1. Line source in front of a rigid half plane. The parabolic cylinder given by Eqs. (5.2) reduces to a rigid half-plane x > 0, y = 0, z = z, when A4 - 0. Since there is no characteristic length we deal with the wave number k. The incident displacement field due to line source of strength A, palced at the point I’,(&, QJ which does not lie on the half plane (Q, > 0), is (see Fig. 6) u&S 7) = (i/4) A fft’(k - &n;o
Ix -
m (-i)” 7 n.
xo
I)
Sn(do)
~,(a3
9n(Q<)
~-n-l(~rl>)~
(5.35)
p,(Eo.To ) FIG.
z = 0.
6. Ox and P0 are the sections of the half-plane and the line source by the vertical plane
30
JAIN
AND
KANWAL
where (T = (-2ik)l12, and 0 is its complex conjugate. The solution u,([, q) for this case is obtained from solutions (5.16) by first writing it in its proper dimensions, namely,
and then letting M--f 0. Since from (5.28) to (5.31) 92n+m
=
S2JO)
0,
=
(-;Jny
)
9-2n-1(0)
=
(i)li2
A)
the required limit is %(5,
rl)
=
-
4
f ?I=0
~2n(4392n(d
~-27&d
g-zn-l(T).
(5.37)
Plane wme. When we take the limits of relations (5.35) and (5.37) as \ 5, \ -+ co, ~7~+ co such that p. -+ co while ro/fo remains constant and equal to tan yo/2, and ‘I2
e i(ko o-?r/4) -+ 1,
(5.38)
we obtain ui((, 7) = exp ik(x cos a + y sin cz) = exp ik[$(t2 - 7”) cos a + & sin LX]
and u,(E, 7) = - (4)“”
set E nto tan2” 2 g2,(u5) B-2n-1(u$,
(5.40)
where we have used the result (5.24) and q. = n + a, 1 a! I < z-/2. Relation (5.40) gives the scattered field when a plane wave moving in the direction perpendicular to z-axis and making an angle CYwith x-axis impinges on the rigid half-plane x > 0, y = 0, z = z. It is convenient to write the left side of (5.39) as a product of two Weber functions. This is achieved with the help of relation go(z) = e-z2/4 and we have u&f, q) = exp ik[+(t2 - +‘) cos a + (17 sin CY]
SCATTERING
31
OF WAVES
Next we use the identity (2+/z 530(z) = %,(iz)
+ .sl(-iz),
the abovle formula becomes
X go (J f cos -5 + 71sin + [ ( 2
)I,
I a! I < 7T/2.
(5.42)
Let us put the formula (5.40) in this form by appealing to the relation
as well a.sthe one obtained from it by changing 01to -01, namely, gz [CT(7j cos $ - 5 sin +j] =
(COS
T)” j.
‘! ‘i:ii
TSO[CT(c cos 2 + 7 sin T)] FEyF’2)
where we have used the identity relation (5.40) is simplified to u,(L q> = - -(27&
t go
[i
~n(uLJ) L@-n(aq),
9,,(-z)
j cdj < 77/2, (5.44)
= g,,(z). By adding (5.43) and (5.44),
a 7j sin 5 - .$ cos f )]~-l[~(~cos~+&sin~)]
+ L?20[IJ (5 cos + -I- r sin +j]
%,
[CT(7 cos 4 - 4 sin T)] 1, 1 011 < z-/2. (5.45)
Finally, we add (5.42) and (5.45) and derive,
- go [U (7 sin $ - E cos %)] 9-l
[U (7 cos 4 + E sin T)] 1, 1 a / < z-/2. (5.46)
595/91/I-3
32
JAIN
AND
KANWAL
To derive the famous Sommerfeld formula we use the identities _
go(&) = pw,
.9’-1(d)
e--ikt2/2
-6t
= 1/2 t/i
s
eia2
d BY
--m
(5.47)
where t is real. Then (5.46) becomes 4cY ?I) = ($1,2
I
exp ik[&(t2 - 77”)cos 01+ EV sin
-~~(fsin(./2)-~C0S(~i2)leis2 X
al
dp,
s-72
- exp ik[$(k” - q”) cos a: - &j sin a] -z/lc?Ssin(a/2)+rlcos(njz))
X
ei@ d/3].
s -m
(5.48)
In terms of polar coordinates .$ = (2p)li2 cos p/2, 7 = (2p)1/2 sin v/2, the above relation becomes the Sommerfeld formula (2ok)%n((m-a)/2)
e@ d/3
eikpcos(m-~)s --m [ (ZdPi2Sin((m+a)/2) _
ei@ d/l].
eikoCos(~+a)
s --m
Stresses on the faces of the half-plane. identities
(5.49)
By applying the formula (5.17) and the
g,‘(z) = - g 9,(z) + ng),-,(z) = ;i gn(z) - ~n+l(z),
(5.50)
and (27r)1’2 q)(z)
= 9Ll(iZ)
(5.51)
+ z%l( AZ),
we obtain from (5.46) the values of the stresses on the upper and low faces x > 0, y = Ok, z = z of the half-plane. They are
= p (-$)1’2
x go [
ik sin 01 1~3~~[a 1 5 1 sin T]
- (0 / [ 1 sin $)-’
I 4> 0,
a.$ cos + ,
(5.52)
SCATTERING
OF
33
WAVES
and
zz
-[T&X,
O+)]e,o
+ 2(ikp sin 01)go [65 sin 51
t < 0.
:< q)
(5.53)
The low-frequency approximation for the displacement field and the stresses for the present case can be obtained from the corresponding formulas (5.32) to (5.34) for the parabolic cylinder. For this purpose, we first write them in the following forms by setting m = KM2 and introducing the original values of the coordina.tes f, 7 with proper dimensions, u(f, 97) = ik(7j -
x g T&
___ g;;
*7) = (p f’~2jl,2 T,;
=
(&)“”
M) [-2
Pik
(-4
cos T + ( sin 01- ?
tan2 5)”
set 5
+ O(W2)],
(5.54) (5.55)
(7 - M)[sin x + O(W)], [-2
(J&y’
cos f
(-4
tan2 T)”
+ O(k1!2)].
Cc?”+ q2)1’2
4M
+ c sin 01 - -
77
set E2
(5.56)
Then by letting A4 -+ 0, we obtain the formulas for the half-plane in the limit. They are, in polar coordinates, u(p, 71) = 2ikp sin 22 [ -
(-&-)I’”
cos q + cos T sin a: +- O((kp)1/2)], (5.57)
TJP, ~1, = pik [sin 5 sin 01+ O((kp)‘/“)],
(5.58)
and Q-&P, p’) = pik [cos 5 sin 01 -
for kp<< 1.
($)1’2
cos $- + O((kp)llz)],
(5.59)
34
JAIN AND KANWAL
6. PARABOLIC CYLINDRICAL
CAVITY
Let us now consider a hollow parabolic cylinder whose surface is stress free. In terms of the notation and configuration of the previous section the incident wave ui due to a line source at x0 = (& , T,,) is given by relation (5.6). The scattered field U, satisfies the boundary value problem
-cc
< f < 03, (6.2)
where we have used the expansion for the Hankel function as given by (5.6). The integral representation formula for u,([, 77)is
where the double layer density is J(t) = ~(6, 1). To obtain the required integral equation we substitute the Weber function expansion for Hi11 in the integrand, differentiate both sides of (6.3) with respect to 77,let 7 --f I and use the boundary condition (6.2). The result is
=-&I$
~?&@0)~n@O%‘(a ~-n-db>,
--co<(
?I=0
(6.4) We solve this integral in the same manner as Eq. (5.13). Indeed we set
use the orthogonality
property (5.15) and find that (6.6)
SCATTERING
OF
35
WAVES
The value of the scattered field u.& 7) is obtained from (6.3), (6.5), and (6.6),
rl > 1, --a3 < ( < GO, (6.7) which obviously satisfies the boundary condition (6.2) for 77 = 1. The total field as well as the total stresses can now be readily calculated. Plane wave. Taking precisely the same limits as we did in the corresponding situation. in the previous section, we derive from relations (5.1) and (6.7), the values of the incident field for a plane wave
~~(5, 77) = exp inz(x cos 01+ y sin LX) = set $ C 2 (tan f)n n=o .
Bn(h.$) L@J&),
(6.8)
where CJI,)= 7r + 01,j o( j < rr/2, while the scattered field is
The result of adding (6.8) and (6.9) is
(6.10) Accordingly,
the values of stress components are
X
[
B,(Xr))
- i ~?a’(a sz-l(b) %,-,Gv
1’
-q > 1,
(6.11)
and
7 > 1. (6.12)
36
JAIN
AND
KANWAL
In the next stage we use the expansion of the Weber functions as given by (5.28) to (5.31) and those for their derivatives and obtain for m < 1, t sin+
a((, 7) = im [ 2 (-g2
+ 2’([” - q2 + 27) cos 01 a, (2n + 3/2)(n !>”
( -4 tan2 $)% +
O(m1/2)],
(6.13)
sin fi2 + 5 cos 01+ 4 set -Z tan -!? n2 2
~~~(5,rl) = pimh [2 (&,“’ x f Vl=O
(2~ + 3f2)(n!)2 (-4 tan2(,/2))” (2n + l)!
(6.14)
+ ow3],
and T,&,
q) = @mh[(l
(6.15)
- T) cos 01 + O@Z~/~)],
where we have dropped all the constant terms upto order O(m) in (6.13). To derive the limiting results for a crack of the shape of a Half-plane crack. half plane defined as x 3 0, z = 0, we use the expansion (5.35) for ui which is given in its dimensional form. Furthermore, we write the scattered field (6.7) also in its dimensional form,
Taking the limit of relation (6.16) when M-+ a,(&
7)
=
-
$
f
(-
1)2n+l
9 zn+l(do>
0, we get
=%+d~~)
g-zn-2(wJ
~-zn-z(4,
(6.17)
?k=O
where we have used the relations
%2,(O)= 0,
J%n+1(0>=
(-1)”
(2n + l)! 2”n! ’
9Y2n--2(0) =
-zy
.
Relation (6.17) gives the scattered field when a line source of strength A is placed at (co, qo). The other quantities of interest can now be readily calculated. To solve this problem we take the (5.35) and (6.17) when 1cl, j + CO, ~~-j co such that
Plane wave incident on a half-plane crack.
limits
of relations
SCATTERING
OF
37
WAVES
r),/& = tan q,/2. We further use the limit (5.38). Then the incident field ui reduces to relation (5.42) as explained there while the scattered field becomes,
a-7 7)==(2;)‘,2 190[i(5rl sin2 - f cos $,)I S1 [U (q cos - 90[u (4 cos 212 + 7 sin E2)] Lie1 [Q (7j cos $- -
f
+
t sin
$)]
4 sin :)]I, (6.18)
where we have used the same steps and notation (5.45). Adding (5.42) and (6.18) we have
+ go [U ([ cos + - q sin f)]
a-l
as we did in deriving
[U (t sin f
relation
+ 7 cos :)]I. (6.19)
Next we use relations (5.47) and derive from (6.19) the formula sponds to Sommerfeld formula (5.49). It is
s
-(2p~)1i2sin(m+(~/2))
+
eik!msb+a)
which
1
e@ dfl .
--3o
corre-
(6.20)
Stresses on the faces of the crack. From relation (6.19) we can derive the stresses ‘on the faces of the crack by using formulas (5.17), which gives
b&G w)lr>O = [+Tdt, 0)1,<,,m = 0. -m
stress component
b&G o+)lz>o = b&5 dlO<,<~ = 4 [ y; Tl=O
= (a)“’
pik 1~0s &-,
(6.21)
is r,, so that
-+I] O
[CT1( / sin 11
+ 2(u 1( I)-’ sin + B0 ~7$sin ; [
1 t>o (6.22)
a.$ cos $ ,
38
JAIN
AND
KANWAL
and
[- . “I [
= 2pik cos c&9,, & sm -2- .9,, ut cos -&] -
[T&X, O+)lZ,,, , (6.23)
where we have used the identities (5.50) and (5.51). Low frequency approximation for the displacement and stress fields, inside the material can be obtained from relations (6.13) to (6.15) by first writing them in the following forms by setting m = kM2 and putting the original values of the coordinates 5, r) with proper dimensions, u([, v) = ik [2 (&)l”
f sin 01+ 1/2(f2 - y2 + 2$4)
cos 01
+ $ (M set 5 tan T 2OD(2n + 3/2)(n !)” (-4 tan2 f)” %=() m + I>!
+ O(k112)], (6.24)
T&,
77) =
x 2 (2n + 3/2)(n !)” (-4 n=() (2n + I>!
tan2 T)”
+ O(kl!z)],
(6.25)
and pik
TJ*(!f, 4 = (P + qy2
[(M - 7) cos a + O(k112)].
Now we let M+ 0 and introduce polar coordinates (2~)l/~ sin v/2. Then relations (6.24) to (6.26) reduce to u(p, y) = 2ikp [(--$-)I’” T&P,
91 =
pik
[(--$-)I’”
(6.26)
5 = (2p)li2 cos v/2,
cos 5 sin a: + k cos y cos a! + O((kp)l/2)], sin $ + cos 01cos T + O((kp)1!2)],
(6.27) (6.28)
and T&P, 91>= -pik
[cos cy.sin $- + O((kp)‘l”)].
(6.29)
REFERENCES
1. M. L. BARON AND J. T. MATHEWS, Diffraction of a pressure wave by a cylindrical cavity in an elastic medium, J. Appl. Mech. 28 (1961), 347-354. 2. M. L. BARON AND J. T. MATHEWS, Diffraction of a shear wave by a cylindrical cavity in an elastic medium, J. Appl. Mech. 29 (1962), 205-207.
SCATTERING
OF WAVES
39
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