Pageoph, Vol. 112, 1974
Birkh~iuser Verlag, Basel
Reflection and Diffraction of SH-Type Waves in Elastic Medium by a Periodic Curved Boundary By SUDARSANBHATTACHARYYA1
Summary - In the present paper, the author considers the effect of periodic irregularities of a boundary on scalar waves in the plane case. By means of Fourier transform, the solution has been obtained in the form of a double integral and this in turn has been approximated by the saddle point method for a geometrical picture.
1. I n t r o d u c t i o n
Because of its closeness to the natural situation, the study of the effect of irregular boundaries on the propagation of waves in an elastic medium has gained much of its importance. As the analytical treatment of the irregularities of the surface in general entails formidable mathematical difficulties, most of the workers, particularly in this branch, concentrated their efforts with considerable success in considering the cases of slightly curved surfaces of different types. SATO [I ] studied the propagation of Love waves in a layer with a sharp change in the thickness, while DE-NOVER [2] considered the same in a layer over a half-space with a sinusoidal interface and K u o and NAVE [3] investigated the propagation of Rayleigh waves in a similar model. OBUKOV [4] considered the effect of a wavy boundary in a new manner. On the other hand MAL [5] and ABUBAKAR [6] also considered the effect of a curved boundary in the presence of a buried line source. All these investigations led to the development of frequency equation of wave motion. Other recent contributions on reflection by irregular surfaces are from ASANO [7, 8], BEHRENS and GOMMLICH [9], BEHRENS e t aL [10], BEHRENS e t aL [11], LEVY and DERESIEWICZ [12] and TWERSKY [13]. One can also anticipate that for such problems the displacement vector consists of two parts. One is the same as that in the case of plane boundaries and the other corresponds to the additional effect due to non-zero curvature of the boundary. In this paper the author investigates the effect of periodic irregularities on scalar waves. It is of much interest to note that the final results do not contain any term involving the curvature of the boundary and as such the results derived here, by Fourier transform technique and saddle point method, are independent of the curvature of the boundary to a considerable degree of accuracy. l) Department of Mathematics, Bengal Engineering College, Howrah-711103, West Bengal, India.
8 38
Sudarsan Bhattacharyya
(Pageoph,
2. Formal method of solution Let x, y, z be the Cartesian coordinate system in the three-dimensional space. For scalar SH-waves the displacement vector/3 has the components (0, ~b, 0) and Oc~/Oy= O. If p be the density of the medium and p its elastic constant, then for a homogeneous elastic medium the only differential equation governing the motion is: a'- ~ a ~ q~ Ox-----~+-~-zZ +tc2(~=O (z>O,--oo
(-oo < v < oo)
d x = ~ (v + sin v)
[cf. OBUKOV[4]]
(2)
where d is half wave-length of the boundary surface. On defining ao
= ~
~bexp (i~v) dv,
--o0 r(u, O =
4~2exp(u);~sin22exp(i~v)d v vT~
(3)
--o0
and using the transformation d z = 7-(1 + U + e"cos v) 2n d x = 2n (v + e" sin v)
(4)
the field equation (1) becomes:
a2~
_ _ + [0r + ~2) + 2X2exp (u) + Z2 exp (2u)] ~ = F(u, 4) au 2
(5)
where X = tcd/2n, and ~ is the parameter of Fourier transform. Relations (4) define a one-to-one mapping for u ~<0. If lul is large enough, the mapping (4) converts the straight lines z = constant into the straight lines u = constant.
Vol. 112, 1974)
Reflection and Diffraction of SH-Type Waves
839
The general solution of the equation (5) may be sought in the form:
where ~(~) ( i = 1,2) are the complementary functions of the homogeneous equation corresponding to (5) and F~(u, ~) is the particular integral. In subsequent analysis the particular integral Fx, which is the contribution due to the non-zero term F(u, ~) in (5), will not come into effect because the presence of the term sin 2v/2 in F(u, ~) in (3) makes it considerably smaller than ~. Hence the total field may be taken as: qS~ A(~) qS(1) + B(4) ~(2) + qSi
(6)
where A(O, B(r are the unknown functions to be ascertained from the boundary condition and ~i is the Fourier transform of the incident field. The appropriate boundary condition for the present problem is: --
0u
= 0
on
u = 0
(7)
The substitution tl = zexp(u) reduces (5) to non-homogeneous confluent hypergeometric equation, whose complementary functions, corresponding to the homogeneous equation, are given by: ~ l ) = exp (-itO t~a Fl(al [el [Z~) ~2) = exp (itl) t l iB F2(a2[c2[~(2)
(8)
where
al= 89 cl = I + 2i~ Zi = 2itl F(alc[x)=~ 0
(a)r x ~, r !(c),
a2 = 8 9 i ~ - iz c2 = 1 - 2ifl Z2 = - 2 i q
~" (Z 2 + 42) 112
( a ) , = a ( a - 1)... 2.1
For large x, F(a]c]x) is of the order ofcx-". Therefore, if we consider only the out-going field as z ~ o% we see that A(~) must be taken to be zero. Hence, using the boundary condition (7), we obtain after a little algebra:
B(r
where X[ = -2i~.
=
bcdexp(bcd/2rc) f (1 + cosv)exp i~v + cosy dv -| ~ _ - - _ . _ _ ~5 ~ _ _ m ~ [ F2(a21e~lz;)(2z)3/2 4(fl + g)exp(-i)~- ifllogz) J
(9)
840
SudarsanBhattacharyya
(Pageoph,
Inserting the value of the unknown function B ( 0 in the expression (6) and taking Fourier inversion, it follows that:
~+i~i
[xdexp(i)~e")] J F2(a21c2[x2)(l+cosvt) ,~ i [ ~ ] - ,. -~o F2(a2lc2[)CO r + )0 + flu + ixd ~ c o s vl 1 dr dv~ 2n
x exp -i~v + ir
(~ > O)
(1O)
Now for large X,
F2(a2lc2lx2)
f2(a2lc2lzl)
exp [0C2- Xl) + u(a2
-
-
C2)]
(1 1)
equation (1O) gives the expression for the resultant field. Integration of the intregal (1 O) will he performed by the saddle point method. Since X is large, for finite 4, the saddle point with respect to v! is v~~ = nn, where n is zero or any integer. Therefore (lO) is approximately: cO+ig
f
~exp(M)(l+cosnzc)
,~ i
exp[ir
2rt[xdcosnrtll/2 -av
~v~-+x2]ae
(12)
~()f + ~/):2 + r --co+i~
where M = 2ix - 89 - ix exp (u) - uix + i - - cos nrr . 2zt The substitution, ~ =
ir
reduces (12) to : e+iao
~b~ - i 2
exp [~x(V - n n ) + u V ~ -- ~1 d~,
exp(M)(1 + cosmr)
2~l~:dcos nrcl"2
f
(13)
~ = 0 and +X are the singularities of the integral under consideration. The doublevalued function (X2 - ~)a/2 is defined a s lim
(~2- - ~2)I/2=
__~
for u > 0,
~t~0
=Z
for u < 0.
(14)
.For our purpose u is always greater than zero which is the consequence of the mapping by (4) from (x, z) plane to the (u, v) plane. Let ~1 = -Zcos(O + iT)
(-oo < T< ~)
Vol. 112, 1974)
Reflection and Diffraction of SH-Type Waves
84 1
where
v-nn=rcos61
u=rsin61.
(15)
Then ~1 = a + i~ = [-X cos O cosh T + ix sin O sinh T]. Without any loss of generality, let us assume that 0 ~< 6} ~< re. Hence, we find that by this substitution the imaginary axis in the T-plane is m a p p e d into the left half when 0 < 61 < 1r/2 and to the right half when n/2 < 61 < it of the hyperbola
whose asymptotes are a/z = + cot 61. Now, when 0 < 61 < 7r/2, we note that the integrand o f the integral (13) vanishes on the arcs Co and Co (Fig. 1). Therefore,
q
[exp(M)(1
+cosmr)]
, . ,
x f e x p [ ~ d v - nn) + u ~ z 2 - ~2] d~a c6
[exp(M)(l +costa0]
0 < 61 < - , 0 < a r g ( z 2 - ~x)~/2 < rc 2 (16)
If n/2 < 61 < lr and 0 < a r g ( x 2 - ~2)112 < 7~, then the integrand in (13) converges to zero on the arcs C1 and C1 (Fig. 1), and hence
exp (M) (1 + cosmr)
x fexp[r
c;
]
Z V ' ~ - 3 2Idea
r
+
)
< 6} < 7r, 0 < arg (Z2 - ~)l/z < rc
r,~x
(l/) It is worthy of note that the first term in (16), which is due to the contribution of the pole, does not occur in (I 7) because of the fact that in pas sing f r o m the contour C to C o, the pole is crossed when 0 < 6} < 7r/2. On the other hand, when n/2 < 61 < n, in passing f r o m C to C / t h e pole is not crossed and as such (17) does not contain the term which follows f r o m the contribution of the pole.
842
(Pageoph,
Sudarsan Bhattacharyya Now applying (15) to (16) and (17) we finally obtain:
~[exp(M)(l+cosmO] [ exp (M)(1 + cosnn)] 2xlxdc~ 1/2 ] e x p ( - i g u ) + ~[-co ~ o s n n - ~ J x f F(T)exp (-xr cosh T)dT
(u > 0, v > nn, 0 < arg (Z2 - ~)1/2 < n)
(18)
0
and q~ ~ ~
[exp(M) (1 + cosmz)]
] • ; F(T) exp (-gr cosh T) dT
(u > 0, v < nrc,0 < arg(g 2 -
~)1/2 < rr) (19)
o
where
F(r)= i c o s h 2 r + ~ s ~ - i - ) ~ ~ _ c o s 2 b ) To evaluate the integral T = ~o F(T)exp(-grcoshT)dT, again we apply the saddle point method. The saddle points are given by (d/dT)coshT= 0 so that T = 0 is such a point. Applying the method of steepest descent, we arrive at: J~
-gr
F ,o, exp(-zr)
(20)
| GOMPLEX~f-PI.A~E
Figure l Transformation of the contour parallel to imaginary axis for different positions of observation
Vol. 112, 1974)
Reflection and Diffraction of SH-Type Waves
843
In view of (20), (19) and (18) become: ~
qb'~-oo
[exp(M)(1 + cosnrO] . . . .
[ 2~~ ~
+
-oo
jexp(-,ZU)
e x p ( M - zr) (1 + cosmr) F(0)
Z(2zrxr)l/2lxdcosnn[1/2
(u > 0, v > mr)
(21)
for (u > 0, v < nrc)
(22)
and ~ q~ ~
e x p ( M - zr)(1 + cosmr) ~ / ~ ~ - ~ s ~ [ 1/2 F(0)
-o0
The first part of (21) gives the reflected field; the second part of (21) and (22) give the diffracted field. We take n here to be any even integer, otherwise (1 + cosnzr) vanishes and makes (21) and (22) meaningless. For numerical purposes we take u = 1 and ~d/21r= 10 and draw curves in the (x-z) plane for different values of v in the following three cases: Casel.
0
Case 2.
2zr < v < 4re
Case 3.
4zr < v < 6zr
and so on. The curves (Fig. 2) so obtained give the zones where one, two etc. reflected waves are obtained. For example, the zones where one, two and three reflected waves are obtained are the areas bounded by the co-ordinate axes and the curve 1, curve 1 and curve 2, and curve 2 and curve 3, respectively.
3
Figure 2 The zones indicating the number of reflected waves
3. Conclusion As the results derived are general in nature, they apply equally to an undulating interface of media with different velocities. The nature and lengths of the undulations may be guessed from the number of reflections observed at different stations situated in a dispersed manner.
844
Sudarsan Bhattacharyya Acknowledgement
The a u t h o r takes this o p p o r t u n i t y o f t h a n k i n g Dr. S. C. DASGUPTA for suggesting this investigation a n d guidance d u r i n g the progress o f the work.
REFERENCES [I] R. SATin,Love waves in case the surface layer is variable in thickness, J. Phys. Earth 9 (1961), 19. [2] J. DE-NOYER, The effect of variations in layer thickness of Love waves, Bull. Seis. Soc. Amer. 51 (1961), 227. [3] J. T. Kuo and J. E. NAVE,Period equation for Rayleigh waves in a layer overlying a half-space with a sinusoidal interface, Bull. Seis. Soc. Amer. 57 (1962), 807. [4] G. G. OBUKOV, The effect of periodic irregularities in a relief on the dispersion curves of seismic surface waves, Izv. Acad. Sci USSR Geophys. Soc. (1963), 340. [5] A. K. MAL, On the frequency equation for Love waves due to abrupt thickening of the crustal layer, Geophys. Pure. Appl. 52 (1962), 21. [6] I. ABUBAKR,Reflection and refraction of plane S H waves at irregular interface, I and II, J. Phys. Earth 10 (1962), 1 and 15. [7] S. AsANo, Reflection and refraction of elastic waves at corrugated boundary surface I, Bull. Earth. Res Inst. 38 (1960), 177. [8] S. ASANO,Reflection and refraction of elastic waves at a corrected interface, Bull. Seis. Soc. Amer. 56 (1966), 201. [9] J. BEI-IRENSand G. GOMMLICH,Model investigations with respect to the interpretation of complicated seismic discontinuity, Z. Geophys. 38 (1972),'659. [10] J. BEHRENS,J. KOZAKand L. WANIEK,Investigation of wave phenomena on corrugated interfaces by means of the Schlieren method, Proceeding of the 12th General Assembly of the European Seismological Commission, Luxembourg, 1971. [11] J. BEHRENS,R. BORTFELD,G. GOMMLICHand K. K6HLER,Interpretation of discontinuities by seismic imaging, Z. Geophys. 38 (1972), 481. [12] A. LEVYand H. DERESIEWICZ,Reflection and transmission of elastic waves in a system of corrugated layers, Bull. Seis. Soc. Amer. 57 (1967), 393. [13] V. TWERSKY,On scattering and reflection ofsoundby rough surfaces, J. Acoust. Soc. Amer. 2 (1957), 128. (Received 25th April 1974)
