RAYLEIGH WAVE SCATTERING AT THE FOOT OF A MOUNTAIN 381

A theoretical study of scattering of seismic waves at the foot of a mountain is discussed here. A mountain of an arbitrary shape and of width a (0≤x≤a, z=0) in the surface of an elastic solid medium (z≥0) is hit by a Rayleigh wave. The method of solution is the technique of Wiener and Hopf. The reflected, transmitted and scattered waves are obtained by inversion of Fourier transforms. The scattered waves behave as decaying cylindrical waves at distant points and have a large amplitude near the foot of the mountain. The transmitted wave decreases exponentially as its distance from the other end of the mountain increases.

P.S. DESHWAL and K.K. MANN medium is a liquid halfspace which reduces the number of elastic parameters, the basic equations and the boundary conditions.Not much theoretical study is available on the problem of scattering of elastic waves due to an irregular boundary of finite dlmenslon Recently Momol (1980, 82) has studied the problem of scattering of Rayleigh waves by semicircular and rectangular discontinuities in the surface of a solid halfspace.The solutions are not exact because of the approximations used in the solution of these problems.
We propose to discuss here the problem of scattering of Raylelgh waves at the foot of a mountain with its base occupying the region 0 x !a, z 0 in the surface of a solid halfspace z 0 Since we are interested in scattering of waves at the foot of a mountain, its shape is immaterial and it is assumed to be rigid such that there is no displacement across the mountain.The method of solution is the Fourier transformation of the basic equations and determination of unknown functions by the technique of Wiener and Hopf (Noble, (1958)).

STATEMENT OF THE PROBLEM.
The problem is two-dimenslonal in zx-plane.The mountain of width a has its base along 0 x a, z 0 in the surface of an elastic solid halfspace z 0 (figure I).The medium is homogeneous, isotropic and slightly dissipative.If the re- tarding force of the medium is proportional to the velocity, then the wave equation is where c is the velocity of propagation and e > 0 is the damping constant.The pote tial function harmonic in time is 2) The equation (2.1) is now where / (m2 + le)/c I + i2 is complex whose imaginary part is small and posi- tive.An incident wave iP0X -B0z #i(x,z) D(2p k2) e e (2.4)   iP0x -60z i(x'z) D(21P080) e e (2.5)   where P0 is a root of the Raylelgh frequency equation 4p286 0, 8 / (p2 ()2), 6 / (p2 k2) (2.6) 80 8(p0), 60 6(p0), strikes at the foot of the mountain from the region x < 0.
The potential (x,z) satisfies the wave equation (V 2 + k2) (x,z) 0 (2.7)Let the total potentials be #t(x'z) (x,z) + #i(x'z) (2.8)   t(x,z) (x,z) + i(x,z) We assume that, for given z, (x,z) and (x,z) have the behaviour exp(-dlxl) as Ix] , d > 0 The Fourier transform _(p,z) f0_ (x,z) e ipx dx, p a + ia 0 (2.12) and its derivatives w.r.t.z are analytic in the strip -d < s 0 < d of the complex plane.The transform of (x,z) has the same behaviour.
The boundary conditions of the problem are The conditions (3.1) and (3.2) subject to (2.3) and (2.Since (x,z) and @(x,z) are bounded as z tends to infinity and their transforms are also bounded.The solutions of (4.1) and (4.2) are (p,z) A(p) e -8z (4o3) @(p,z) B(p) e (4.4) The signs in radicals for 8 and 6 are such that their real parts are positive for all p.We use the notations (p) @(p) for (p,0) (p,O) etc. From (4.3) we obtain 'Cp)/B -Cp) The left hand member of (4.6) is analytic in a 0 > -d and the right hand member in a 0 > d They represent an entire function.Further each member tends to zero as Pl By Liouville's theorem, the entire function is identically zero.Equating each member to zero, it is found that Similarly, we find from (4.4) that x )e dx dx 0 (4.17) Since w 3#/3z 3/3x is continuous, the first integral in (4.17) is integrated by parts to obtain 3@) ipx k2 8 3 -2ip /_0 (38z x e dx _(p) -2(z x)0 or 8i 3i) -2ipI(P) + (2p2-k2) -(P) (3z x 0 + 2ip(i) 0 2BoD(k2-2pp (4.18)  Inversion of Fourier transforms will give various waves. 5. VARIOUS WAVES.
The factor exp(-ipx) exp(-iex) exp(eoX in the inverse transforms makes the inte- grals vanish at infinity in the upper part of the complex plane if x < 0 and in the lower part if x > 0. For waves in the region x < O, we have To find the integrand, the wave equation (2.3) is integrated from x to x 0 after multiplying it by exp(ipx), to find out d2_/dz 2 82_ ffi-($/x) 0 + ip() 0 (5. 2E is obtained from here as the left hand member is known from (4.8) and (4.20).We find the integrand in (5.3) to be 2ipD (5.9) The pole at p P0 contributes (5.10) This represents the reflected wave in the region x < 0. This does not depend upon the width a of the mountain.The reflected shear wave in the region is found to be 4P0B0 -iP0x -6 z l(X'Z) [0) [(2p-k 2) F(Po) 4iPoBoD(2p-k2B060 )] e e 0 (5.11) The integrand in (5.9) has branch points at p k and p k.The branch cuts are given by the conditions Re(B) 0 Re (6).As discussed by Ewtng and Press (1957), the parts of branch cuts are hyperbolic as shown in figure 2. For contribution along the branch cut we put p + iu, u being small as the main contribution is around the branch point.Along the cut, Re(B) 0 and Im(8) changes signs along two sides of the cut.Since B is imaginary, 82 is negative.Therefore 82 =( + iu)2_ ()2 2iU(l + i2) U 2 (5.12) From here B + +i i 0 (5.13)Integrating (5.3) along two sides of the branch cut, we find f [H (u) sin z 22u + H2(u) u/-cos z22u e-UXdu (5.14) u is small, Hi(u) and H2(u are expanded around u 0 and only HI(0) and H2(0) are retained.The following Laplace integrals (Oberhettinger, (1973)) are used sin a e -pt dt aCn exp(-a2/4p)/2p 3/2 (5.15) cos a t/{--e -pt dt (p/2 a2/4) exp(-a2/4p)/p 5/2 (5.16)The scattered waves for the region x < 0 are obtained to be (5.17) when x >>z, then r /(x 2 + z2) x+z2/2x (5.18)   and the scattered wave is of the form 2 (x,z) GO exp (2r)/r (5.19)This represents a cylindrical wave.On the free surface (z--0), the wave has the behaviour of exp(k2x)/x 3/2 We now find the waves transmitted to the other side of the mountain.The poten- tial for the region x > a is given by We take the Fourier transform of the wave equation (2.3) from x a to x to find out d2+a/dZ2 82+a (a/ax) a e ipa ip() a e ipa (5.23) Changing p to -p and subtracting the resulting equation from (5.23)   it is found that (d2/dz 2 82)[+a (P z)e -ipa +a(-p,z)e ipa] iPoa -80z The pole P -P0 in the integrand in (5.22) The first term cancels exactly the incident wave and the second term is the transitional wave in the region x > a. 6. CONCLUSIONS.
The integrals along the branch cuts_at p -, k lead to the scattered waves as obtained in (5.19).They behave as exp(-2r)/r where r is the distance from the scatterer.These are cylindrical waves which die at distant points from the foot of the mountain.On the free surface (z 0), the scattered waves have the form exp(-2x)/x3/2 which is large at the points near the scatterer.Thus the energy of the scattered waves is very large close to the scatterer and dminishes as the wave moves away from it.The RAYLEIGH WAVE SCATTERING AT THE FOOT OF A MOUNTAIN 389 transmitted wave in (5.29) depends upon width a of the mountain.As the distance from the other end of the mountain increases, the transmitted wave decreases exponentially and dies out at distant points.The reflected waves are given by (5.10) and (5.11) which do not depend upon the width of the mountain.Numerical results for the amplitude of the scattered wave have been computed or Poisson's solids for which k 3 k at a point (r I/2 km, z 0) in the region < 0 of the free surface.The results are obtained for q 0 and 1.8932 k.There is a sharp increase in the amplitude (fig.3) when the wave number k increases through small values.
The results of this paper have their application to underground nuclear explosions carried out on either side of the mountains llke Himalayas.It helps calculating the amount of energy reflected and scattered at the foot of the mountain and the amount of energy which is transmitted to the other side of the mountain.
X z C FIG. 2

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integral (5.1) is in the strip -d < < d The contour is in the o upper half of the complex plane where _(-p,z) is analytic and hence o f _(-p,z) e -ipx dp (