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\title{ASYMPTOTIC STUDY OF THE ELASTIC BOTTOM EFFECTS
 IN UNDERWATER SOUND PROPAGATION}
\author{G. N. Makrakis}
{Institute of Applied and \\
Computational Mathematics \\
FORTH-HELLAS, P.O. Box 1527 \\
71110 Heraklion, Crete \\
GREECE}


\noindent
We present a theoretical and asymptotic investigation of the
Green's function for the elliptic system governing the propagation
of time-harmonic acoustic waves in a horizontally stratified
fluid-layer with free surface, which overlays a homogeneous,
isotropic and linearly elastic half space. Employing
the Neumann to Dirichlet map for the elastic half space,
we reduce the problem to an equivalent one in the layer, where
the pressure satisfies a non-local boundary condition at the bottom.
The reduced problem is further transformed by Hankel transform in
the horizontal variables, to a non-selfadjoint boundary value
problem for a second-order ordinary differential equation over the depth.
The well posedness of the depth problem is investigated applying analytic
Fredholm theory, and a generalized eigenfunction expansion of its
solution is constructed. The Hankel inversion of this expansion leads to
a representation of the Green's function as a sum of a modal series,
and a couple of line integrals which account for the correct elastodynamic
radiation in the bottom. It is finally shown that for a  bottom with
small shear modulus, the constructed Green's function is a regular
perturbation of that for a fluid bottom. For the later case, the procedure
of asymptotic expansion reveals a new alternative representation, which
contains a simpler line integral than that in the Pekeris' and EJP
representations.



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