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\title{THE EXCITATION AND PROPAGATION OF THE `ANALOGOUS HEAD
WAVES' DUE TO A TOTALLY INTERNALLY REFLECTED PLANE-WAVE INCIDENT UPON A
CONVEX FLUID-SOLID BOUNDARY}
\author{Z.M. Rogoff and R.H. Tew}{
Department of Theoretical Mechanics, University of Nottingham\\
University Park, Nottingham NG7 2RD, UK\\
e-mail \hspace{1cm} etxzr@vme.nottingham.ac.uk}

Much attention has been devoted to the acoustic interaction between
compressible fluids and adjacent elastic solids. Here, we focus upon the
sub-class of problems associated with light-fluid loading which brings a
small real parameter into the analysis. Within this field of study the
excitation of the head
waves due to a circular wave impinging upon a planar fluid-solid interface
has been considered. The intention of this paper is to consider the reciprocal
problem, that of plane wave insonification of a curved boundary, where
the angles of which incoming rays excite the `analogous head waves' are those
of total
internal reflection. We observe that the curvature in the former problem
which arises through the incident wave has been transferred to the latter
through the curved nature of the boundary. This problem introduces a
further small real parameter, this being proportional to the ratio of the
acoustic wavelength to the radius of curvature at the point of excitation.

The problem of plane wave insonification of such a convex boundary
possesses other sub-problems, total internal reflection being just
one failure of the ``geometrical optics'' solution.  The ray
solution will also fail at the angles corresponding to free mode excitation
and grazing incidence.

Due to the two distinct wavespeeds of the solid there exist two critical
angles of total internal reflection which correspond to distinct points
of the boundary. About each such point we construct an `inner' diffraction
problem within a wavelength of the boundary where full account is taken of
curvature. We solve this by multiple scales methods, where the solution
is found to be given in terms of parabolic cylinder functions. The
solution matches the outer ray picture as well as generating the
`analogous head waves', a result of the Stokes phenomenon of the parabolic
cylinder functions. To establish the
`analogous head waves' well removed from the points of excitation we
must investigate the whispering gallery modes which are excited within
the solid.





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