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\title{Dilatation at Shear Wave in Smoothly Inhomogeneous Elastic Medium}
\author{Alexey P. Kiselev}
{Institute of Mechanical Engineering Research, Russian Academy of Sciences,\\
Vasilievsky Ostrov Bolshoy Prospect 61, 199178, St.Petersburg, Russia}

      Discussing linear wave propagation in inhomogeneous media
it is common to have in mind the case of constant coefficients and
to start with plane waves. In isotropic homogeneous
elastic material two types of body waves can propagate. The fast
wave is called longitudinal, or pressure, or dilatation;
the slow one is transverse, or shear, or rotational.
No longitudinal displacement occurs at plane shear wave,
and its dilatation vanishes identically.
For non-plane wave in homogeneous medium (of which spherical waves
can be an example) so-called additional components are normally
present; for slower wave that is a longitudinal displacement appears
(e.g. [1,2])
but dilatation remains zero.

      Ray theory, or geometrical acoustics, generalizes plane
waves to inhomogeneous medium. Wave fields are
described by ray series asymptotics, involving, for
time harmonic processes, expansions with respect to
inverse powers of frequency. For the slow wave the leading term
provides no contribution to dilatation, but the first correction
does, just that describing the anomalous longitudinal
displacement.

      Now we shall present the result for dilatation of the slow
wave in frame of the ray theory. The material is characterized by
smooth volume density $\rho=\rho(\hbox{\bf r})$ and smooth wave speeds
$ a = a(\hbox{\bf r})$ and $ b = b(\hbox{\bf r})$, $a > b$~;
$\hbox{\bf r} = (x,y,z) $.
A time harmonic displacement $ \hbox{\bf U}(\hbox{\bf r})
\exp(-i\omega t) $
is described by the series
$$
\hbox{\bf U}=\left\{ \hbox{\bf U}^0+\frac{\hbox{\bf U}^1}{-i\omega}+ ...
\right\}
\exp\{i \omega (\tau -t)\},\hskip 10pt \hbox{\bf U}^0\cdot
\hbox{\rm grad\,}\tau=0$$

\noindent where $\omega $   is angular frequency and
$ \tau=\tau(\hbox{\bf r}) $ is a geometrical travel time of the
slow wave.
The rays are assumed given.
Let $ \hbox{\bf n} = \hbox{\bf n}(\hbox{\bf r}) $ and
$ K = K(\hbox{\bf r}) $ be unit normal vector to the ray
coming to the point $\hbox{\bf r}$ and its curvature. The
expression for the dilatation is
$$
{\rm div} \hbox{\bf U} = \hbox{\bf U}^0
\cdot \hbox{\bf S} \, \exp\{ i\omega(\tau-t)\}
$$
with
$$ \hbox{\bf S}= \hbox{\bf S}(\hbox{\bf r})=
\frac{b^2}{b^2-a^2}
\left\{ \frac{\hbox{\rm grad\,}\rho}{\rho}+
2\frac{\hbox{\rm grad\,} b}{b} -
{2K \hbox{\bf n}} \right\}$$
Higher order terms with respect to the small parameters
$$a\frac{|\hbox{\rm grad\,}\rho|}{\omega \rho},\hskip 5pt
\frac{|\hbox{\rm grad\,} a|}{\omega a} \hbox{\hskip 5pt \rm and \hskip 5pt}
\frac{|\hbox{\rm grad\,} b|}{\omega b}$$ are omitted.
Similar expression for rotation at the fast wave have been
given in [3].

      The above results can be interesting to geophysicists
believing [4] that direct measurements of dilatation and rotation
could be helpful in wave discrimination.

      The work was supported by a grant of the Russian Ministry of
Higher Education.

\begin{thebibliography}{99}
\bibitem{kis1} V.M.Babich, A.P.Kiselev. Non-geometrical waves - are there
  any ? An asymptotic description of some 'nongeometrical' phenomena
  in seismic wave propagation. Geophys.J.Int. {\bf 99}, 412-420, 1989.
\bibitem{kis2} A.P.Kiselev, Yu.V.Roslov. Use of additional components for
  numerical modelling of polarization anomalies of elastic body
  waves. Sov.Geol.Geophys. {\bf 32}, 105-114,1991.
\bibitem{kis3} A.A.Gvozdev. On conditions at fronts of elastic waves
propagating in inhomogeneous medium. Prikl.Math.Mech. {\bf 23},
  395-398,1959 (in Russian).
\bibitem{kis4} J.E.White. Underground Sound, Elsevier,1983.
\end{thebibliography}
\end{document}
