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\title{THE UMOV ENERGETIC EQUATION AND BERRY PHASE IN PROCESS OF PROPAGATION
OF WAVES ALONG A WEAKLY NONHOMOGENEOUS PIEZOELECTRIC LAYER}
\author{Vasilii M. Babich}{
Matem.Institute of Russian Academy of Sciences St.Petersburg Branch\\
Fontanka 27, St.Petersburg 191011, Russia}
\coauthor{Valery V. Lucyanov}{
St.Petersburg State  University, Mat.-Mech. Faculty\\
Bibliotechnaya 2, Staryi Peterhoff, St.Petersburg 198904, Russia}

  The interest to piezoelectric waveguides and piezoelectric layers
increases in connections with their technical applications. The
applications inspire the publications of papers on wave propagations
in waveguides and layers. There are no ideal waveguides or layers.
Very often they are weakly bent or nonhomogeneous in longitudinal
direction.
The word weakly means, that corresponding properties
vary very little on the distances of the wave  length.

The process of
the wave propagation in the such structures allows a theoretical
investigation by the asymptotic methods. Corresponding theory
 was developed in the case  of elastic isotropic layers [1-3],
 elastic anisotropic waveguides and layers [4]. It was outlined
a general theory of the problems of this kind [5].


The equations of the piezoelectric media are peculiar kind of equations:
they do not belong to classical types of partial differential equations.
It would be possible to denote this type of equations
 \lq\lq elliptico-hyperbolic equations\rq\rq. In spite of it the methods
of the papers [1-5] with some modifications are applicable  here.

The theory, developed in our paper, gives the possibility to describe
wave packets, propagating with group velocity along piezoelecrtic layers.
Energetic equation --- having a form of classical hydrodynamic
 continuity equation --- allows to obtain the amplitude of the wave
 in explicit form. A version of this equation was suggested by Russian
physicist N.A.Umov in the seventies of last century [6]. An increment of the
phase along the rays is a version of well known Berry phase.

\begin{thebibliography}{99}
\bibitem{babich1} Woodhouse J.H.
Surface waves in laterally varying layered structures.
Geophys.J.R.Astr.Soc.37.461-490.1974.
\bibitem{babich2} Babich V.M., Chikhachev B.A., Yanovskaya T.B.
The surface waves in
 nonhomogeneous in vertical direction with weak horizontal nonhomogeneity.
Izvestiya AN SSSR.Fizika Zemli. no 4.24-31 1976 (in Russian)
\bibitem{babich3} Babich V.M., Grigoryeva N.S.
Space-time ray method of calculation of waves
in weak nonhomogeneous layered medium. In book  \lq\lq Interference waves
in layered medium \rq\rq.LOMI AN SSSR,v 99.5-18.1980 (in Russian)
\bibitem{babich4} Tromp J., Dahlen F.A.
Surface wave propagation in slowly varying
ani\-so\-tro\-pic waveguide.Geophys.J.Int.113.239-249.1993.
\bibitem{babich5} Tromp J., Dahlen F.A.
The Berry phase of a slowly varying waveguide.
Proc.R.Soc.Lond.A.437.379-342.1992.
\bibitem{babich6} Umov N.A.
Selected papers. M. p.325.1950
\end{thebibliography}

\end{document}
