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\begin{document}



\title{
MATHEMATICAL  MODELS  OF  THE  SLOWLY  VARYING   WAVEGUIDES   WITH
PENETRABLE BOUNDARIES}
\author{M.E.Abramyan, S.L.Edelstein}{
        Mechanics and Mathematics Department, \\
	Rostov State University, \\
        ul.Zorge 5, 344104, Rostov-na-Donu, Russia\\
        E-MAIL: edel@mmf.unird.ac.ru}


     The  report  presents  the   mathematical   theory   of  
slowly range-depending hydro-acoustic  waveguides  with the  
boundaries  described   by   means   the  pseudodifferential   
operators determined through impedance (see also [1,2]).  It 
is  a  general way for modeling penetrable boundaries.
 
     A theory of adiabatic normal   waves   is   constructed  
for this  model (see also [2,3]). The  principal   advantage  
of  the mathematical  model  consists  in  the   possibility   
to   use experimental data on impedance  directly   in   the  
mathematical model without solving an  inverse  problem.  We 
mainly discuss an algorithm of an impedance  reconstruction.  
This algorithm only needs some  experimental   data,   which   
can     be   measured  inside  the  waveguide.   By    means  
mathematical  simulation  we estimate   abilities   of   the  
approach  for   waveguide   mode calculation (see also [4]). 
We illustrate some applied aspects of the technic for  ocean 
acoustic waveguides. 

     The model is  also analyzed by  means of a  method   of  
a  parabolic equation (see also [2]). 

\begin{thebibliography}{99}
\bibitem{ed1} Edelshtein S.L. On the Helmholtz Equation in a  Strip  with 
Penetrable Boundary, Funkzionalnyi analiz  i  ego  prilozgenia 
1994, v.28,  No  3,  92-94  (English  translation:  Functional 
Analysis and its Application, V. 28, No. 3, 1994).
\bibitem{ed2} Edelstein S.L. Asymptotic decomposition of boundary value 
problems for Helmholtz equation in a strip with "penetrable" 
boundaries. Deposited VINITI \#2915-B95 03 of November 1995  - 
35p.
\bibitem{ed3} Edelshtein  S.L.  Mode  asymptotics  of  a  slowly  varying 
waveguide with penetrable boundaries,  Akusticheskii  zhurnal, 
1990, 36, No. 2, 355-359 (English  translation:Soviet  Physics 
Acoustics 36 (1990), No. 2,  193-195) .
\bibitem{ed4} Abramyan M.E., Lisnevskii S.S., Edelstein S.L.  On  modes 
models  of  slowly  varying  waveguides   with   "penetrable" 
boundaries. Deposited VINITI \#2916-B95 03 of November 1995  - 
28p.
\end{thebibliography}




\end{document}
