# Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

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## Abstract

ThePekeris differential operator is defined by where and where

$$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$

*x*=(*x*_{1},*x*_{2},...*x*_{ n })∈*R*_{ n },*∇*=(∂/∂*x*_{1}, ∂/∂*x*_{2},...∂/∂*x*_{ n }), and the functions*c*(*x*_{n}),*σ*(*x*_{n}) satisfy$$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$

$$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$

*c*_{1},*c*_{2},*ϱ*_{1},*ϱ*_{2}, and*h*are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speed*c*_{1}and density*ϱ*_{1}which overlays a bottom having sound speed*c*_{2}and density*ϱ*_{2}.In this paper it is shown that the operator

*A*, acting on a class of functions u (x) which are defined for x_{n}≧0 and vanish for x_{n}=0, defines a selfadjoint operator on the Hilbert space where*R*_{+}^{ n }={*x*∈*R*^{ n }:*x*_{ n }>0} and*dx*=*dx*_{1}*dx*_{2}...*dx*_{n}denotes Lebesgue measure in R_{+}^{n}. The spectral family of*A*is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. When*c*_{1}≧*c*_{2}each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c_{1}< c_{2}, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0<x_{n}*h*by total reflection at the interface x_{n}=*h*.## Keywords

Reflection Neural Network Complex System Positive Constant Spectral Analysis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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