# 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## Preview

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