Archive for Rational Mechanics and Analysis

, Volume 60, Issue 3, pp 259–300 | Cite as

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

  • Calvin Wilcox


ThePekeris differential operator is defined by
$$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$
wherex=(x1,x2,...x n )∈R n ,=(∂/∂x1, ∂/∂x2,...∂/∂x n ), and the functionsc(xn),σ(xn) satisfy
$$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
$$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
wherec1,c2,ϱ1,ϱ2, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc1 and densityϱ1 which overlays a bottom having sound speedc2 and densityϱ2.
In this paper it is shown that the operatorA, acting on a class of functions u (x) which are defined for xn≧0 and vanish for xn=0, defines a selfadjoint operator on the Hilbert space
whereR + n ={xR n :x n >0} anddx =dx1dx2...dxn denotes Lebesgue measure in R + n . The spectral family ofA 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. Whenc1c2 each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c1< c2, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0<xnh by total reflection at the interface xn=h.


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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Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Calvin Wilcox
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake City

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