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
Article

Abstract

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
$$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
and
$$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
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.

Keywords

Reflection Neural Network Complex System Positive Constant Spectral Analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    DeRham, G., Variétés différentiables, 2nd Ed. Paris: Hermann 1960.Google Scholar
  2. 2.
    Kato, T., Perturbation Theory for Linear Operators. New York: Springer 1966.Google Scholar
  3. 3.
    Lions, J.L., & E.Magenes, Non-Homogeneous Boundary Value Problems and Applications. New York: Springer 1972.Google Scholar
  4. 4.
    Pekeris, C.L., Theory of propagation of explosive sound in shallow water. Geol. Soc. Am., Memoir27 (1948).Google Scholar
  5. 5.
    Riesz, F., & B.Sz.-Nagy, Functional Analysis. New York: Ungar Publishing Co., 1955.Google Scholar
  6. 6.
    Schwartz, L., Théorie des Distributions, vol. I. Paris: Hermann 1950.Google Scholar
  7. 7.
    Titchmarsh, E.C., Introduction to the Theory of Fourier Integrals. Oxford: University Press 1937.Google Scholar
  8. 8.
    Wilcox, C.H., Initial-boundary value problems for linear hyperbolic partial differential equations of the second order. Arch. Rational Mech. Anal.10, 361–400 (1962).Google Scholar
  9. 9.
    Wilcox, C.H., Transient electromagnetic wave propagation in a dielectric waveguide. Proceedings, Conference on the Mathematical Theory of Electromagnetism, Istituto Nazionale di Alta Matematica, Rome 1974 (to appear).Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

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

Personalised recommendations