Bottom Interaction Effects on Normal Modes: an Algebraic Approach
We assume that the speed of sound in the water and in the bottom of the ocean is a function of only the depth, and not the range. We also assume that the ocean and its bottom is eventually underlaid with a rigid interface. This problem can be solved by the method of normal modes, involving the eigenvalues and eigenfunctions of a depth dependent ordinary differential equation. We investigate the changes in these eigenvalues and eigenfunctions that result from changes in the depth dependent sound speed within the ocean and its bottom, using a perturbation approach. We formulate the perturbation in terms of an algebraic eigenvalue problem, and we show that it is equivalent to the usual power series expansion in a small parameter.
KeywordsPower Series Normal Mode Sound Speed Fourier Coefficient Algebraic Formulation
Unable to display preview. Download preview PDF.
- 2.E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part II, Oxford University Press, Oxford, England, 1958.Google Scholar
- 3.D. H. Wood, M. Duston, and G. Verma, Changes in eigenvalues due to bottom interaction using perturbation theory, Proceedings of the 11th IMACS World Congress, Oslo, Norway, August 5–9, 1985 (to appear).Google Scholar
- 5.H. Lev-Ari, Cholesky factorization of structured matrices, Second SIAM Conference on Applied Linear Algebra, Raliegh, North Carolina, April 29 — May 2, 1985.Google Scholar
- 6.P. J. Davis, Circulant Matrices, John Wiley and Sons, New York, 1979.Google Scholar
- 7.U. Grenander and G. Szego, Toeplitz Forms and Their Applications, Chelsea Publishing Company, New York, 1984.Google Scholar