Improved methods for determining eigenfunctions in multi-layered normal-mode problems

Abstract

This paper derives and illustrates the utility of the derivative of a determinant that arises from the solution of the wave equation in a multi-layered medium. The horizontal wavenumbers at which the determinant is zero are the eigenvalues of the solution. The derivative of the determinant with respect to the horizontal wavenumber therefore can be an aid in finding the roots (zeros). More importantly, the eigenfunction normalization constant is shown to be a function of this derivative. Two types of boundary are discussed, an impedance boundary and an infinite half-space. The normalization using the derivative of the determinant is shown to apply to both types. The derivative of the determinant is also useful in evaluating mode group velocities. An algorithm for the efficient determination of the derivative of a determinant is given. This algorithm includes a rapid computational technique that applies special properties of matrix operations and utilizes the sparseness of the determinant.