A Legendre Galerkin Technique for Approximating the Eigenvalues of the Depth Separated Wave Equation in Underwater Acoustics

The mean weighted residual error of a candidate eigenpair, in underwater acoustics, is taken to be zero for every member of a set of test functions. The solution of this weak formulation of the problem is approximated in the Galerkin method when the test functions are restricted to a finite dimensional subspace. The error is determined by the approximation theoretic characteristic of the finite dimensional subspace of polynomials. A Legendre polynomial basis is constructed that satisfies the boundary and interface conditions of the problem, in a process called recombination. In cases when corners exist in the water sound speed profile, the eigenvalue error bounds decay as [Formula: see text] where M is the number of recombined polynomials in the subspace. The asymptotic error behavior is demonstrated in an example.