Eigenvalue methods in the solution of acoustical scattering problems from submerged objects

Abstract

Waterman's extended boundary condition (EBC) method has proven to be extremely successful in dealing with scattering from submerged targets. An alternate and useful computational method (which is related to Enskog's method) separately considers the Heimholtz integral equation at exterior and interior points, as does the EBC method. However, the interior problem can be shown to transform to an eigenvalue problem, which produces eigenstates that span the space of the displacement on the object surface. The eigenstates can be used directly to solve the exterior problem and yield a numerically stable and convergent solution. This new method avoids problems encountered by the EBC method. For example, in the usual EBC approach, the unknown surface terms are expanded on a known, but to some extent arbitrary, basis set, with unknown expansion coefficients. The number of expansion terms on the surface and those of the incident partial waves must match. Computationally, this requirement can often dictate there be many more incident partial waves than strictly required for convergence of the incident field. This leads to small‐incidence high‐order components, that, in turn, render the resulting matrix problem ill‐conditioned. The method used in this study is presented with several representative numerical examples including objects of aspect ratios of 30 to 1 for kL/2 values to 120.