Improving the parabolic equation solution for problems involving sloping fluid‐solid interfaces.

Abstract

Several approaches are being investigated for improving the elastic parabolic equation for problems involving sloping fluid‐solid interfaces. Approaches based on single scattering and energy conservation provide accurate solutions for problems involving sloping fluid‐fluid and solid‐solid interfaces, but the mixed‐media problem has proven to be more challenging. The energy‐conservation approach has been applied previously by deriving a linear equivalent to the nonlinear expression for energy flux. One of the approaches that are currently being investigated is based on going back to the nonlinear expression. Although it would not be practical to solve the full nonlinear scattering problem, promising results have been obtained for the fluid‐fluid case with this approach by correcting the amplitude at only one grid point near the interface. With this approach, the nonlinear problem reduces to the evaluation of a quadratic function. Another approach that is being investigated is based on an alternative formulation that involves the vertical displacement and a quantity that is proportional to the normal stress on a horizontal interface. In these variables, the interface conditions across a horizontal interface are first order, and this may facilitate the extension of the single‐scattering solution to the mixed‐media problem. [Work supported by the Office of Naval Research.]