A parabolic equation for penetrable rough surfaces: using the Foldy-Wouthuysen transformation to buffer density jumps

The goal of this effort is to design a parabolic equation (PE) that can be fully integrated into modern rough surface scattering theory. The Foldy-Wouthuysen transformation is used to design a PE that addresses this challenge. The paradigm employed is based on a non-relativistic theory of the quantum Lamb shift, in which a parabolic equation (the Schrödinger equation) was used to model a field near a rough surface (the world line of the hydrogen nucleus advected by vacuum fluctuations). With the acoustic field serving as the prototypical classical field, the PE derived using the Foldy-Wouthuysen transformation exploits higher-order boundary conditions to buffer density discontinuities in a manner precisely dictated by the formalism. This is significant because the techniques used in rough surface scattering theory ultimately rely on perturbation theory, conformal mappings, a local method of images, or some similar distortion of the range-independent problem. This type of distortion is not possible with techniques currently employed at a density jump, and so ad hoc rules have been used instead. The new PE allows interfaces where the density jumps (such as the ocean bottom) to be distorted into rough ones, and so it is fully compatible with rough surface scattering theory.