Using the Foldy-Wouthuysen Transformation to Derive Acoustic Parabolic Equations that Properly Dress Discontinuities

摘要

This effort develops a firm theoretical foundation for the acoustic parabolic equation (PE) used in the presence of penetrable rough interfaces. As the interface roughness extends through the wavelength scale, it induces Bragg scattering (i.e., behaves like a diffraction grating). This is a nontrivial, phase-sensitive problem that involves theoretical and computational challenges that go beyond those found in problems to which the PE is most typically applied. A satisfactory formalism that addresses Bragg scattering should fully integrate the PE with field and rough surface scattering theories. In this report, the Foldy-Wouthuysen (FW) transformation is used to design a PE formalism that addresses this challenge. Along an interface where the density jumps, the full-wave problem predicts a jump in the downrange flux, but the PE generated by the FW transformation buffers this discontinuity by absorbing it into the higher-order boundary conditions. The new formalism is free of the ad hoc fixes that have characterized the PE methods currently used in the vicinity of a jump in the density, and it is ideally suited for the modeling of (forward) scattering from multiscale rough surfaces. The formalism can also be used to generate stochastic equations in circumstances where it has been impossible until now to do so.