High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium

Abstract

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. $(Δ+ E + x_2)u(x_1,x_2) = 0$ where $E$ is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 100 nodes, with an effort that is independent of the frequency parameter $E$. By combining with high-order Nystrom quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop.