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

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