  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. $(\Delta + E + x_2)u(x_1,x_2) = 0$ 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.
