Reflection and refraction of sound waves at a smooth boundary of two liquids

The problem of the scattering of sound waves by the boundary between two liquids has an elementary solution in a case of flat boundary and can be reduced to a solution of an integral equation with the singular oscillating kernel. In practical applications, the vast majority of boundaries are treated as random surfaces and approximate methods produce only some statistical moments of the sound field. When the boundary is smooth and non-flat there is no simple method that can produce physically consistent results. This work presents an approach, which exploits a smoothness of the boundary to build an orthogonal coordinate system and rewrite wave equations at both sides of the surface in this system. For a given incident field of sound pressure and particle velocities by satisfying the boundary conditions and applying the split-step method and FFT, a reflected and refracted field of sound pressure was built as well as particle velocity in both media, producing physically reliable interference patterns.