A wave impinging on a plane interface between two media must satisfy the kinematic boundary condition
If the wave field is described as a superposition of functions of the form , the condition (1) implies that for incident and scattered waves the component of tangential to the boundary ()must be the same. If we divide by the frequency of the incident wave and define a vector of magnitude 1/v (where v is the phase velocity)
Auld (1990) examines condition (3) by looking at the corresponding slowness surfaces (a plot of as a function of its direction), as Figure shows. In this figure the slowness surfaces for both P and S waves are represented for each isotropic medium. The vector is the projection on the interface of the point in the slowness surface that corresponds to the incident P wave phase. From the construction of Figure it is possible to derive easily Snell's law that gives the angles of the scattered phases for both P and S waves:
Snell's law tells us how a given phase changes its direction when it crosses the interface between two media. It also tells us how rays change direction when crossing an interface, but the medium must be isotropic for this to be true, like in Figure . In isotropic media, since rays and waves travel in the same direction with the same velocity, boundary conditions valid for waves are also used to predict the behavior of rays. In anisotropic media, however, this simplification is no longer valid because in general rays and waves travel with different velocities in different directions. This is shown in Figure .