A wave impinging on a plane interface between two media must satisfy the kinematic boundary condition

(1) |

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)

(2) |

(3) |

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:

(4) |

Figure 1

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 In anisotropic media, however, this simplification is no longer valid because in general rays and waves travel with different velocities in different directions.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 .

Figure 2

11/18/1997