BOUNDARY CONDITIONS

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

$${\bf v}_{1} \ =\ {\bf v}_{1}.$$ (1)

that says that the particle velocity ${\bf v}$must be continuos at all points on the boundary between medium 1 and medium 2 (Auld, 1990).

If the wave field is described as a superposition of functions of the form $\exp (i {\bf k r})$, the condition (1) implies that for incident and scattered waves the component of ${\bf k}$ tangential to the boundary (${\bf k_{\parallel}}$)must be the same. If we divide ${\bf k}$ by the frequency $\omega$ of the incident wave and define a vector ${\bf \tilde{k}}$ of magnitude 1/v (where v is the phase velocity)

$${\bf \tilde{k}} \ =\ \frac{\bf k}{\omega} \ =\ \frac{{\bf k}}{\Vert {\bf k} \Vert} \frac{1}{v},$$ (2)

the continuity of ${\bf k_{\parallel}}$ implies the continuity of ${\bf \tilde{k}_{\parallel}}$, usually called ray parameter ${\bf p}$:

$${\bf p_{1}} \ =\ {\bf p_{2}}.$$ (3)

Since we know that ${\bf p}$ is parallel to the interface, the ray parameter is usually considered as an scalar p with a positive or negative sign in the front to indicate the sign of angle of the incident phase with respect to the normal.

Auld (1990) examines condition (3) by looking at the corresponding slowness surfaces (a plot of $\Vert {\bf \tilde{k}} \Vert$ 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 $CA = AB = {\bf p}$ 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:

$$\begin{eqnarray} p & = & \frac{1}{v_{1_P}} \sin(\theta_{P_i}) \ =\ \frac{1}{v_{... ...S}} \sin(\theta_{S_r}) \ =\ \frac{1}{v_{1_P}} \sin(\theta_{P_r}).\end{eqnarray}$$ (4)

 

slow-surf
Figure 1 Plane wave scattering at a plane boundary between two isotropic media. From this construction Snell's law can be easily derived.

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 .

 

slow-surf-ani
Figure 2 Plane wave scattering at a plane boundary between two anisotropic media. The slowness surfaces are separated and nonspherical. Ray directions (dashed arrows) and phase directions (continuos arrows) are no longer the same.(Modified from Aki and Richards, 1980.)