Zoeppritz equations

Definition

Let two homogeneous isotropic elastic half-spaces $z < 0$ and $z > 0$ be welded along the plane $z = 0$. Let medium $j \in \set{1, 2}$ have density $\rho_j$, P-speed $\alpha_j$ and S-speed $\beta_j$. Let a plane P-wave of unit amplitude travel in medium 1 in the $xz$-plane and hit the interface at the angle $\theta_1$ from the normal. It produces a reflected P-wave at angle $\theta_1$ and a reflected SV-wave at angle $\varphi_1$. It also produces a transmitted P-wave at angle $\theta_2$ and a transmitted SV-wave at angle $\varphi_2$, with $$\frac{\sin\theta_1}{\alpha_1} = \frac{\sin\varphi_1}{\beta_1} = \frac{\sin\theta_2}{\alpha_2} = \frac{\sin\varphi_2}{\beta_2}$$ by Snell's law. Let $R_P, R_S, T_P, T_S$ be their displacement amplitudes. The Zoeppritz equations are the four linear equations in $R_P, R_S, T_P, T_S$ stating that, at $z = 0$:

1. the horizontal displacement $u_x$ is continuous;
2. the vertical displacement $u_z$ is continuous;
3. the normal stress $\sigma_{zz}$ is continuous;
4. the shear stress $\sigma_{xz}$ is continuous.

Here $\sigma$ is the Cauchy stress, given by Hooke's law in each medium.