# Dictionary:Zoeppritz’s equations

(zō’ pritz) Equations that express the partition of energy when a plane wave impinges on an acoustic-impedance contrast. In the general case for an interface between two solids when the incident angle is not zero, four waves are generated: reflected P-wave and S-wave and transmitted P-wave and S-wave. The partition of energy among these is found from four boundary conditions which require continuity of normal and tangential displacement and stress. Using the symbols given in Figure S-12, Snell’s law states:

_{P1})/(

*V*

_{P1})=(sinθ

_{S1})/(

*V*

_{S1})=(sinθ

_{P2})/(

*V*

_{P2})=(sinθ

_{S2})/(

*V*

_{S2}) ;

this defines all angles. For an incident plane P-wave of unity amplitude, the continuity conditions yield the four Zoeppritz equations shown in Figure Z-1.

Figure Z-1 also shows the variation of energy with angle for several sets of parameters. Beyond the critical angles for P- and S-waves, the respective refracted waves vanish. The increase in reflection energy near the critical angle is sometimes referred to as the **wide-angle phenomenon** and is sometimes exploited in seismic surveying. The same relationships in terms of potentials are called **Knott’s equations**. See Sheriff and Geldart (1995, 73–75). Because no provision was made in the equation’s derivation for the head waves, these equations do not give head-wave amplitudes or correct values beyond the critical angle.