Dictionary:Zoeppritz’s equations

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(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:

(sinθP1)/(VP1)=(sinθS1)/(VS1)=(sinθP2)/(VP2)=(sinθS2)/(VS2) ;

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.