Zoeppritz’s equations for incident SV- and SH-waves

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Problem 3.4a

3.4a Derive Zoeppritz’s equations for an SV-wave incident on a solid/solid interface.

Solution

Figure 3.1a defines the positive directions of displacements except that the incident P-wave is replaced by an incident SV-wave whose positive direction is down and to the left (the same as that of ). Using the same symbols as in equations (3.1b,c), we define the following functions:

where and are the same as in equation (3.1d,e). We get the following expressions for the displacements and using equations (3.2a,b,c,d), where the terms in are replaced with terms in :


The boundary conditions require the continuity of , , and at . Continuity of and gives

For the normal stress, we have


Thus, continuity of requires that

Continuity of the tangential stress, , gives

We can simplify the equations for and by noting that

where is the raypath parameter [see equation (3.1a)]. Also,

We can now write the four equations in the form

Problem 3.4b

3.4b Derive the Zoeppritz equations for an incident SH-wave.

Solution

For an SH-wave traveling in the -plane, the wave motion involves only displacement parallel to the -axis where . We take the incident, reflected, and refracted waves in the form [see equations (3.1b,c,d,e)]


The boundary conditions require that the tangential displacement and tangential stress be continuous at . The first condition gives


(3.4a)

The tangential stress is (note that ), where

Recalling that we can take , [see equation (3.2g)], we get

So


(3.4b)

Solving equations (3.4a,b), we find

The absence of P-waves is important in SH-wave studies.

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