Problem 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
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
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(3.4a)
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The tangential stress is
(note that
), where
Recalling that we can take
,
[see equation (3.2g)], we get
So
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(3.4b)
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Solving equations (3.4a,b), we find
The absence of P-waves is important in SH-wave studies.
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Also in this chapter
External links
find literature about Zoeppritz’s equations for incident SV- and SH-waves
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