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

{{publication
 | name    = Problems in Exploration Seismology and their Solutions
 | image   = [[File:897.Def.Lz-cropped.png|frameless|180px|link=Problems in Exploration Seismology and their Solutions]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = [[Problems in Exploration Seismology and their Solutions]]
 | part    = 
 | chapter = 3
 | pages  = 47 - 77
 | author  = [[Lloyd P. Geldart]] and [[Robert E. Sheriff]]
 | doi     = 10.1190/1.9781560801733
 | pdf     = 
 | store   = https://shop.seg.org/shop/products/detail/897
 | isbn    = ISBN 9781560801153
}}
== 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 <math>B_{2})</math>). Using the same symbols as in equations (3.1b,c), we define the following functions:

<math>
\begin{align}
\chi _{0} =B_{0} e^{\mathrm{j}\omega \zeta _{0}^{\prime} },\qquad \chi _{1} &=B_{1} e^{\mathrm{j}\omega \zeta _{1}^{\prime} }, \qquad \chi _{2} =B_{2} e^{\mathrm{j}\omega \zeta _{2}^{\prime} } ;\\
\phi _{1} &=A_{1} e^{\mathrm{j}\omega \zeta _{1} }, \qquad \phi _{2} =A_{2} e^{\mathrm{j}\omega \zeta _{2} }.
\end{align}
</math>

where <math>\zeta _{i}</math> and <math>\zeta _{i}^{'}</math> are the same as in equation (3.1d,e). We get the following expressions for the displacements <math>u_{i}</math> and <math>w_{i}</math> using equations (3.2a,b,c,d), where the terms in <math>A_{0}</math> are replaced with terms in <math>B_{0}</math>:


<math>
\begin{align}
w_{1} &=-\chi_{0} \sin\delta_1 - \chi_{1} \sin \delta_{1} + \phi_{1} \cos \theta_{1}, \\
w_{2} &= \qquad\qquad-\chi_{2} \sin \delta_{2} - \phi_2 \cos \theta_{2}; \\
u_{1} &=-\chi_{0} \cos \delta_{1} + \chi_{1} \cos \delta_{1} +\phi_{1} \sin \theta _{1}, \\
u_{2} &= \qquad\qquad-\chi_{2} \cos \delta_{2} + \phi_2 \sin \theta _{2}.
\end{align}
</math>

The boundary conditions require the continuity of <math>w</math>, <math>u</math>, <math>\sigma _{zz}</math> and <math>\sigma _{zz}</math> at <math>z=0</math>. Continuity of <math>w</math> and <math>u</math> gives

<math>
\begin{align}
-B_{0} \sin \delta_{1} -B_{1} \sin \delta_{1} +A_{1} \cos \theta_{1} &=-B_{2} \sin \delta_{2} -A_{2} \cos \theta_{2},\\
-B_{0} \cos \delta_{1} +B_{1} \cos \delta_{1} +A_{1} \sin \theta_{1} &=-B_{2} \cos \delta_{2} +A_{2} \sin \theta_{2}.
\end{align}
</math>

For the normal stress, we have

<math>
\begin{align}
\sigma_{zz} &=\lambda \Delta +2\mu \varepsilon_{zz} = \lambda \left(\frac{\partial u}{\partial x} +\frac{\partial w}{\partial z} \right)+2\mu \left(\frac{\partial w}{\partial z} \right)=\lambda \left(u_{x} +w_{z} \right)+2\mu w_{z} \\
&=\left(\lambda +2\mu \right)\left(u_{x} +w_{z} \right)-2\mu u_{x} =\rho\alpha ^{2} \left(u_{x} +w_{z} \right)-2\rho\beta ^{2} u_{x}.
\end{align}
</math>


Thus, continuity of <math>\sigma _{zz}</math> requires that

<math>
\begin{align}
&\rho_{1} \alpha_{1}^{2} [(-B_{0} \cos \delta_{1} +B_{1} \cos \delta_{1} +A_{1} \sin \theta _{1})+(B_{0} \cot \delta_{1} \sin \delta_{1} \\
&\qquad\qquad\quad-B_{1} \cot \delta_{1} \sin \delta_{1} +A_{1} \cot \theta _{1} \cos \theta _{1} )]\\
&\qquad\quad-2\rho_{1} \beta _{1}^{2} \left(-B_{0} \cos \delta_{1} +B_{1} \cos \delta_{1} +A_{1} \sin \theta _{1} \right)\\
&\quad=\rho_{2} \alpha_{2}^{2} \left[\left(-B_{2} \cos \delta_{2} +A_{2} \sin \theta _{2} \right)+\left(B_{2} \cot \delta_{2} \sin \delta_{2} +A_{2} \cot \theta _{2} \cos \theta _{2} \right)\right]\\
&\qquad\quad-2\rho_{2} \beta _{2}^{2} \left(-B_{2} \cos \delta_{2} +A_{2} \sin \theta _{2} \right).
\end{align}
</math>

Continuity of the tangential stress, <math>\sigma _{xz} =\mu \varepsilon _{xz} =\mu \left(u_{z} +w_{x} \right)</math>, gives

<math>
\begin{align}
&\mu _{1} [\left(B_{0} \cot \delta_{1} \cos \delta_{1} +B_{1} \cot \delta_{1} \cos \delta_{1} +A_{1} \cot \theta _{1} \sin \theta _{1} \right)\\
&\qquad\qquad\qquad +\left(-B_{0} \sin \delta_{1} -B_{1} \sin \delta_{1} +A_{1} \cos \theta _{1} \right)]\\
&\quad =\mu _{2} [\left(B_{2} \cot \delta_{2} \cos \delta_{2} -A_{2} \cot \theta _{2} \sin \theta _{2} \right)+(-B_{2} \sin \delta_{2} -A_{2} \cos \theta _{2})].
\end{align}
</math>

We can simplify the equations for <math>\sigma _{zz}</math> and <math>\sigma _{xz}</math> by noting that

<math>
\begin{align}
&\rho\alpha^{2} (-\cos \delta+ \cot\delta \sin \delta) + 2\rho\beta ^{2} \cos \delta=2\rho\beta ^{2} \cos \delta=\left(W/p\right) \sin2\delta,\\
&\rho\alpha^{2} \left(\sin \theta +\cot \theta \cos \theta \right)-2\rho\beta^{2} \sin \theta = \rho\left[\left(\alpha ^{2} /\sin \theta \right)-2\beta ^{2} \sin \theta \right]\\
&\quad = \rho\sin \theta \left(1/p^{2} -2\sin ^{2} \delta/p^{2} \right)=\left(\rho\alpha /p\right)\cos 2\delta=\left(Z/p\right) \cos 2\delta,
\end{align}
</math>

where <math>p</math> is the raypath parameter [see equation (3.1a)]. Also,

<math>
\begin{align}
\mu \left(\cot \delta\cos \delta-\sin \delta\right)=\rho\beta ^{2} \left(\frac{\cos ^{2} \delta-\sin ^{2} \delta}{\sin \delta} \right)=\left(W/p\right)\cos 2\delta.
\end{align}
</math>

We can now write the four equations in the form

<math>
\begin{align}
A_{1} \cos \theta _{1} -B_{1} \sin \delta_{1} +A_{2} \cos \theta _{2} +B_{2} \sin \delta_{2} &=B_{0} \sin \delta_{1}, \\
A_{1} \sin \theta _{1} +B_{1} \cos \delta_{1} -A_{2} \sin \theta _{2} +B_{2} \cos \delta_{2} &=B_{0} \cos_l, \\
A_{1} Z_{1} \cos2\delta_1 -B_{1} W_{1} \sin2\delta_1 -A_{2} Z_{2} \cos2\delta_2 -B_{2} W_{2} \sin2\delta_2 &=-B_{0} W_{1} \sin2\delta_1, \\
\left(\beta _{1} /\alpha _{1} \right)A_{1} W_{1} \sin 2\theta _{1} +B_{1} W_{1} \cos 2\delta_{1} +\left(\beta _{2} /\alpha _{2} \right)A_{2} W_{2} \sin 2\theta _{2} &-B_{2} W_{2} \cos2\delta_2 \\
&=-B_{0} W_{1} \cos 2\delta_{1}.
\end{align}
</math>

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

=== Solution ===
For an SH-wave traveling in the <math>xz</math>-plane, the wave motion involves only displacement <math>v</math> parallel to the <math>y</math>-axis where <math>v=v\left(x,\; z,\; t\right)</math>. We take the incident, reflected, and refracted waves in the form [see equations (3.1b,c,d,e)]

<math>
\begin{align}
v_{1} &=C_{0} e^{\mathrm{j}\omega p\left(x-z\cot \delta_{1} \right)} + C_{1} e^{\mathrm{j}\omega p\left(x+z\cot \delta_{1} \right)},\\
v_{2} &=C_{2} e^{\mathrm{j}\omega p\left(x-z\cot \delta_{2} \right)}.
\end{align}
</math>


The boundary conditions require that the tangential displacement and tangential stress be continuous at <math>z=0</math>. The first condition gives

{{NumBlk|:|<math>
\begin{align}
C_{0} +C_{1} =C_{2},\quad \mathrm{or}\quad C_{1} -C_{2} =-C_{0}.
\end{align}
</math>|{{EquationRef|3.4a}}}}

The tangential stress is <math>\sigma _{yz}</math> (note that <math>\sigma _{xz} =0</math>), where

<math>
\begin{align}
\sigma _{yz} =\mu \varepsilon _{yz} =\mu \left(\frac{\partial v}{\partial z} +\frac{\partial w}{\partial y} \right)=\mu \frac{\partial v}{\partial z}.
\end{align}
</math>

Recalling that we can take <math>\partial /\partial x=+1</math>, <math>\partial /\partial z=\pm \cot \delta_{i}</math> [see equation (3.2g)], we get

<math>
\begin{align}
\mu_{1} \left(-C_{0} \cot \delta_{1} +C_{1} \cot \delta_{1} \right)=-\mu _{2} C_{2} \cot \delta_{2},
\end{align}
</math>

So

{{NumBlk|:|<math>
\begin{align}
\mu _{1} C_{1} \cot \delta_{1} +\mu _{2} C_{2} \cot \delta_2 = \mu _{1} C_{0} \cot \delta_1.
\end{align}
</math>|{{EquationRef|3.4b}}}}

Solving equations (3.4a,b), we find

<math>
\begin{align}
\frac{C_{1} }{C_{0}} &= \frac{\rho_{1} \beta _{1}^{2} \cot \delta_{1} -\rho_{2} \beta _{2}^{2} \cot \delta_{2}}{\rho_{1} \beta _{1}^{2} \cot \delta_{1} + \rho_{2} \beta _{2}^{2} \cot \delta_{2}} =\frac{\left(W_{1} \cos \delta_{1} -W_{2} \cos \delta_{2} \right)}{\left(W_{1} \cos \delta_{1} +W_{2} \cos \delta_{2} \right)}, \\
\frac{C_{2} }{C_{0}} &= \frac{2\mu _{1} \cot \delta_{i} }{\left(\mu _{1} \cot \delta_{i} +\mu _{2} \cot \delta_{i} \right)} = \frac{2W_{1} \cos \delta_{1}}{\left(W_{1} \cos \delta_{1} +W_{2} \cos \delta_{2} \right)}.
\end{align}
</math>

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

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== Also in this chapter ==
* [[General form of Snell’s law]]
* [[Reflection/refraction at a solid/solid interface and displacement of a free surface]]
* [[Reflection/refraction at a liquid/solid interface]]
* [[Reinforcement depth in marine recording]]
* [[Complex coefficient of reflection]]
* [[Reflection and transmission coefficients]]
* [[Amplitude/energy of reflections and multiples]]
* [[Reflection/transmission coefficients at small angles and magnitude]]
* [[Magnitude]]
* [[AVO versus AVA and effect of velocity gradient]]
* [[Variation of reflectivity with angle (AVA)]]

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