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

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B_{2}) ). Using the same symbols as in equations (3.1b,c), we define the following functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta _{i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \zeta _{i}^{'} are the same as in equation (3.1d,e). We get the following expressions for the displacements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u_{i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w_{i} using equations (3.2a,b,c,d), where the terms in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{0} are replaced with terms in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B_{0} :


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

The boundary conditions require the continuity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{zz} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{zz} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=0 . Continuity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u gives

$ {\begin{aligned}-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{aligned}} $

For the normal stress, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}


Thus, continuity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{zz} requires that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

We can simplify the equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{zz} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{xz} by noting that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): p is the raypath parameter [see equation (3.1a)]. Also,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

We can now write the four equations in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Problem 3.4b

Derive the Zoeppritz equations for an incident SH-wave.

Solution

For an SH-wave traveling in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): xz -plane, the wave motion involves only displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v parallel to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -axis where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v=v\left(x,\; z,\; t\right) . We take the incident, reflected, and refracted waves in the form [see equations (3.1b,c,d,e)]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}


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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} C_{0} +C_{1} =C_{2},\quad \mathrm{or}\quad C_{1} -C_{2} =-C_{0}. \end{align} (3.4a)

The tangential stress is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{yz} (note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sigma _{xz} =0 ), where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Recalling that we can take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \partial /\partial x=+1 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \partial /\partial z=\pm \cot \delta_{i} [see equation (3.2g)], we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

So


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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} (3.4b)

Solving equations (3.4a,b), we find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

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

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