Reflection/refraction at a solid/solid interface and displacement of a free surface

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 3.2a

Derive the Zoeppritz equations for a P-wave incident on a solid/solid interface.

Background

The normal and tangential displacements plus the normal and tangential stresses must be continuous when a P-wave is incident at the angle $ \theta _{1} $ on an interface between two solid media (see problem 2.10).

Solution

We use the functions in equations (3.1b,c,d,e) to represent the displacements of the waves, the positive direction of displacement for the waves being shown in Figure 3.1a. (We omit the factor $ e^{-\mathrm {j} \omega t} $ because the boundary conditions do not depend upon the time $ t $, hence this factor cancels out).

We first derive the equations expressing the continuity of normal and tangential displacements, $ w $ and $ u $. These equations are obtained by resolving the various wave displacements into $ z $- and $ x $-components. Thus,


$ {\begin{aligned}w_{1}&=-A_{0}\cos \theta _{1}e^{\mathrm {j} \omega \zeta _{0}}+A_{1}\cos \theta _{1}e^{\mathrm {j} \omega \zeta _{1}}-B_{1}\sin \delta _{1}e^{\mathrm {j} \omega \zeta _{1}^{\prime }},\end{aligned}} $ (3.2a)


$ {\begin{aligned}w_{2}&=\qquad \qquad -A_{2}\cos \theta _{2}e^{\mathrm {j} \omega \zeta _{2}}-B_{2}\sin \delta _{2}e^{\mathrm {j} \omega \zeta _{2}^{\prime }},\end{aligned}} $ (3.2b)


$ {\begin{aligned}u_{1}&=A_{0}\sin \theta _{1}e^{\mathrm {j} \omega \zeta _{0}}+A_{1}\sin \theta _{1}e^{\mathrm {j} \omega \zeta _{1}}+B_{1}\cos \delta _{1}e^{\mathrm {j} \omega \zeta _{1}^{\prime }},\end{aligned}} $ (3.2c)


$ {\begin{aligned}u_{2}&=\qquad \qquad A_{2}\sin \theta _{2}e^{\mathrm {j} \omega \zeta _{2}}-B_{2}\cos \delta _{2}e^{\mathrm {j} \omega \zeta _{2}^{\prime }}.\end{aligned}} $ (3.2d)

At the interface, $ z=0 $ and $ w_{1}=w_{2} $, $ u_{1}=u_{2} $. The exponentials all reduce to $ e^{\mathrm {j} \omega x} $, hence cancel out, and we get for the normal and tangential displacements, respectively,


$ {\begin{aligned}\left(-A_{0}+A_{1}\right)\cos \theta _{1}-B_{1}\sin \delta _{1}=-A_{2}\cos \theta _{2}-B_{2}\sin \delta _{2},\end{aligned}} $ (3.2e)


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} \left(A_{0} +A_{1} \right)\sin \theta _{1} +B_{1} \cos \delta_{1} = A_{2} \sin \theta_{2} -B_{2} \cos \delta_{2}. \end{align} (3.2f)

To apply the boundary conditions for the normal and tangential stresses, we differentiate equations (3.2a,b,c,d) with respect to 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/":): x 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/":): z . Equations (3.1d,e) show that the differentiation with respect to 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/":): x 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/":): z multiplies each function by 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/":): \mathrm{j}\omega p and either 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/":): \pm \mathrm{j}\omega p\cot \theta _{i} or $ \pm \mathrm {j} \omega p\cot \delta _{i} $. The common factor 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/":): \mathrm{j}\omega p will cancel in the end, so we simplify the derivation by taking


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} \partial /\partial x=1,\quad \partial /\partial z=\pm \cot \theta _{i}\quad \mathrm{or}\quad \pm \cot \delta_{i}. \end{align} (3.2g)

From equations (2.1b,c,e,h,i) we get for the normal and tangential stresses:

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(u_{x} +w_{z} \right)+2\mu w_{z} =\lambda u_{x} +\left(\lambda +2\mu \right)w_{z}, \\ \sigma _{xz} &=\mu \left(u_{z} +w_{x} \right), \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/":): u_{x} , 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_{z} , 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_{x} , 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_{z} are partial derivatives with respect to $ x $ 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/":): z . This allows us to find the normal and tangential stresses in each medium and equate them 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 . The result for the normal stresses 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/":): \begin{align} \lambda _{1} \left[\left(A_{0} +A_{1} \right)\sin \theta _{1} +B_{1} \cos \delta_{1} \right]+\left(\lambda _{1} +2\mu _{1} \right)\left(A_{0} +A_{1} \right)\cos \theta _{1} \cot \theta _{1} -B_{1} \cos \delta_{1}\\ =\lambda _{2} \left(A_{2} \sin \theta _{2} -B_{2} \cos \delta_{2} \right)+\left(\lambda _{2} +2\mu _{2} \right)\left(A_{2} \cos \theta _{2} \cot \theta _{2} +B_{2} \cos \delta_{2} \right). \end{align}

Writing 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/":): \lambda =\left(\lambda +2\mu \right)-2\mu =\rho\alpha ^{2} -2\rho\beta ^{2} (see equations (9,6) and (9,7) in Table 2.2a) and recalling 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/":): \sin 2x=2\sin x\cos x , 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/":): \cos 2x=\cos^{2} x-\sin^{2} x , the equation can be changed to 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} \left(A_{0} +A_{1} \right)Z_{1} \cos2\delta_1 -B_{1} W_{1} \sin2\delta_1 =A_{2} Z_{2} \cos 2\delta_{2} +B_{2} W_{2} \sin2\delta_2, \end{align} (3.2h)

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/":): Z_{i} =\rho_{i} \alpha _{i} , 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} = \rho_{i} \beta _{i} ; 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_{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} are called impedances.

In the same way we get for the tangential stresses the equation

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

This can be simplified using equation (3.1a) to give


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} &\left(-A_{0} +A_{1} \right)\left(\beta _{1} /\alpha _{1} \right)W_{1} \sin2\theta_1 + B_{1} W_{1} \cos2\delta_1& \\ &\qquad =-A_{2} \left(\beta _{2} /\alpha _{2} \right)W_{2} \sin 2\theta_2 + B_{2} W_{2} \cos 2\delta_2. \end{align} (3.2i)

Equations (3.1e,f,h,i) are known as the Zoeppritz equations. For ease of reference, we have collected them below:


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} &\left(-A_{0} +A_{1} \right)\cos \theta _{1} -B_{1} \sin \delta_{1} = -A_{2} \cos \theta _{2} -B_{2} \sin \delta_2, \end{align} (3.2e)


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} &\left(A_{0} +A_{1} \right)\sin \theta _{1} +B_{1} \cos \delta_{1} = A_{2} \sin \theta _{2} -B_{2} \cos \delta_{2}, \end{align} (3.2f)


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} &\left(A_{0} +A_{1} \right)Z_{1} \cos2\delta_1 -B_{1} W_{1} \sin2\delta_1 =A_{2} Z_{2} \cos 2\delta_{2} +B_{2} W_{2} \sin2\delta_2, \end{align} (3.2h)


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} &\left(-A_{0} +A_{1} \right)\left(\beta _{1} /\alpha _{1} \right)W_{1} \sin2\theta_1 +B_{1} W_{1} \cos 2\delta_{1} \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad=-A_{2} \left(\beta _{2} /\alpha _{2} \right)W_{2} \sin 2\theta _{2} +B_{2} W_{2} \cos2\delta_2. \end{align} (3.2i)

Problem 3.2b

3.2b Derive the equations below for the tangential and normal 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 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 , of a free surface for an incident P-wave of amplitude 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} :


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} u/A_{0} &=\left[2/\left(m+n\right)\right]\left(m\sin \theta+\cos \delta\right) e^{\mathrm{j}\omega \left(px-t\right)}, \\ w/A_{0} &=\left[-2/\left(m+n\right)\right]\left(n\cos \theta+\sin \delta\right) e^{\mathrm{j}\omega \left(px-t\right)}, \end{align} (3.2e)

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/":): m=(\beta/\alpha) \tan2\delta , 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/":): n=\left(\alpha /\beta \right)\cos2\delta/\sin2\theta , $ \theta $ 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/":): \delta being the angles of incidence of the P- and S-waves, respectively.

Solution

To determine the displacements at a free surface, we start by disregarding equations (3.2e,f) because there are no constraints on displacements at a free surface. After setting 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_{2}=0=B_{2} , we are left with


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} Z \cos2\delta -BW \sin2\delta &=-A_{0} Z \cos2\delta, \end{align} (3.2j)


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} \left(\beta /\alpha \right)W \sin2\theta +BW \cos2\delta &=A_{0} \left(\beta /\alpha \right)W \sin2\theta, \end{align} (3.2k)

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/":): Z=\rho\alpha , 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=\rho\beta , 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/":): \beta /\alpha =W/Z , and we have dropped unnecessary subscripts. These equations can be written


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} -mB=-A_{0}, \qquad A_{1} +nB=A_{0}, \end{align} (3.2l)

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} m&=\left(W\sin 2\delta\right)/\left(Z\cos 2\delta\right)=\left(\beta /\alpha \right) \tan2\delta =\left(\sin \delta\tan 2\delta\right)/\sin \theta, \end{align} (3.2m)


$ {\begin{aligned}n&=\left(\cos 2\delta \right)/[\left(\beta /\alpha \right)\sin 2\theta ]=\left(\cos 2\delta \right)/\left(2\cos \theta \sin \theta \right).\end{aligned}} $ (3.2n)

The solution of equations (3.2Failed 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/":): \ell ) 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/":): \begin{align} A_{1} /A_{0} = \left(m-n\right)/\left(m+n\right),\quad B_{1} /A_{0} =2/\left(m+n\right). \end{align}

In equations (3.2a,c) we set 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 , the factor 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/":): e^{\mathrm{j}\omega px} drops out, and 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} u/A_{0} &= \left(1+A_{1} /A_{0} \right)\sin \theta +\left(B/A_{0} \right)\cos \delta, \end{align} (3.2o)


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/A_{0} &= \left(-1+A_{1}/A_{0} \right)\cos \theta -\left(B/A_{0} \right)\sin \delta. \end{align} (3.2p)

We now reinsert the values 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/":): A_{1} /A_{0} 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/":): B_{1} /A_{0} in terms of m 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/":): n , and equations (3.2o,p) become


$ {\begin{aligned}u/A_{0}&=\left\{\left[1+\left(m-n\right)/\left(m+n\right)\left]\sin \theta +\right[2/\left(m+n\right)\right]\cos \delta \right\}e^{\mathrm {j} \omega \left(px-t\right)}\\&=\left\{\left[2/\left(m+n\right)\right]\left(m\sin \theta +\cos \delta \right)\right\}e^{\mathrm {j} \omega \left(px-t\right)},\end{aligned}} $ (3.2q)


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/A_{0} &=\left\{\left[-2/\left(m+n\right)\right]\left(n\cos \theta +\sin \delta\right)\right\}e^{\mathrm{j}\omega \left(px-t\right)}. \end{align} (3.2r)

Problem 3.2c

3.2c Show that the displacements of a free surface at normal incidence are

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} u/A_{0} =0,\qquad w/A_{0} =-2. \end{align}

Solution

For normal incidence at the surface, 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 , 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/":): \theta =0=\delta . Equations (3.2j,k) give 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_{1} /A_{0} =-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/":): B/A_{0} =0 . Substituting in equations (3.2o,p), 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/":): u/A_{0} =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/":): w/A_{0} =-2 .

Problem 3.2d

3.2d Show that the displacements of a free surface of a solid, 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/":): \theta=45^{\circ} , $ \alpha =3.0 $ km/s, 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/":): \beta /\alpha =1/\sqrt{2} , 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 =1/\sqrt{2} , are

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} u/A_{0} =1.793,\quad w/A_{0} =-1.035. \end{align}

Solution

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/":): \theta =45^{\circ} , 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/":): \alpha =3 km/s, 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/":): \left(\beta /\alpha \right)=1/\sqrt{2} , 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/":): \sin\delta =\left(\beta /\alpha \right) \sin \theta =(1/\sqrt{2} )^{2} , that 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/":): \sin\delta =1/2 , 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/":): \delta=30^{\circ} . From the definitions of m and n, we get

$ {\begin{aligned}m&=\left(\beta /\alpha \right)\tan 2\delta =(1/{\sqrt {2}})\tan 60^{\circ }=1.225,\\n&=\left(\alpha /\beta \right)(\cos 2\delta /\sin 2\theta )={\sqrt {2}}\left(\cos 60^{\circ }/\sin 90^{\circ }\right)=0.707.\end{aligned}} $

Equations (3.2q,r) now give (omitting the factor 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/":): e^{\mathrm{j}\omega \left(px-t\right)} ).

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} u/A_{0} &=\left[2/\left(1.225+0.707\right)\right]\left(1.225\sin 45^{\circ} +\cos 30^{\circ} \right)=1.793,\\ w/A_{0} &=\left[-2/\left(1.225+0.707\right)\right]\left(0.707\cos 45^{\circ} +\sin 30^{\circ} \right)=1.035. \end{align}

Problem 3.2e

3.2e Show that the displacements at the surface of the ocean are 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/A_{0} =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/":): w/A_{0} = -2\cos \theta .

Solution

In a fluid 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 and equation (3.2j) 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/":): A_{1} /A_{0} =-1 , so equations (3.20,p) show 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/":): u/A_{0} =0 and $ w/A_{0}=-2\cos \theta $.

Continue reading

Previous section Next section
General form of Snell’s law Reflection/refraction at a liquid/solid interface
Previous chapter Next chapter
Theory of Seismic Waves Geometry of seismic waves

Table of Contents (book)

Also in this chapter

External links

find literature about
Reflection/refraction at a solid/solid interface and displacement of a free surface
Retrieved from "http://wiki.seg.org/w/index.php?title=Reflection/refraction_at_a_solid/solid_interface_and_displacement_of_a_free_surface&oldid=141063"