Reflection and transmission coefficients

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Problem 3.7a

Calculate the reflection and transmission coefficients, $ R $ and $ T $ of equations (3.6a,b), for a sandstone/shale interface (for a wave incident from the sandstone) for the following:

  1. $ V_{ss}=2.43 $ km/s, $ V_{sh}=2.02 $ km/s, $ \rho _{ss}=2.08 $ g/cm$ ^{3} $, and $ \rho _{sh}=2.23 $ g/cm$ ^{3} $;
  2. $ V_{ss}=3.35 $ km/s, $ V_{sh}=3.14 $ 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/":): \rho_{ss} =2.21 g/cmFailed 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/":): ^{3} , 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/":): \rho_{sh} =2.52 g/cmFailed 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/":): ^{3} .
  3. What are the corresponding values in nepers and decibels?

Background

A wave has kinetic energy due to the velocity of the medium and potential energy due to the strains in the medium. For a harmonic wave 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/":): \psi =A \sin \omega t , the particle velocity 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/":): \partial \psi /\partial t=\omega A\cos \omega t and the kinetic energy/unit volume 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/":): \frac{1}{2} \rho\omega^{2} A^{2} \cos ^{2} \omega t , 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/":): \rho being the density. As the wave progresses, the energy changes back and forth between kinetic and potential. When the potential energy is zero, the kinetic energy is a maximum and therefore equals the total energy. The maximum value comes when 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 ^{2} \omega t=+1 , so the total energy/unit volume$ {}=E={\frac {1}{2}}\rho \omega ^{2}A^{2} $, 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/":): E is the energy density (problem 2.3).

The coefficients 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/":): R 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/":): T in equations (3.6a,b) give ratios of the relative amplitudes of the reflected and transmitted waves. We denote the fractions of the incident energy that are reflected and transmitted 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/":): E_{R} 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/":): E_{T} ; “energy” as used here denotes the amount of energy flowing through a unit area normal to the wave direction per unit time (the intensity). The energy density (i.e., energy/unit volume) is given by equation (3.3k). The energy flowing through a unit area normal to the wave direction per unit time is equal to the energy density times the velocity, 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/":): \left(\frac{1}{2} \rho\omega ^{2} A^{2} \right)\alpha for a P-wave. Therefore


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} E_{R} =R^{2},\qquad E_{T} =\frac{\rho_{2} \alpha _{2} \omega ^{2} A_{2}^{2} }{\rho_{1} \alpha _{1} \omega ^{2} A_{0}^{2} } =\left(\frac{Z_{2} }{Z_{1} } \right)T^{2} =\frac{4Z_{1} Z_{2} }{(Z_{1} +Z_{2} )^{2}}. \end{align} (3.7a)

The coefficients 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_{R} and $ E_{T} $ are sometimes referred to as reflection and transmission energy coefficients to distinguish them from 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/":): R 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/":): T . Note that both 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_{R} 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/":): E_{T} are independent of the direction of travel through the interface.

Solution

i) Using equations (3.6a,b), 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} Z_{ss} =2.08\times 2.43=5.05,\qquad Z_{sh} =2.23\times 2.02=4.50 \end{align}

(where the units are g. km/cm3. s). Then,

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} R=\left(4.50-5.05\right)/\left(4.50+5.05\right)=-0.55/9.55=-0.058. \end{align}

(The minus sign denotes a phase reversal; see problem 3.6.) Because the incident wave is in the sandstone, 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} T=2\times 5.05/9.55=1.06. \end{align}

Note that the amplitude of the transmitted wave is larger than that of the incident wave when 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/":): R is negative (see problem 3.6c).


ii) 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_{ss} = 2.21 \times 3.35=7.40, \ Z_{sh} = 2.52 \times 3.14 = 7.91 .


Then,

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} R&=\left(7.91-7.40\right)/\left(7.91+7.40\right)=0.51/15.3=0.033,\\ T&=2\times 7.40/15.3=0.967. \end{align}


iii) From problem 2.17, we have nepers ln(amplitude ratio) and 1 neper 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/":): {}=8.686 dB. Therefore,

for (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/":): \begin{align} R&=\ln \left(0.058\right)=-2.8\ \mathrm{nepers} =-24\ \mathrm{dB},\\ T&=\ln \left(1.06\right)=0.058\ \mathrm{nepers}\ =0.51\ \mathrm{dB}. \end{align}

For (ii),

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} R&=\ln \left(0.033\right)=-3.4\ \mathrm{nepers}\ =-30\ \mathrm{dB}, \\ T&=\ln \left(0.967\right)=-0.034\ \mathrm{nepers}\ =-0.29\ \mathrm{dB}. \end{align}

Negative values of nepers 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/":): dB merely mean that the values are less than unity.

Problem 3.7b

Calculate the energy coefficients 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_{R} 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/":): E_{T} for cases (i) and (ii) in part (a).

Solution

We use equation (3.7a) to get,

for (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/":): \begin{align} E_{R} &=R^{2} =(0.058)^{2} =0.0034,\\ E_{T} &=\left(Z_{2} /Z_{1} \right)T^{2} =\left(4.50/5.05\right)\times 1.06^{2} =1.00. \end{align}


For (ii),

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} E_{R} &=0.033^{2} =0.001,\\ E_{T} &=(\left(7.91/7.40\right)\times 0.967^{2} =1.00. \end{align}

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/":): E_{R} +E_{T} =1 (within the accuracy of the calculations).

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