Ghost amplitude and energy

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Problem 6.7a

If the source depth is $ c\lambda $ (where $ \lambda $ is the wavelength) and $ 0<c<+1 $ in equation (6.7a), discuss the conditions under which the amplitude of $ \psi _{P} $ is zero.

Background

The low-velocity layer (LVL) is discussed in problem 4.16, ghosts in problem 3.8. When a ghost is superimposed on a downgoing wave, it affects not only the waveshape but also the directivity. In Figure 6.7a, $ S $ is a point source at a depth $ c\lambda $ and $ I $ is the image point (see problem 4.1) for energy reflected at the surface. For a ghost arriving at point $ P $, the virtual path is $ IP $. If the source emits the wave $ \psi _{P}=A{\rm {\;cos\;}}(\kappa r-\omega t) $ and the reflection coefficient at the surface is $ -1 $, the combined primary wave plus ghost at point $ P $ is

$ {\begin{aligned}\psi _{P}=A{\rm {\;cos\;}}(\kappa r_{1}-\omega t)-A{\rm {\;cos\;}}(\kappa r_{2}-\omega t).\end{aligned}} $

Using the identity $ {\rm {\;cos\;}}x-{\rm {\;cos\;}}y=-2{\rm {\;sin\;}}[(x+y)/2]{\rm {\;sin\;}}[(x-y)/2] $, we get

$ {\begin{aligned}\psi _{P}=-2A{\rm {\;sin\;}}[\kappa (r_{1}+r_{2})/2-\omega t]{\rm {\;sin\;}}[\kappa (r_{1}-r_{2})/2].\end{aligned}} $

When $ r\gg c\lambda $, $ r_{1}\approx r-c\lambda {\rm {\;cos\;}}\theta $, $ r_{2}\approx r+c\lambda {\rm {\;cos\;}}\theta $; also $ c\lambda =2\pi c/\kappa $, so we get


$ {\begin{aligned}\psi _{P}=2A{\rm {\;sin\;}}(\kappa r-\omega t){\rm {\;sin\;}}(2\pi c{\rm {\;cos\;}}\theta )\\\qquad =[2A{\rm {\;sin\;}}(2\pi c\ {\rm {cos}}\ \theta )]{\rm {\;cos\;}}(\kappa r-\omega t-\pi /2).\end{aligned}} $ (6.7a)

Transmissivities $ T\uparrow $ and $ T\downarrow $ are defined in problem 3.6 where equation (3.6c) shows that


$ {\begin{aligned}T\uparrow +T\downarrow =2,\;T\uparrow T\downarrow =E_{T}.\end{aligned}} $ (6.7b)

Absorption is discussed in problem 2.18.

Figure 6.7a.  Directivity of source plus ghost.

Solution

Equation (6.7a) gives for the amplitude of the primary wave plus ghost,


$ {\begin{aligned}A^{*}=2A{\rm {\;sin\;}}(2\pi c\ {\rm {cos}}\ \theta ).\end{aligned}} $ (6.7c)

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/":): 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/":): 2\pi c{\rm \; cos\; }\theta =n\pi , 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/":): {\rm \; cos\; }\theta =n/2c, n=0, \pm 1,\pm 2 , .... 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/":): n=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/":): {\rm \; cos\; }\theta = 0 , i. 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/":): \theta =\pm 90^{\circ} , and the waves are traveling horizontally. 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/":): |n|\ge 2 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/":): c<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/":): {\rm \; cos\; }\theta > 1 so there are no appropriate 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/":): \theta .

Problem 6.7b

For a source below the base of the LVL, compare the amplitude and energy of ghosts generated at the base of the LVL and at the surface of the ground, given that the velocities and densities just below and within the LVL are $ V_{H}=1.9\ {\rm {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_{H} =2.0\ {\rm g/cm}^{3} , 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_{W} =0.40\ {\rm km/s} 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_{W} =1.6\ {\rm g/cm}^{3} , respectively.

Solution

We assume small incidence angles so that equations (3.6a,b) are valid. 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} Z_{1} =2.0 \times 1.9=3.8; Z_{2} =0.40 \times 1.6=0.64 \ ({\rm g.km/cm^{3} s}). \end{align}

At the base of the LVL,

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=(0.64-3.8)/(0.64+3.8)=-3.2/4.4=-0.71, \end{align} 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} =0.50, \end{align} 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\uparrow =2\times 3.8/(3.8+0.64)=1.71\;,\; T\downarrow =0.29, \end{align} 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_{T} =T\uparrow \times T\downarrow =1.71\times 0.29=0.50. \end{align}

Assuming equal (unit) amplitudes for waves leaving the source in different directions, the ghost produced at the base of the LVL has 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/":): -0.71 and energy 0.50. The ghost produced at the surface has amplitude $ T\uparrow T\downarrow \times (-1)=-0.50 $ and energy 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} =0.50^{2} = 0.25 . The amplitude of the ghost from the base of the LVL 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/":): 0.71/0.50=1.4 times that of the ghost from the surface while the ratio of the energies 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/":): 0.50/0.25=2.4.

Problem 6.7c

Assume that the LVL 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} \lambda in thickness and 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/":): \eta \lambda =0.6\ {\rm dB} for the LVL; now what are the ratios of the ghost amplitudes and energies?

Solution

The surface ghost has to travel a distance 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 farther than the ghost from the base of the LVL during which its amplitude is reduced by 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^{-\eta x} =e^{-\eta \lambda} =e^{-0.60} =0.55 . The previous amplitude was 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/":): -0.47 , so with absorption this becomes 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/":): -0.26 , and energy becomes 0.22 $ \times 0.55^{2}=0.067 $. The ratios of the amplitudes and energies in part (b) now become 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/":): 0.73/0.26=2.8 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/":): 0.53/0.067=7.9 , the dB values being 8.9 and 18.0 dB, respectively.

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