Ghost amplitude and energy

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Problem 6.7a

If the source depth is (where is the wavelength) and in equation (6.7a), discuss the conditions under which the amplitude of 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, is a point source at a depth and is the image point (see problem 4.1) for energy reflected at the surface. For a ghost arriving at point , the virtual path is . If the source emits the wave and the reflection coefficient at the surface is , the combined primary wave plus ghost at point is

Using the identity , we get

When , , ; also , so we get


(6.7a)

Transmissivities and are defined in problem 3.6 where equation (3.6c) shows that


(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,


(6.7c)

For , , that is, , .... For , , i. e., , and the waves are traveling horizontally. When and , so there are no appropriate values of .

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 , , and , respectively.

Solution

We assume small incidence angles so that equations (3.6a,b) are valid. Then

At the base of the LVL,

Assuming equal (unit) amplitudes for waves leaving the source in different directions, the ghost produced at the base of the LVL has amplitude and energy 0.50. The ghost produced at the surface has amplitude and energy . The amplitude of the ghost from the base of the LVL is times that of the ghost from the surface while the ratio of the energies is

Problem 6.7c

Assume that the LVL is in thickness and that for the LVL; now what are the ratios of the ghost amplitudes and energies?

Solution

The surface ghost has to travel a distance farther than the ghost from the base of the LVL during which its amplitude is reduced by the factor . The previous amplitude was , so with absorption this becomes , and energy becomes 0.22 . The ratios of the amplitudes and energies in part (b) now become and , the dB values being 8.9 and 18.0 dB, respectively.

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