Reinforcement depth in marine recording

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Problem 3.5a

3.5a For a source at a depth , show that the maximum amplitude of a downgoing incident wave and its reflection at the surface of the sea occurs at the depth , where is the angle of incidence, by expressing the pressure in the form used in equations (3.1b,d) and applying appropriate boundary conditions.

Solution

Since the interface is liquid/vacuum, only two waves exist, the incident and reflected P-waves. Taking the z-axis positive downward, we take in the form

There is only one boundary condition, namely that at . This gives Using Euler’s formulas (see Sheriff and Geldart, 1995, problem 15.12a), we get

upon inserting the time factor. The amplitude of the combined incident and reflected waves is

It is a maximum when , that is, when

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