Far- and near-field effects for a point source

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Problem 2.13a

Show that for harmonic waves of the form


(2.13a)

the displacement is

(2.13b)

Background

If we set in equation (2.9a), we obtain the result , where is a solution of the P-wave equation. Furthermore if is independent of latitude and longitude, the wave equation reduces to equation (2.5c), and the solution of problem 2.5c shows that equation (2.13a) is a P-wave solution of equation (2.5c).

Solution

Since in equation (2.13a) is a solution of equation (2.5c), it represents a spherically symmetrical P-wave and therefore the only displacement is along . If we take the x-axis along , equation (2.9d) shows that

Problem 2.13b

Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance .

Solution

The two terms are of equal importance when the two amplitudes are equal, that is, when or .

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Disturbance produced by a point source Rayleigh-wave relationships
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