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The farfield term of eq. (10)

(11) 
permits an interesting generalization. Wielandt (1975) has derived the following formula for the Pwave displacement radiated vertically downward from an explosion at the seabottom:

(12) 
Z_{1} is the acoustic impedance of the water and Z_{2} that of the solid; z is the depth below the seabottom. The delay time associated with the radius of the source is neglected. It can be seen that (12) is essentially the same formula as (11), except for the transmission coefficient
Z_{1}/(Z_{1}+Z_{2}) which equals 1/2 when the two impedances are the same. The two formulae then become identical for vertical radiation. This is remarkable because eq. (12) was derived under the assumption of cylindrical rather than spherical symmetry. Also, eq. (12) remains valid even when the dynamics of the source are highly nonlinear, such as for explosions; so the same must be true for eq. (11). This simple equation can thus be used to estimate the seismic farfield of explosive sources. Of course absorption in the medium and additional transmission coefficients or spreading factors may have to be taken into account.
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Erhard Wielandt
20010921