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Since intensity is power per unit area, and since the area of a sphere increases as the square of its radius, the intensity falls off as the inverse square of the radius (or range) in order that power remains constant. The corresponding TL is defined as:

This relationship is valid for an isotropic deep ocean with no absorption effects.

An analogous expression can be derived for cylindrical spreading. This spreading law would be appropriate for a duct, or in shallow water, where the water is homogeneous and the boundaries are perfect reflectors. Then, referring to Figure 3.7(b):

where H is the depth of the duct or of the water column. Since power is conserved, P1 = I1A1 = I2A2 and:

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