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# Spherical Waves

Consider a spherically symmetric wavefunction , where is a standard spherical coordinate (Fitzpatrick 2008). Assuming that this function satisfies the three-dimensional wave equation, (537), which can be rewritten (see Exercise 3)

 (540)

it can be shown (see Exercise 3) that a sinusoidal spherical wave of phase angle , wavenumber , and angular frequency , has the wavefunction

 (541)

Here, is the amplitude of the wave. The associated wavefronts are a set of concentric spheres that propagate radially outward, from their common center ( ), at the phase velocity . The wave amplitude attenuates as . Such behavior can again be understood as a consequence of energy conservation. (The area of a constant surface scales as , and the power flowing such a surface is proportional to .) The spherical wave specified in expression (541) is such as would be generated by a point source located at .

Next: Oscillation of an Elastic Up: Multi-Dimensional Waves Previous: Cylindrical Waves
Richard Fitzpatrick 2013-04-08