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Spherical Waves

Consider a spherically symmetric wavefunction $ \psi(r,t)$ , where $ r=(x^2+y^2+z^2)^{1/2}$ 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)

$\displaystyle \frac{\partial^2\psi}{\partial t^2} = v^2\left(\frac{\partial^2\psi}{\partial r^2} + \frac{2}{r}\,\frac{\partial\psi}{\partial r}\right),$ (540)

it can be shown (see Exercise 3) that a sinusoidal spherical wave of phase angle $ \phi$ , wavenumber $ k$ , and angular frequency $ \omega=k\,v$ , has the wavefunction

$\displaystyle \psi(r,t) =\frac{ \psi_0}{r}\,\cos(\omega\,t-k\,r-\phi).$ (541)

Here, $ \psi_0/r$ is the amplitude of the wave. The associated wavefronts are a set of concentric spheres that propagate radially outward, from their common center ($ r=0$ ), at the phase velocity $ \omega/k=v$ . The wave amplitude attenuates as $ r^{-1}$ . Such behavior can again be understood as a consequence of energy conservation. (The area of a constant $ r$ surface scales as $ A\propto r^2$ , and the power flowing such a surface is proportional to $ \psi^2\,A$ .) The spherical wave specified in expression (541) is such as would be generated by a point source located at $ r=0$ .


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