Spherical Waves

Consider a spherically-symmetric (about the origin) wavefunction $\psi(r,t)$ , where $r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}$ is a standard radial spherical coordinate (Fitzpatrick 2008). Assuming that this function satisfies the three-dimensional wave equation, (7.9), which can be rewritten (see Exercise 3)

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

(7.12)

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

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

(7.13)

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, according to which the power flowing across the various $r={\rm const.}$ surfaces must be constant. (The area of a constant- $r$

surface scales as $A\propto r^{\,2}$ , and the power flowing across such a surface is proportional to $\psi^{\,2}\,A$ .) The spherical wave specified in expression (7.13) is such as would be generated by a point source located at $r=0$