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)

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

$$\psi(r,t) =\frac{ \psi_0}{r}\,\cos(\omega\,t-k\,r-\phi).$$ (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 $r=0$.