Cylindrical Waves

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

$$\frac{\partial^{\,2}\psi}{\partial t^{\,2}} = v^{\,2}\left(\frac{... ...partial \rho^{\,2}} + \frac{1}{\rho}\,\frac{\partial\psi}{\partial\rho}\right),$$ (7.10)

it can be shown (see Exercise 2) that a sinusoidal cylindrical wave of phase angle $\phi$, wavenumber $k$, and angular frequency $\omega=k\,v$, has the approximate wavefunction

$$\psi(\rho,t) \simeq \frac{\psi_0}{\rho^{1/2}}\,\cos(\omega\,t-k\,\rho-\phi)$$ (7.11)

in the limit $k\,\rho\gg 1$. Here, $\psi_0/\rho^{1/2}$ is the amplitude of the wave. The associated wavefronts (i.e., the surfaces of constant phase) are a set of concentric cylinders that propagate radially outward, from their common axis ($\rho=0$), at the phase velocity $\omega/k=v$. See Figure 7.3. The wave amplitude attenuates as $\rho^{\,-1/2}$. Such behavior can be understood as a consequence of energy conservation, according to which the power flowing across the various $\rho={\rm const.}$ surfaces must be constant. (The areas of such surfaces scale as $A\propto \rho$. Moreover, the power flowing across them is proportional to $\psi^{\,2}\,A$, because the energy flux associated with a wave is generally proportional to $\psi^{\,2}$, and is directed normal to the wavefronts.) The cylindrical wave specified in expression (7.11) is such as would be generated by a uniform line source located at $\rho=0$. See Figure 7.3.

Figure 7.3: A cylindrical wave.