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Cylindrical Waves

Consider a cylindrically symmetric wavefunction $ \psi(\rho,t)$ , where $ \rho= (x^2+y^2)^{1/2}$ is a standard cylindrical coordinate (Fitzpatrick 2008). Assuming that this function satisfies the three-dimensional wave equation, (537), which can be rewritten (see Exercise 2)

$\displaystyle \frac{\partial^2\psi}{\partial t^2} = v^2\left(\frac{\partial^2\psi}{\partial \rho^2} + \frac{1}{\rho}\,\frac{\partial\psi}{\partial\rho}\right),$ (538)

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

$\displaystyle \psi(\rho,t) \simeq \frac{\psi_0}{\rho^{1/2}}\,\cos(\omega\,t-k\,\rho-\phi)$ (539)

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 40.) 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 (539) is such as would be generated by a uniform line source located at $ \rho=0$ . (See Figure 40.)

Figure 40: A cylindrical wave.
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next up previous
Next: Spherical Waves Up: Multi-Dimensional Waves Previous: Three-Dimensional Wave Equation
Richard Fitzpatrick 2013-04-08