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# Cylindrical Waves

Consider a cylindrically symmetric wavefunction , where 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)

 (538)

it can be shown (see Exercise 2) that a sinusoidal cylindrical wave of phase angle , wavenumber , and angular frequency , has the approximate wavefunction

 (539)

in the limit . Here, 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 ( ), at the phase velocity . (See Figure 40.) The wave amplitude attenuates as . Such behavior can be understood as a consequence of energy conservation, according to which the power flowing across the various surfaces must be constant. (The areas of such surfaces scale as . Moreover, the power flowing across them is proportional to , because the energy flux associated with a wave is generally proportional to , 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 . (See Figure 40.)

Next: Spherical Waves Up: Multi-Dimensional Waves Previous: Three-Dimensional Wave Equation
Richard Fitzpatrick 2013-04-08