Consider the wave equation

(1)

where v is the speed of the wave. In cylindrical coordinates with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation

(2)

The solutions are Bessel functions. For large r, however, they can be approximated as

(3)

Note that, unlike the plane and spherical waves, cylindrical waves cannot assume an arbitrary functional form.

Plane Wave, Spherical Wave, Wave