# Are there solutions of three dimensional wave equation that are not cylindrical, spherical or plane?

Wave's equation in three dimensions can be written as

$${ \partial^2 \xi \over \partial t^2 } = c^2 \nabla^2 \xi \tag{1}$$

Plane waves are solution of $(1)$ of course, but are there other types of waves that do satisy $(1)$?

Firstly do all cylindrical and spherical waves satisfy $(1)$?

If we have spherical symmetry it can be shown that $r\xi$ satisfy the one dimensional waves equation, that is

$${\partial^2 r\xi \over \partial t^2} = c^2 {\partial^2 r\xi \over \partial x^2}$$

And from here we can conclude that $r \xi=f(x \pm vt)$. A similar thing can be done with cylindrical waves.

But are there solutions of $(1)$ that cannot be reduced to solve the one dimensional waves equation? Like are there other wave types (not plane, cylindrical or spherical) that satisy $(1)$?

• I'm told that there are known solutions to the singly ionized hydrogen molecule. – dmckee Sep 18 '16 at 18:15
• What does that mean? – user45664 Oct 1 '16 at 20:13