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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 $(1)$?

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  • $\begingroup$ I'm told that there are known solutions to the singly ionized hydrogen molecule. $\endgroup$ – dmckee Sep 18 '16 at 18:15
  • $\begingroup$ What does that mean? $\endgroup$ – user45664 Oct 1 '16 at 20:13
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What about a superposition of a cylindrical, a spherical, and a plane wave? In general, there are lots and lots of solutions that don't fit into those three categories, they're just very had to find nice expressions for in those three coordinate systems.

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Solving the wave equation in a way that allows reduction of dimensions can be done with coordinate systems that are orthogonal coordinate systems. That is, coordinate systems where the subdimensions are linearly independent of each other. When that is the case the wave equation can be decomposed into a system of one dimensional equations by doing separation of variables.

Plane waves solutions result from decomposing the wave equation this way in 3-D cartesian coordinates. Spherical and cylindrical coordinates are also orthogonal coordinate systems in 3-D space, so spherical and cylindrical waves result from a similar approach to decomposition of the wave equation.

There are also other 3-D orthogonal coordinate systems where the wave equation can be solved this way, such as oblate spheroidal coordinates and parabolic coordinates (there are other orthogonal systems besides those,those are just examples).

Now in a coordinate system that is not orthogonal, the separation of variables approach will not work. Solution of the wave equation in such systems would require different techniques, possibly numerical.

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