# General conditions for a wave to be considered a standing wave?

Introductory physics textbooks usually describe standing waves on a string as superpositions of left and right traveling harmonic waves:

$$\psi = \psi_1 + \psi_2 = A\sin(\frac{2\pi x}{\lambda}-\omega t)+A\sin(\frac{2\pi x}{\lambda}+\omega t).$$

I figure that this can be generalized to arbitrary periodic waves. At least I see no reason why they should be harmonic, but I have never seen any textbook where such a thing is described. (Perhaps cause I have never found a nice, moderately advanced textbook that deals with waves in an in depth way).

It is much more difficult for me to figure out how to generalize this to two or three space dimensions. I suspect the general definition should probably be related to the Helmholtz equation $\nabla^2 \phi + k^2 \phi = 0$, but I'm not exactly sure how. Are standing waves defined as solutions of the wave equation that fulfill some special properties? Are they always periodic functions of the form $F = f(\mathbf{r})g(t)$? Is periodicity necessary? (I have a hard time imagining why, for instance, two compactly supported waves travelling in opposite directions on a string should be considered a single standing wave, that is why I am inclined to ask for periodicity, but maybe this is not the case).

The bottom line is, I have never understood what are the properties a solution to the wave equation should satisfy in order to be considered a standing wave, in a general way, preferably in 2 or 3 dimensions.

Note: I can see my question is related to this closed question, Mathematical defintion of a standing wave?, I hope I managed to make my question clear enough.

I'm not sure if there's a widely agreed-upon rigorous definition in arbitrary dimensions, but I always think of them as any function that can be written as a product $A({\bf x}) f(t)$, where $A$ and $f$ are both real and $f$ is periodic.
• But separable doesn't necessarily imply a non-travelling wave? e.g. $e^{i k x}e^{i \omega t}$ This definition otherwise gets to the heart of it imo, that is a stationary wave profile, that changes amplitude uniformly in space, as a fn of time. I think your defn works if one or both of $A$, $f$ is restricted to being real? – CDCM Aug 8 '17 at 6:38
• @CDCM Yeah, I edited my answer to include the requirement that $A$ and $f$ be real. – tparker Aug 8 '17 at 7:04