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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.

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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.

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  • $\begingroup$ 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? $\endgroup$
    – CDCM
    Aug 8, 2017 at 6:38
  • $\begingroup$ @CDCM Yeah, I edited my answer to include the requirement that $A$ and $f$ be real. $\endgroup$
    – tparker
    Aug 8, 2017 at 7:04
  • $\begingroup$ I now believe my question might have a problem though. Is it actually the case that there can be non harmonic standing waves in a one dimensional string? I have been thinking a little bit more about it and it seems to me that the solutions to the wave equation obtained via separation of variables in 1D are all sine-cosine products. General solutions should be linear combinations of these solutions, but I can't see how you could construct a function that is a linear combination of these solutions and is also separable. Can superpositions of standing waves be standing waves then? $\endgroup$
    – Ignacio
    Aug 8, 2017 at 18:34
  • $\begingroup$ It has also occurred to me that the sum of two arbitrary waves with equal shape but traveling in opposite directions wouldn't necessary be a separable function, or would it? $\endgroup$
    – Ignacio
    Aug 8, 2017 at 18:39
  • $\begingroup$ @Ignacio Linear combinations of standing waves definitely do not in general form other standing waves (so the set of standing waves do not form a vector subspace). You're right that all the standing-wave solutions to the 1D wave equation are just sines and cosines with harmonic time dependence, but in higher dimensions or for different "wave-like" differential equations, you can get more general standing waves. $\endgroup$
    – tparker
    Aug 8, 2017 at 21:41

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