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I know that a standing wave is the superposition of two waves of equal amplitude and wavelength, moving in opposite directions. But I am looking for a more mathematical defintion of such a wave. The best I can come up with is that (I am foucsing on 1d) its displacement can be written as: $$y=f(x)g(t)$$ But I am not sure if this is right? If not can you please provide me with a mathematical defintion (along with a source if possible) thanks.

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    $\begingroup$ What about your first sentence is not mathematical? You can write down the formula for a wave, and then write down the general formula for a standing wave, can you not? $\endgroup$ – ACuriousMind♦ Feb 10 '15 at 19:22
  • $\begingroup$ @ACuriousMind It doesn't seem general enough, and ristricted to a special case. I wouln't know how to formulate it mathematically in the general case (e.g. with non-harmonic waves) $\endgroup$ – Quantum spaghettification Feb 10 '15 at 19:26
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    $\begingroup$ I do not believe there is a treatment of this for the general anharmonic case. $\endgroup$ – ACuriousMind♦ Feb 10 '15 at 19:29
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    $\begingroup$ Your formula does not work, because the travelling wave $e^{i(kx-\omega t)}$ can be written as a product of your form---$e^{ikx}e^{-i\omega t}$ $\endgroup$ – Brian Moths Feb 10 '15 at 20:58
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    $\begingroup$ @NowIGetToLearnWhatAHeadIs Good point! Perhaps Joseph's definition would work if $f$ and $g$ are required to be real (or if $fg$ must be real). Technically, only the real part of your wave describes the displacement, and it cannot be written in Joseph's form. $\endgroup$ – pwf Feb 10 '15 at 21:51

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