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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.

Sofia
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Quantum spaghettification
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    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? – ACuriousMind Feb 10 '15 at 19:22
  • @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) – Quantum spaghettification Feb 10 '15 at 19:26
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    I do not believe there is a treatment of this for the general anharmonic case. – ACuriousMind Feb 10 '15 at 19:29
  • @ACuriousMind Ok assuming that my first statment can be put into mathetmacial terms. Even so can we say a stationary wave is a wave that satifies the equation I have given? – Quantum spaghettification Feb 10 '15 at 19:32
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    Your formula does not work, because the travelling wave $e^{i(kx-\omega t)}$ can be written as a product of your form---$e^{ikx}e^{-i\omega t}$ – Brian Moths Feb 10 '15 at 20:58
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    @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. – pwf Feb 10 '15 at 21:51
  • @NowIGetToLearnWhatAHeadIs This is simply not true : a propagating wave will always be described by the real part of $e^{i(kx-\omega t)}$ i.e. $\cos(kx-\omega t)$. If you dont take the real part at the end, nothing that you state has any physical meaning. That exactly what pwf is stating : everything at the end must be real. – dolun Feb 11 '15 at 13:26
  • I guess I had a broader interpretation of what a wave was. I was thinking of something like a wavefunction where the phase is physically meaningful. – Brian Moths Feb 11 '15 at 22:45

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