What is the mathematical proof for this?
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$\begingroup$ It is not always true: For instance, if the waves have different amplitude or different frequency they will not form a standing wave $\endgroup$– user126422Commented Feb 2, 2017 at 1:53
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$\begingroup$ @AlbertAspect How can you show this? $\endgroup$– JobHunter69Commented Feb 2, 2017 at 2:14
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$\begingroup$ program it and view the result $\endgroup$– user18764Commented Feb 2, 2017 at 2:36
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2$\begingroup$ -1. No research effort. A citation for "I heard it was true always" would be useful! See Standing waves due to two counter-propagating travelling waves of different amplitude. $\endgroup$– sammy gerbilCommented Feb 2, 2017 at 2:52
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$\begingroup$ @sammygerbil Your link does not include mathematical derivation. I heard it was true from a TA which is totally irrelevant. "No research effort" Wrong, unless you can pull up my search history and prove otherwise $\endgroup$– JobHunter69Commented Feb 2, 2017 at 3:27
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1 Answer
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Wave going into positive direction has the equation:
$$p(x)=\sin(x+\omega t)$$
Into the negative direction:
$$n(x)=\sin(x-\omega t)$$
Their sum:
\begin{align} p(x)+n(x)&=\sin(x+\omega t)+\sin(x-\omega t) \\ &=\sin(x)\cos(\omega t) + \cos(x)\sin(\omega t) + \sin(x)\cos(\omega t) - \cos(x)\sin(\omega t)\\ &=2 sin(x)cos(\omega t) \end{align}
As you can see, the wave on the bottom is a standing sine wave, whose amplitude is multiplied by the original frequency of the moving waves.
