Do two waves propagating in different directions on the same line induce a standing wave? [duplicate]

What is the mathematical proof for this?

marked as duplicate by Bill N, John Rennie, SRS, Kyle Kanos, AccidentalFourierTransformFeb 2 '17 at 12:59

• It is not always true: For instance, if the waves have different amplitude or different frequency they will not form a standing wave – user126422 Feb 2 '17 at 1:53
• @AlbertAspect How can you show this? – Goldname Feb 2 '17 at 2:14
• program it and view the result – user18764 Feb 2 '17 at 2:36
• -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. – sammy gerbil Feb 2 '17 at 2:52
• @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 – Goldname Feb 2 '17 at 3:27

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.