Standing Waves

Standing Waves

If you have a solid understanding of what Traveling waves are (click here if you need a refresher) then when you add up a sine wave moving to the right with a wave moving to the left, you get a standing wave.

$ y(x,t) = \sin(kx-\omega t) + \sin(kx + \omega t) $

Using $ \sin(a) + \sin(b) = 2 \sin((a+b)/2) \cos((a-b)/2) $ and simplifying the above equation we get :

$y(x,t) = 2\sin(kx)\cos(\omega t) $

A plot of this looks like the following:

Standing wave
Red circles denote the nodes where the amplitude of vibration is 0. Anti nodes on the other hand are the regions of maximum amplitude oscillations.

For the most part when one is referring to standing waves, it is customary to just talk about the resultant wave that you see above.

But one should understand that the way you form this is by taking a right moving wave and adding it up with a left-moving wave:

Blue - Traveling wave moving to the right. Red- Traveling wave moving to the left. Adding Blue+ Red = Standing wave
Blue – Traveling wave moving to the right. Red- Traveling wave moving to the left

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