Standing Waves


Two waves one propagating in the positive x-direction ( $y_1 = A\sin(\omega t - b x)$ purple) and the second one ($y_2 = A\sin(\omega t + b x)$ red) in the negative x-direction. When added together (green), they produce a standing (do not propagate) wave.
Explanation
$y_1 + y_2 = A\sin(\omega t - b x) + A\sin(\omega t + b x) = A\sin(\omega t)\cos(b x) - A\cos(\omega t)\sin(b x) + A\sin(\omega t)\cos(b x) + A\cos(\omega t)\sin(b x) = 2 A\sin(\omega t)\cos(b x)$
When the two propagating waves are added, we obtain $2 A\sin(\omega t)\cos(b x)$ which is a standing wave (that does not propagate) with its amplitude changing with the time.
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