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Illustration 17.4: Superposition of Traveling Waves
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In Illustration 17.3 we considered the superposition of two traveling pulses. In this Illustration we consider a superposition of two traveling sinusoidal waves. (In Illustration 16.5 and Illustration 16.6 we consider the addition of multiple periodic functions in a Fourier series). Restart.
Let's begin by considering Animation 1, which represents two waves traveling on a string (position is given in meters and time is given in seconds). As you play the animation, focus on x = 0 m. Until each wave arrives at x = 0 m, the amplitude there is zero. Watch what happens during the time that the two waves overlap, t ≥ 8 s. They add together in the way you would expect. Given that the two waves always have the opposite amplitude at x = 0 m, the superposition of the two traveling waves at x = 0 m will always be zero. This point that never moves is called a node. The resulting wave is called a standing wave. It is created when we have two identical waves traveling in opposite directions in a particular medium (here the medium is a string, but we can set up standing waves in air as well).
What does the superposition in Animation 2 look like at t ≥ 8 s? The two waves add together and exactly cancel at x = 0 m. As time goes on, the waves "reappear" (they were always there) and move along the string as if they had not "run into" each other. Given that the two waves always have the same amplitude at x = 0 m, the superposition of the two traveling waves at x = 0 m will always be changing. This point is called an anti-node. The resulting wave is still a standing wave. Note that it is shifted in comparison to Animation 1.
In Animation 3, we have a traveling wave that is incident on a wall located at x = 15 m. The wave travels and is then reflected by the wall. By reflected we mean that the direction of propagation of the wave changes (the right-moving wave is now a left-moving wave) and its amplitude is now the negative of what it was before. So we now have a right-moving wave and a left-moving wave that resemble the superposition shown in Animation 1. In that animation, the node was at x = 0 m; here in Animation 3 the node is at x = 15 m.
Illustration authored by Mario Belloni.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.