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One of the most interesting phenomena we can explore is that of a superposition of waves. In this Illustration we consider a superposition of two traveling pulses, while in Illustration 17.4 and Illustration 17.7 we consider the superposition of two traveling waves. (In Illustration 16.5 and Illustration 16.6 we considered the addition of multiple periodic functions in a Fourier series) Restart.
A superposition of two waves is nothing more than the arithmetic sum of the amplitudes of the two underlying waves. We can represent the amplitude of a transverse wave by a wave function, y(x, t). Notice that the amplitude, the value y, is a function of position on the x axis and the time. If we have two waves moving in the same medium, we call them y1(x, t) and y2(x, t), or in the case of this animation, f(x, t) and g(x, t). Their superposition, arithmetic sum, is written as f(x, t) + g(x, t).
This may seem like a complicated process, so we often focus on the amplitude at one point on the x axis, say x = 0 m (position is given in meters and time is given in seconds). So now let's consider Animation 1, which represents waves traveling on a string. The top panel represents the right-moving Gaussian pulse f(x, t), the middle panel represents g(x, t), the left-moving Gaussian pulse, and the bottom panel represents what you would actually see: the superposition of f(x, t) and g(x, t). As you play the animation, focus on x = 0 m. Until the tail of each wave arrives at x = 0 m, the amplitude there is zero. Watch what happens during the time that the two waves overlap. They add together in the way you would expect. As time goes on, the waves "separate" and move along the string as if they had not "run into" each other.
What does the superposition in Animation 2 look like at t = 10 s? The two waves add together and exactly cancel there. 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.
Illustration by Wolfgang Christian and Mario Belloni.