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If I jerk a wire from the free end fixed at an end and then fix the free end, it will produce a pulse travelling back and forth between the two ends. So in a string fixed at both ends, if we pluck it at some place, why does it not produce a pulse originating from the place I plucked and it travels back and forth?

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It does produce a pulse. However, the speed of sound in a string is very large so the pulse reaches the ends of the string very quickly. This means that in a second many reflections occur and you only hear the "normal modes" of the pulse which constructively add and therefore survive. In particular, the lowest mode, which typically gives the largest contribution to a Gaussian-like pulse (in a Fourier decomposition sense). The lowest mode is what gives the "note" of the string.

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It does, there are some great high-speed videos on the web of exactly this taking place when a stretched bass guitar string is violently plucked by a heavy-metal musician. Note that the reason that a hard pluck on a stretched string does not produce a square wave pulse is that the end conditions of the plucked string are that the displacement at the ends is zero and a square pulse does not meet this criterion.

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