Why are sine-wave vibrations so common?

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a “sine wave” or “sinusoidal” even if it is a cosine, or a sine or cosine shifted by some arbitrary horizontal amount.)It may not be surprising that it wiggle of this general sort, but why is it a specific mathematically perfect shape? Why is it not a sawtooth shape like (b) or some other shape like (c)? The mystery deepens as we find that a vast number of apparently unrelated vibrating systems show the same mathematical feature.