Wave decompositions

I'm trying to figure out whether a wave -- or at least one with some physical basis, like sound or light -- has some essential "canonical" decomposition into sinusoid waves (in the time-frequency domain). That is, it seems like all such physical waves are originally built out of combinations of specific primitive sinusoid waves (vibrations of different frequencies combining/interfering, etc.), but is there any mathematical way to recover that information? We have the wavelet and chirplet and short-time Fourier transforms and such, but are they all approximating some inherent "right" answer, or does a plain waveform not contain enough information to determine that exactly, so that the best you can do is decide which of the various transforms provides the most useful interpretation of a wave in a given situation?

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