Why should physicists bother about wavelets? Why not leave them to the mathematicians and engineers?

Physicists are sometimes reluctant to learn about wavelets because they cannot be interpreted in physical terms as easily as sines and cosines and their frequencies. This is understandable enough: the ‘harmonic oscillator’ has been with us for more than three centuries, and continues to play its important role. But as we hope to show in the chapters that follow, wavelets can also be of great help in uncovering the presence or absence of certain frequencies in a physical phenomenon. Wavelet analysis is not replacing frequency analysis, but is rather an important refinement and expansion of it: Fourier analysis analyses a signal globally, whereas wavelet analysis looks into the signal locally.

Let us illustrate this is in musical terms. If you listen to a classical symphony you hear several parts, usually three to four. Each of them has its own main key: e.g. C minor, Eb major, etc. The Fourier power spectrum of the symphony will of course reveal the dominating keys: groundtones, and their harmonics. Frequencies of other chords which occur more fleetingly during modulations and variations in the piece of music, will also show up. If you would play the parts in a different order, the power spectrum would not change at all, but to the listener it becomes a very different piece, and more so if you interchange parts within the parts, at an ever finer scale: you have changed the musical score drastically.