# Dispersion relation for non-harmonic waves

This question is related to my previous one. The entire linear theory of waves is built on dispersion relations, which represent the algebraic dependence of frequency on wave number. That is we decompose a complex wave motion (Fourier expansion is most often used) into simple harmonic sine waves, and look at the dependence of the speed of each wave on the wave number. This is a classic description of dispersion. But is there a theory of dispersion relations built not on the basis of harmonic functions that inevitably arise when using Fourier and Laplace analysis, but on the basis of other elementary types of waves? For example, we could use a wavelet transform based on the Haar wavelet, and decompose the original composite wave packet into elementary rectangular functions, each of which will have its own characteristic period and characteristic length. Moreover, using time as a parameter, we could trace how the scale (spectral) characteristics for nonlinear waves change, and thus control how the dispersion changes with time. Does something like this exist? Google turned up nothing.

• Commented Mar 30 at 16:04
• They two questions are related to each other :-) The whole idea was to use another classes of functions in order to generalise dispersion relation approach. E.g., wavelets could be useful for frequency-modulated waves, where Foureir analysis is inapplicable.. But it's just occured to me that complex exponentials are the only such functions that turn differential equations into algebraic equations. So, apparentely, there are no other options :-) Commented Mar 30 at 22:31