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.
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$\begingroup$ Is this question the previous one? $\endgroup$– Kyle KanosCommented Mar 30 at 16:04
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$\begingroup$ 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 :-) $\endgroup$– shamil khalCommented Mar 30 at 22:31
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There is no sense to exclude some other basis of decomposition. From the mathematical point of view there is no special type of functions that is used for studying the waves. Nevertheless, some scholars emphasize to consider that mathematical functions are different. Shaping and dispersion of the linear system isn't always described in the linear terms including derivatives of the solutions of the equation of motion. It rather occurs that using the geometrical method one can represent discontinuous evolution of a linear wave in a completely different representation than in the representation using the differential & integral calculus. It is argued that the question of a dispersion relation cannot have a definite answer because it implies that the evolution has discontinuous jumps at the boundaries of the fragments (it defines the scattering matrix on these fragments). Therefore, I think, saying that the linear theory is related to the linear dispersion branch of the wave packet evolution equation is a misnomer. The theory of fractional differences describes the dispersive dynamics better. In some sense, the phase speed and group velocity are just the first two components of the four-velocity, and they aren't directly related to the wave characteristics (special relativity).
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$\begingroup$ Do you mean that the fractional derivative can be used to solve the problem of discontinuities in wave propagation? en.wikipedia.org/wiki/Fractional_calculus If so, then this is a very interesting generalization of dispersion properties. Could you share a couple of links to relevant articles? :-) $\endgroup$ Commented Mar 30 at 22:51