Abstract
Some properties of solutions of initial value problems and mixed initial-boundary value problems of a class of wave equations are discussed. Wave modes are defined and it is shown that for the given class of wave equations there is a one to one correspondence with the roots ω i (k) or k j (ω) of the dispersion relation W(ω, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W 12 (−∞, ∞); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. It is found that such solutions are stable, even if some of the singularities of the functions k j (ω) lie in the upper half of the ω plane. The implications of this result for the Kramers-Kronig relations are discussed.
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Broer, L.J.F., Peletier, L.A. Some properties of the solutions of wave equations. Appl. sci. Res. 21, 138–161 (1969). https://doi.org/10.1007/BF00411602
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DOI: https://doi.org/10.1007/BF00411602