Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media

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Abstract
For the dispersion of waves in a homogeneous medium there exist the Kramers–Kronig relations for the wave number K(ω) = ω/c(ω). The usual mathematical proof of such relations depends on assumptions for the asymptotic behavior of c(ω) at high frequency, which for electromagnetic waves in dielectrics can be evaluated from the microphysical properties of the medium. In this paper such assumptions are removed and the necessary asymptotic behavior is shown to follow the representation of K(ω) as a Herglotz function. From the linear, causal, and passive properties of the media we thus establish the Kramers–Kronig relations for all linear wave disturbances including acoustic, elastic, and electromagnetic waves in inhomogeneous as well as homogeneous media without any reference to the microphysical structure of the medium.

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