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The important consequence of the Kramers-Kronig relations (KKrs) is thatdissipative behavior in material media inevitably implies the existence ofdispersion, i.e., a frequency dependence in the constitutive equations.Basically, the relations are the frequency-domain expression of causality andcorrespond mathematically to pairs of Hilbert transforms. The relations havemany forms and can be obtained with diverse mathematical tools. Here, twodifferent demonstrations are given in the electromagnetic case, illustratingthe eclectic mathematical apparatus available for this purpose. Then, we applythe acoustic (mechanical)-electromagnetic analogy to obtain the elasticversions. Finally, we discuss the concepts of stability and passivity andprovide a novel algorithm to compute the relations numerically by using thefast Fourier transform.

Kramers-Kronig relations and the analogy between electromagnetic and mechanical waves