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

The important consequence of the Kronig-Kramers relations (KKrs) is that dissipative behaviour in material media inevitably implies the existence of dispersion, i.e. a frequency dependence in the constitutive equations. Basically, the relations are the frequency-domain expression of causality and correspond mathematically to pairs of Hilbert transforms. The relations have many forms and can be obtained with diverse mathematical tools. Here, two different demonstrations are given in the electromagnetic case, illustrating the eclectic mathematical apparatus available for this purpose. Then, we apply the acoustic (mechanical)-electromagnetic analogy to obtain the elastic versions. One major consequence is wave propagation attenuation and pulse spreading, that is, the progressive widening of a pulse as it propagates through a medium [vacuum seems to be the only "medium" where this does not occur (electromagnetic dispersion), while mechanical waves do not propagate]. Therefore, we derive KKrs that relate the wave velocity to the attenuation and quality factors. Finally, we discuss the concepts of stability and passivity and provide a novel algorithm to compute the relations numerically by using the fast Fourier transform.