The attenuation of waves in a dispersive medium

The attenuation of a one-dimensional wave in a dispersive medium is investigated theoretically. By assuming that a finite high-frequency limit to the phase velocity exists, the real and imaginary parts of the phase velocity are interrelated by means of standard Kramers-Krönig dispersion relations. It is shown that the assumption of an attenuation coefficient which increases as the square root of the frequency leads to a particularly simple integral representation for the propagated wave form. The propagation of a number of physically interesting wave shapes is investigated. In particular, the propagation of an initially rectangular pulse is compared with the corresponding result in which the frequency dependence of the real part of the phase velocity is neglected. It is shown that, insofar as the results may be applied to seismic propagation in the earth, the effect of including dispersion in the real part of the phase velocity is quite likely to be undetectable.