Wave-speed dispersion associated with an attenuation obeying a frequency power law

An attenuation scaling as a power of frequency, |ω|(β), over an infinite bandwidth is neither analytic nor square-integrable, thus calling into question the application of the Kramers-Kronig dispersion relations for determining the frequency dependence of the associated phase speed. In this paper, three different approaches are developed, all of which return the dispersion formula for the wavenumber, K(ω). The first analysis relies on the properties of generalized functions and the causality requirement that the impulse response, k(t), the inverse Fourier transform of -iK(ω), must vanish for t