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

作者：Buckingham Michael J^*

来源：Journal of the Acoustical Society of America, 2015, 138(5): 2871-2884.

DOI：10.1121/1.4932030

An attenuation scaling as a power of frequency, vertical bar omega vertical bar(beta), 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 wave-number, K(omega). 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(omega), must vanish for t<0. Second, a wave equation is introduced that yields the phase-speed dispersion associated with a frequency-power-law attenuation. Finally, it is shown that, with minor modification, the Kramers-Kronig dispersion relations with no subtractions (the Plemelj formulas) do in fact hold for an attenuation scaling as vertical bar omega vertical bar(beta), yielding the same dispersion formula as the other two derivations. From this dispersion formula, admissible values of the exponent beta are established. Physically, the inadmissible values of beta, which include all the integers, correspond to attenuation-dispersion pairs whose Fourier components cannot combine in such a way as to make the impulse response, k(t), vanish for t<0. There is no upper or lower limit on the value that b may take.

• 出版日期2015-11