Thomas L. Szabo
Imaging Systems, Hewlett Packard, 3000 Minuteman Rd., Andover, MA 01810
The Kramers--Kronig relations are used frequently to obtain velocity dispersion from attenuation frequency data and vice versa. For many cases of practical interest, attenuation is described by a power law equation, (alpha)((omega))=(alpha)[sub 0]|(omega)|[sup y], where (alpha)[sub 0] and y are arbitrary real constants and (omega) is angular frequency. If y(greater than or equal to)1, these relations fail because the Paley--Weiner theorem is no longer satisfied. Based on a time domain expression of causality, new time-causal relations, with the aid of generalized functions, provide a way of obtaining dispersion relations for the y(greater than or equal to)1 cases. For y<1, the time-causal relations reduce to the Kramers--Kronig relations. The new results for attenuation of the power law form indicate that dispersion is maximum for y=1, and it falls symmetrically to zero at y=0 and y=2. These results are in contrast to the approximate nearly local Kramers--Kronig theory [M. O'Donnell et al., J. Acoust. Soc. Am. 69, 696--701 (1981)], which predicts dispersion for values of y=0 to y=2. The two theories agree at y=1, but deviate elsewhere. Experimental data confirming the new theory will be given.