Differential forms of the Kramers-Kronig dispersion relations

Differential forms of the Kramers-Kronig dispersion relations provide an alternative to the integral Kramers-Kronig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Kronig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (/spl alpha/(/spl omega/) = /spl alpha//sub 0/ + /spl alpha//sub 1/ |/spl omega/|/sup y/). Dispersion predictions using the differential dispersion relations are compared to the measured dispersion for a series of specimens (two polymers, an egg yolk, and two liquids) exhibiting attenuation obeying a frequency power law (1.00 /spl les/ y /spl les/ 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Kronig dispersion prediction is found to be identical to the (integral) Kramers-Kronig dispersion prediction.