ABSTRACT Differential forms of the Kramers-Krönig dispersion relations provide an alternative to the integral Kramers-Krönig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Krönig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (α(ω)=α0+α1|ω|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≤ y≤ 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Krönig dispersion prediction is found to be identical to the (integral) Kramers-Krönig dispersion prediction.
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