Causal determination of acoustic group velocity and frequency derivative of attenuation with finite-bandwidth Kramers-Kronig relations

Abstract

Kramers-Kronig (KK) analyses of experimental data are complicated by the extrapolation problem, that is, how the unexamined spectral bands impact KK calculations. This work demonstrates the causal linkages in resonant-type data provided by acoustic KK relations for the group velocity (c(g)) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number) without extrapolation or unmeasured parameters. These relations provide stricter tests of causal consistency relative to previously established KK relations for the phase velocity (c(p)) and attenuation coefficient (alpha) (components of the undifferentiated acoustic wave number) due to their shape invariance with respect to subtraction constants. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the entire infinite spectrum, but differ when restricted to bandlimited spectra. Using experimental data from suspensions of elastic spheres in saline, the accuracy of finite-bandwidth KK predictions for c(g) and alpha' is demonstrated. Of the multiple methods, the most accurate were found to be those whose integrals were expressed only in terms of the phase velocity and attenuation coefficient themselves, requiring no differentiated quantities.