Finite‐bandwidth Kramers–Kronig relations for the determination of acoustic group velocity and frequency derivative of attenuation

Abstract

Kramers–Kronig (K–K) analyses of experimental data are complicated by the extrapolation problem, which is concerned with how the unexamined spectral bands impact K–K calculations. In this talk the causal linkages in resonant‐type data are demonstrated without extrapolation through acoustic K–K relations for the group velocity and the derivative of the attenuation coefficient (components of the derivative of the complex wave number). Due to their shape‐invariant predictions with respect to subtraction constants, these new relations provide stricter tests of causal consistency than previously established K–K relations for the phase velocity and attenuation coefficient. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the infinite spectrum, but differ when restricted to finite spectral windows. Using experimental data from microsphere suspensions, the accuracy of finite‐bandwidth K–K predictions for group velocity, and the attenuation derivative are demonstrated. Of the multiple methods, the most accurate forms were found to be those whose integrals were expressed only in terms of the phase velocity, and attenuation coefficient themselves requiring no differentiated quantities.