Causality and acoustic group velocity in dispersive media

Abstract

This talk examines two specific issues related to causality and the group velocity of acoustic pulses in dispersive media. First, the causal prediction of group velocity from attenuation with finite-bandwidth Kramers-Kronig (K-K) relations is discussed. Without extrapolating beyond the measurement bandwidth, the causal linkages of resonant-type data are established using expressions derived from the acoustic K-K relations to predict group velocity and slope of attenuation (frequency derivative of attenuation). These predictions provide a stricter test of causal consistency than the determination of the phase velocity and attenuation coefficient due to their shape-invariance with respect to subtraction constants. Secondly, conditions under which the group velocity is the velocity of the peak in the envelope of a acoustic pulse are described. The spatial and temporal signatures of acoustic pulses with arbitrarily large (e.g., superluminal) and negative group velocities are also demonstrated. These signatures are shown to be consistent with both simple and relativistic causality.