The dispersion formula and the Green's function associated with an attenuation obeying a frequency power law

An attenuation obeying a frequency power law scales as $$ , where ω is angular frequency and β is a real constant. A recently developed dispersion formula predicts that the exponent β can take only certain values in well defined, disjoint intervals. It is shown here that these admissible values of β are consistent with the physical requirement, stemming from the second law of thermodynamics, that the work done during the passage of a wave must always be positive. Since the dispersion formula, which is derived from the strain-hardening wave equation, is a causal transform, it is expected that the associated Green's function should also satisfy causality for all the permitted values of β. Such is not the case, however: the Green's function is maximally flat at the time of source activation, and hence is causal, but only for values of β in the interval (0.5, 1). This restriction supersedes the weaker constraints on β derived from the dispersion formula alone. For the previously admissible values of β outside the interval (0.5, 1), although the dispersion formula satisfies causality, the Green's function is non-causal. Evidently, causality may be satisfied by the dispersion formula but violated by the Green's function.