|Title||The dispersion formula and the Green's function associated with an attenuation obeying a frequency power law|
|Publication Type||Journal Article|
|Year of Publication||2018|
|Journal||Journal of the Acoustical Society of America|
|Type of Article||Article|
|Keywords||acoustics; Audiology & Speech-Language Pathology; causality; equations; marine-sediments; media; relaxation; wave-propagation|
An attenuation obeying a frequency power law scales as vertical bar omega vertical bar(beta), where omega is angular frequency and beta is a real constant. A recently developed dispersion formula predicts that the exponent beta can take only certain values in well defined, disjoint intervals. It is shown here that these admissible values of beta 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 beta. 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 beta in the interval (0.5, 1). This restriction supersedes the weaker constraints on beta derived from the dispersion formula alone. For the previously admissible values of beta 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. (C) 2018 Acoustical Society of America.