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The dispersion formula and the Green's function associated with an attenuation obeying a frequency power law
Ist Teil von
The Journal of the Acoustical Society of America, 2018-08, Vol.144 (2), p.755-765
Ort / Verlag
United States
Erscheinungsjahr
2018
Quelle
American Institute of Physics (AIP) Journals
Beschreibungen/Notizen
An attenuation obeying a frequency power law scales as
|
ω
|
β
, 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.
Sprache
Englisch
Identifikatoren
ISSN: 0001-4966
eISSN: 1520-8524
DOI: 10.1121/1.5049809
Titel-ID: cdi_pubmed_primary_30180676
Format
–
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