Transient propagation in media with classical or power-law loss

Abstract

This paper addresses the problem of small-signal transient wave propagation in media whose absorption coefficient obeys power-law frequency dependence, i.e., alpha infinity omega n. Our approach makes use of previously derived relations between the absorption and dispersion based on the Kramers-Kronig relations. This, combined with a recently obtained solution to a causal convolution wave equation enable expressions to be obtained for one-dimensional transient propagation when n is in the range 0 < n < 3. For n = 2, corresponding to no dispersion, straightforward analytical expressions are obtained for a delta-function and a sinusoidal step function sources and these are shown to correspond to relations previously derived. For other values of n, the effects of dispersion are accounted for by using Fourier transforms. Examples are used to illustrate the results for normal and anomalous dispersive media and to examine the question as to the conditions under which the effects of dispersion should be accounted for, especially for wideband ultrasound pulses of the type used in B-mode tissue imaging. It is shown that the product of the attenuation and total propagation path can be used as a criterion for judging whether dispersion needs to be accounted for.

Analytical time-domain Green’s functions for power-law media

“…The pulse distortion in the fat-like medium indicates that the effects of dispersion become significant for large depths, which is consistent with the conclusion reached in Ref. 27. demonstrate that for observation points only one wavelength from the radiating source, Eq.…”

“…Equation ͑9͒ corresponds to the phase velocities computed via the Kramers-Kronig relations 14,27 and the time-causal theory. 13 In addition, Eq.…”

Parametric internal waves in a compressible fluid

“…Stokes transient problem has two types of solution : (a) solutions in the form of series [22,23] or integrals [24] and (b) closed-form approximations [25]. Cobbold et al [26] has recently stated that a difficulty arises with the approximate, closed-form type of solution : " approximate solutions to such problems do not satisfy causality in the strict sense ." Many examples of breakdown of causality for transient solutions of Stokes equation may be found in White [27].…”

Acoustic and photoacoustic characterization of micron-sized perfluorocarbon emulsions

“…24 The increase in attenuation αðfÞ with frequency is generally described using a power law fit, such as αðfÞ ¼ α 0 f n , where the attenuation coefficient α 0 and the power law exponent n describe how the attenuation changes with frequency. 27 A direct relationship between sound dispersion and ultrasonic attenuation can be accurately described using the Kramers-Kronig relation 28 cðfÞ…”