When sound propagates in a lossy fluid, causality dictates that in most cases the presence of attenuation is accompanied by dispersion. The ability to incorporate attenuation and its causal companion, dispersion, directly in the time domain has received little attention. Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)] showed that attenuation and dispersion in a linear medium can be accounted for in the linear wave equation by the inclusion of a causal convolutional propagation operator that includes both phenomena. Szabo's work was restricted to media with a power-law attenuation. Waters et al. [J. Acoust. Soc. Am. 108, 2114-2119 (2000)] showed that Szabo's approach could be used in a broader class of media. Direct application of Szabo's formalism is still lacking. To evaluate the concept of the causal convolutional propagation operator as introduced by Szabo, the operator is applied to pulse propagation in an isotropic lossy medium directly in the time domain. The generalized linear wave equation containing the operator is solved via a finite-difference-time-domain scheme. Two functional forms for the attenuation often encountered in acoustics are examined. It is shown that the presence of the operator correctly incorporates both, attenuation and dispersion.
In the low-kilohertz frequency range, acoustic transmission in shallow water deteriorates as wind speed increases. Although the losses can be attributed to two environmental factors, the rough sea surface and the bubbles produced when breaking-or spilling waves are present, the relative role of each is still uncertain. For simplicity, in terms of an average bubble population, the time-and space-varying assemblage of microbubbles is usually assumed to be uniform in range and referred to as ''the subsurface bubble layer.'' However the bubble population is range-and depth-dependent. In this article, results of an experiment ͓Weston et al., Philos. Trans. R. Soc. London, Ser. A 265, 507-606 ͑1969͔͒ involving fixed source and receivers, and observations during an extended period of time under varying weather conditions are re-examined by exercising a numerical model that allows for the dissection of the problem. Calculations are made at 2-and 4-kHz. It is shown that at these frequencies and at wind speeds capable of generating breaking waves the main mechanism responsible for the excess loss in the shallow-water waveguide is the patchy nature of the subsurface bubble field. Refraction and attenuation within the pockets of high void fraction are minor contributors to the losses.
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