Buckingham Michael J
Marine Physical Laboratory, Scripps Institution of Oceanography, University of California-San Diego, 9500 Gilman Drive, La Jolla, California 92093-0238,
J Acoust Soc Am. 2008 Oct;124(4):1909-20. doi: 10.1121/1.2973231.
Acoustic wave propagation in a dispersive medium may be described by a wave equation containing one or more dissipation terms. Three such equations are examined in this article: van Wijngaarden's equation (VWE) for sound propagating through a bubbly liquid; Stokes' equation for acoustic waves in a viscous fluid; and the time-dependent diffusion equation (TDDE) for waves in the interstitial gas in a porous solid. The impulse-response solution for each of the three equations is developed and all are shown to be strictly causal, with no arrivals prior to the activation of the source. However, the VWE is nonphysical in that it predicts instantaneous arrivals, which are associated with infinitely fast, propagating Fourier components in the Green's function. Stokes' equation and the TDDE are well behaved in that they do not predict instantaneous arrivals. Two of the equations, the VWE and Stokes' equation, satisfy the Kramers-Kronig dispersion relations, while the third, the TDDE, does not satisfy Kramers-Kronig, even though its impulse-response solution is causal and physically realizable. The Kramers-Kronig relations are predicated upon the (mathematical) existence of the complex compressibility, a condition which is not satisfied by the TDDE because the Fourier transform of the complex compressibility is not square-integrable.
在色散介质中的声波传播可以用一个包含一个或多个耗散项的波动方程来描述。本文研究了三个这样的方程：用于描述声音在含气泡液体中传播的范·温加登方程（VWE）；用于描述粘性流体中声波的斯托克斯方程；以及用于描述多孔固体中孔隙气体中波的含时扩散方程（TDDE）。推导了这三个方程各自的脉冲响应解，结果表明它们都是严格因果的，在源激活之前没有波的到达。然而，VWE是非物理的，因为它预测了瞬时到达，这与格林函数中无限快传播的傅里叶分量有关。斯托克斯方程和TDDE表现良好，它们不会预测瞬时到达。其中两个方程，即VWE和斯托克斯方程，满足克拉默斯 - 克勒尼希色散关系，而第三个方程TDDE不满足克拉默斯 - 克勒尼希关系，尽管其脉冲响应解是因果的且在物理上是可实现的。克拉默斯 - 克勒尼希关系基于复压缩性的（数学）存在性，而TDDE不满足这一条件，因为复压缩性的傅里叶变换不是平方可积的。
