复杂介质中任意阶频率依赖耗散声波的分数阶导数模型

实验现象表明,声波在复杂介质中传播时,其衰减往往呈现频率的任意次幂律依赖现象.鉴于复杂介质的力学和物理性质的记忆性和长程相关性,频率幂律依赖的声波衰减现象难以用经典的声波方程描述,因为经典的阻尼波方程和近似热黏性波方程只能分别描述与频率无关和频率二次方依赖的声衰减.近年来,带有分数阶导数项的声波方程已被成功用于描述这一声衰减现象.基于课题组对声波衰减分数阶导数建模的研究,对已有的分数阶导数声波方程的研究进展及获得的成果做一个系统的综述,重点讨论这些模型的力学本构、统计力学解释等.简述了软物质中声波传播的时间分数阶导数唯象模型和本构模型,空间分数阶导数唯象模型和本构模型,并深入讨论了各种模型之间的联系与区别:介绍了分数阶导数声波模型在多孔介质中的成功应用,该部分内容涉及了均匀和非均匀多孔介质,刚性固体骨架和可变形固体骨架多孔介质等;通过空间分数阶扩散方程与Levy稳定分布之间的联系,给出了频率幂律依赖指数的变化区间为[0,2]的统计力学解释.最后,讨论了声波传播耗散行为的分数阶导数建模领域仍然存在的问题,并对今后的研究方向进行了探讨和展望.

FRACTIONAL DERIVATIVE MODELING OF FREQUENCY-DEPENDENT DISSIPATIVE MECHANISM FOR WAVE PROPAGATION IN COMPLEX MEDIA 1)

The existing experimental data indicate that the attenuations of acoustic waves propagating in complex media always exhibit a non-integer power-law dependence on frequency. Such phenomenon is di cult to be characterized by traditional damping wave equation or approximate thermo-viscous wave equation, which can only describe the frequency independent or frequency-squared dependent attenuation, respectively. With the dynamic development and wide applications of fractional calculus, wave equations with fractional derivative terms have been successfully applied to depicting the frequency dependent attenuation. Based on the research achievements of our group, this paper aims at presenting a review of the various fractional derivative wave equations, discussing the corresponding mechanical constitutive relationships and statistical interpretation, and laying the foundation for the in-depth study in the future. The time-and space-fractional derivative wave equations for soft matters are introduced, which can be classified into two groups:the constitutive models and the phenomenological models. The connections and di erences between such models are also discussed. Then, the successful applications of fractional derivative in modeling wave propagation in porous media are also summarized. The statistical interpretation for the power-law dependent exponent covering[0, 2] is presented via linking the space-fractional diffusion equation with Lévy stable distribution. Finally, the key problems in such area for future explorations are highlighted.