Laboratoire de Probabilites et Modeles Aleatoires and Laboratoire Jacques-Louis Lions, Universite Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France.
J Acoust Soc Am. 2010 Jan;127(1):62-72. doi: 10.1121/1.3263608.
This paper considers multiple scattering of waves propagating in a non-lossy one-dimensional random medium with short- or long-range correlations. Using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation, which corresponds to an effective medium with a frequency-dependent attenuation that obeys a power law with an exponent between 0 and 2. The exponent is related to the Hurst parameter of the medium, which is a characteristic parameter of the correlation properties of the fluctuations of the random medium. Moreover the frequency-dependent attenuation is associated with a special frequency-dependent phase, which ensures that causality and Kramers-Kronig relations are satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.
本文考虑了在具有短程或长程相关性的无损耗一维随机介质中传播的波的多次散射。利用随机均匀化理论，可以证明脉冲传播由有效确定性分数阶波动方程描述，该方程对应于具有频率相关衰减的有效介质，其衰减遵循幂律，指数在 0 到 2 之间。该指数与介质的赫斯特参数有关，赫斯特参数是随机介质的波动相关特性的特征参数。此外，频率相关衰减与特殊的频率相关相位相关联，这确保了因果关系和凯拉克斯-克朗尼克关系的满足。在时域中，有效波动方程具有分数阶导数的线性积分微分方程的形式。
