We investigated a diffusion-like equation with a bounded speed of signal propagation (the so called telegrapher’s equation) in a random media. We discuss some properties of the mean-value solution in a well-defined perturbation theory. The frequency-dependent effective-velocity of propagation is studied in the long and short time regime. We show that due to the wave-like character of telegrapher’s equation the effective-velocity is a complex dispersive function in time. Exact results and asymptotic perturbative long-time behaviors (for an exponential space-correlated binary disorder) are presented, showing their agreement and corroborating the goodness of the effective medium approximation in continuous system.

中文翻译：

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随机介质中的有限速度扩散

我们研究了在随机介质中具有有限信号传播速度的类扩散方程（所谓的电报方程）。我们讨论了定义明确的扰动理论中平均值解的一些特性。在长短时间范围内研究了与频率相关的有效传播速度。我们表明，由于电报方程的波状特征，有效速度是一个复杂的时间色散函数。给出了精确结果和渐近微扰长期行为（对于指数空间相关的二元无序），显示了它们的一致性并证实了连续系统中有效介质近似的优点。