# Abstract

We have critically tested the application of the

diffusion approximationto describe the propagation of ultrasonic waves through a random, strongly scattering medium. The transmission of short ultrasonic pulses has been measured through a concentrated suspension of glass beads immersed in water. The transmitted sound field is found to exhibit temporal fluctuations with a period determined by the width of the incident pulse. Provided that appropriate boundary conditions are used to account for the reflectivity of the interfaces,the time dependence of the ensemble-averaged transmitted intensity is shown to be well described by the diffusion equation.

我们对应用**扩散近似**来描述超声波在随机强散射介质中的传播进行了严格测试。我们测量了短超声波脉冲在浸入水中的玻璃珠浓缩悬浮液中的传播情况。研究发现，传播声场表现出时间波动，其周期由入射脉冲的宽度决定。只要使用适当的边界条件来考虑界面的反射性，就可以用扩散方程很好地描述系综平均透射强度的时间依赖性。

This enables us to determine both the diffusion coefficient for the sound waves as well as the inelastic absorption rate. The consistency of these results is established by varying the experimental geometry; while the transmitted pulse shape changes markedly, the values for the diffusion coefficient and absorption rate obtained through a description using the diffusion approximation remain unchanged. We have also measured the absolute transmitted intensity ofthe sound as the sample thickness is varied; this provides an accurate measure of the transport mean free path and thus also the energy transport velocity. These results convincingly demonstrate the validity of using the diffusion approximation to describe the propagation of sound waves through strongly scattering media.

这使我们能够确定声波的扩散系数和非弹性吸收率。通过改变实验几何形状可以确定这些结果的一致性；虽然透射脉冲的形状发生了明显变化，但通过使用扩散近似描述获得的扩散系数和吸收率值保持不变。我们还测量了样品厚度变化时声音的绝对传输强度；这提供了传输平均自由路径的精确测量，从而也提供了能量传输速度。这些结果令人信服地证明了使用扩散近似来描述声波在强散射介质中传播的可靠性。

# Introduction

The description of the propagation of classical waves through strongly scattering media is a problem of considerable importance to many areas of physics; it is also a problem of great difficulty and a full understanding has as yet remained elusive, despite considerable research effort. However, much progress has been achieved in recent years through the study of the propagation of

electromagnetic wavesthrough strongly scattering materials. To a considerable extent, this progress has been based on the success of the diffusion approximation in describing the propagation. Within this picture,the phase information of the scattering processes is neglectedand the propagation of the average energy density is approximated as a diffusive process.

描述经典波在强散射介质中的传播是一个对物理学许多领域都相当重要的问题；同时也是一个难度很大的问题，尽管研究人员付出了大量努力，但至今仍无法完全理解这个问题。不过，近年来通过研究**电磁波**在强散射材料中的传播，已经取得了很大进展。在很大程度上，这一进展是基于扩散近似在描述传播方面的成功。在这种情况下，__散射过程的相位信息被忽略，平均能量密度的传播被近似为一个扩散过程__。

The diffusion coefficient is $D = v_{e}l^{*}/3$, where the transport mean free path $l^{*}$ is the mean distance traveled before the direction of propagation is randomized and v, is the velocity at which the energy is transported. The solution of the diffusion equation determines the distribution of multiple scattering paths; each of these paths is then assigned a phase based on its total length. This approach has been particularly successful with electromagnetic waves, in- cluding light and microwaves; it correctly accounts for a wide variety of fascinating phenomena, from the enhancement of the backscattered radiation [2,3] to the correlations of the transmitted intensity with variations in the incident frequency [4], the angle of the sample [5], or the temporal position of the scatterers [6,7].