Abstract
Time reversal (TR) focusing of ultrasound in granular packings is experimentally investigated. Pulsed elastic waves transmitted from a compressional or shear transducer source are measured by a TR mirror, reversed in time and back-propagated. We find that TR of ballistic coherent waves onto the source position is very robust regardless driving amplitude but provides poor spatial resolution. By contrast, the multiply scattered coda waves offer a finer TR focusing at small amplitude by a lens effect.
However, at large amplitude, these TR focusing signals decrease significantly due to the vibration-induced rearrangement of the contact networks, leading to the breakdown of TR invariance. Our observations reveal that granular acoustics is in between particle motion and wave propagation in terms of sensitivity to perturbations. These laboratory experiments are supported by numerical simulations of elastic wave propagation in disordered 2D percolation networks of masses and springs, and should be helpful for source location problems in natural processes.
对颗粒堆积中超声波的时间反转（TR）聚焦进行了实验研究。从压缩或剪切传感器源发射的脉冲弹性波通过 TR 镜测量、时间反转和反向传播。我们发现，弹道相干波在源位置上的 TR 非常稳健，与驱动振幅无关，但空间分辨率较低。相比之下，通过透镜效应，多重散射的尾随波可在小振幅上提供更精细的 TR 聚焦。
然而，当振幅较大时，由于振动引起的接触网络重新排列，这些 TR 聚焦信号会显著降低，从而导致时间反演对称性的破坏。我们的观察结果表明，就对扰动的敏感性而言，颗粒声学介于粒子运动和波传播之间。这些实验室测试得到了由质量和弹簧组成的无序二维渗滤网络中弹性波传播数值模拟的支持，应该有助于解决自然过程中的声源定位问题。
Introduction
In a non-dissipative medium, the wave equation is symmetric in time. Therefore, for every wave diverging from a pulsed source, there exists in theory a wave, the time-reversed wave, that precisely retraces all its original paths in a reverse order and converges in synchrony at the original source as if time were going backwards. This time-symmetry exists even in a strongly heterogeneous medium where waves are strongly reflected, refracted, or scattered.
在无耗散介质中，波方程在时间上是对称的。因此，对于从脉冲源发散的每一个波，理论上都存在一个波，即时间反向波，它以相反的顺序精确地回溯其所有原始路径，并同步汇聚到原始源，就像时间在倒退一样。这种时间对称性甚至存在于波被剧烈反射、折射或散射的强异质介质中。
In the early nineties, an original method for generating such a time-reversed wave was proposed in acoustics: a pulsed wave is sent from a source, propagates in an unknown media and is captured at a transducer array termed a “Time Reversal Mirror (TRM)”. Then the waveforms received at each transducer are reversed in time and sent back, resulting in a wave converging at the original source regardless of the complexity of the propagation medium. TRMs have now been implemented in a variety of physical scenarios from hundreds of Hz in ocean acoustics and MHz Ultrasonics to GHz Microwaves. Common to this broad range of scales is a remarkable robustness exemplified by observations that the more scattering the medium, the sharper the focus.
九十年代初，声学领域提出了一种产生这种时间反转波的独创方法：从声源发出脉冲波，在未知介质中传播，并被称为 “时间反转镜（TRM）“的传感器阵列捕获。然后，将每个传感器接收到的波形在时间上反转并发送回去，这样，无论传播介质的复杂程度如何，都会产生汇聚到原始声源的波形。TRM 现已应用于各种物理场景，从海洋声学中的数百赫兹、兆赫超声波到千兆赫微波。在这一广泛的尺度范围内，TRM 具有显著的鲁棒性，这体现在介质散射越大，聚焦越清晰。
For the last decade the time reversal focusing concept has also been a very active research in seismology, especially for seismic source imaging and source location of seismic events that exhibit no compressional (P-) and shear (S-) wave arrivals, such as tremor, glacial earthquakes and Earth hum. In that case the real Earth, i.e., the medium where the wave field is generated and propagates, and the virtual Earth, i.e., the velocity model in which the time-reversed wave is numerically back-propagated, are however different.
近十年来，时间反转聚焦概念在地震学领域也是一项非常活跃的研究，特别是在震源成像和震源定位方面，这些研究主要针对没有压缩波（P 波）和剪切波（S 波）到达的地震事件，如震颤、冰川地震和地球嗡嗡声。在这种情况下，真实地球（即波场产生和传播的介质）和虚拟地球（即时间反向波在其中进行数值反向传播的速度模型）是不同的。
As a model system for athermal amorphous media or seismic fault gouges, the granular medium constitutes a particular case among strongly scattering systems. Dry granular media are collections of macroscopic grains that interact through repulsive and frictional contact forces.
For given values of macroscopic control parameters, such as packing density and confining pressure, granular media exhibit many microstates characterized by highly heterogeneous contact force networks that can rearrange under driving. These media whose features range from the microscopic scale (grain) to the mesoscopic scale (force-chain) and the macroscopic scale (bulk), may be modelled either as particulate or continuum materials.
作为热非晶介质或地震断层沟的模型系统，颗粒介质是强散射系统中的一种特殊情况。干燥的颗粒介质是宏观颗粒的集合，它们通过排斥力和摩擦接触力相互作用。
在宏观控制参数（如堆积密度和约束压力）给定的情况下，颗粒介质会表现出许多微观状态，其特征是高度异质的接触力网络，在驱动力的作用下可以重新排列。这些介质的特征范围从微观尺度（颗粒）到介观尺度（力链）和宏观尺度（块体），既可以被模拟为颗粒材料，也可以被模拟为连续材料。
Elastic waves that propagate from grain to grain provide a unique probe of the contact force networks. Generally speaking, one distinguishes between the longwavelength coherent (P- and S-) waves and the short-wavelength scattered waves scattered by the heterogeneous force chains, often referred to as coda waves. The study of the TR focusing of elastic waves in a granular medium raises two challenging issues.
First, no wave equation is available at the scale of the force chains.
在晶粒间传播的弹性波为接触力网络提供了独特的探测手段。一般来说，我们将弹性波分为长波长相干波（P 波和 S 波）和由异质力链散射的短波长散射波（通常称为尾波）。颗粒介质中弹性波的 TR 聚焦研究提出了两个具有挑战性的问题。
首先，没有力链尺度的波方程。
This issue is related to a fundamental question: at what scale is the continuum elasticity applicable in a contact network?
Secondly, one may wonder whether time-reversal invariance still holds in a fragile granular medium, beyond a certain wave amplitude where the wave itself not only acts as a probe but also as a pump, leading to the acoustic fluidization of the jammed media via the rearrangement of the contact network. This situation is fundamentally different from those previously reported where a perturbation was introduced in the continuous medium between the forward and backward propagation steps.
这个问题与一个基本问题有关：连续弹性在什么尺度上适用于接触网络？
其次，我们可能会问，在脆弱的颗粒介质中，时间反转不变性是否仍然成立？超过一定的波幅后，波本身不仅起探针作用，还起到了泵的作用，通过接触网络的重新排列导致堵塞介质的声学流体化。这种情况与之前报道的在前进和后退传播步骤之间在连续介质中引入扰动的情况有着本质区别。
In this work, we address the above issues by experimentally investigating timereversal focusing of ultrasonic waves in glass bead packings under external load. The robustness of TR invariance is tested with a specifically developed TRM as a function of the source amplitude. A particular attention is paid to the spatial extent of the rearrangement caused by the large-amplitude driving.
在这项工作中，我们通过对外部负载下玻璃珠堆积中超声波的时间反向聚焦进行实验研究，以解决上述问题。我们使用专门开发的 TRM 测试了 TR 不变性与声源振幅的函数关系。特别关注了大振幅驱动所引起的重新排列的空间范围。
Experiments
A sketch of the experimental setup is shown in Figure 1a. Our granular materials consists of dry monodisperse glass beads of diameter $d = 1.5$ or $3\text{mm}$, confined in a cylindrical container of diameter $D = 150\text{mm}$ with rigid walls (i.e., an oedometer cell) which is filled to a height of $H\approx 55\text{mm}$ with a packing density of about $\phi\approx 0.62$. A static uniaxial stress $P\approx 85\text{kPa}$ is applied to the granular packing. To perform the time-reversal experiment, a compressional or shear transducer is placed in contact with the granular packing at the top of the cell and used as a source excited by a 3-cycle tone burst centered at $100\text{kHz}$. We have developed a specific time-reversal mirror (TRM) with sixteen identical transducers, compressional or shear. Six other transducers surrounding the source (with a pitch of $20\text{mm}$ between neighbouring transducers) are used to measure the extension of the time-reversed focal spot. The diameters of these transducers are $a = 12\text{mm}$, which are sufficiently larger than the bead size to ensure an efficient detection of transmitted elastic waves. To investigate nonlinear effects, we vary the source amplitude from $5$ to $300\text{V}$, corresponding to a vibration displacement $u_{0}\approx 1-60\text{nm}$.
