Pressure wave propagation in a granular bed Stephen R. Hostler * and Christopher E. Brennen California Institute of Technology, Pasadena, California 91125, USA Received 28 February 2005; revised manuscript received 13 July 2005; published 21 September 2005 The transmission of pressure waves in granular materials is complicated by the heterogeneity and nonlin- earity inherent in these systems. Such waves are propagated through particle contacts primarily along the “force chains” which carry most of the load in granular materials. These fragile and ephemeral chains coupled with irregular particle packing lead to the observed heterogeneity. Nonlinearity in these systems is largely the result of the force-deformation characteristic at particle contacts. Through experiments and simulations, we study the effects of heterogeneity and nonlinearity on the properties of pressure waves through a granular bed. DOI: 10.1103/PhysRevE.72.031303 PACS numbers: 45.70.-n, 62.30.+d I. INTRODUCTION Wave propagation within a granular medium is a compli- cated process that is important not only in a broad range of technological and natural contexts but also because of the potential it has for the interrogation of the state of that me- dium. While there are circumstances in which the interstitial fluid plays an important role in the wave propagation see, for example, the fluidized bed research of Gregor and Rumpf 1, Musmarra et al. 2, and Weir 3 or the shock waves of Ben-Dor et al. 4 we choose to focus here on the simpler circumstances of a relatively static bed in which the intersti- tial fluid plays a negligible role. Rather the wave propagation involves transmission through the particles and from particle to particle through the contact points. Experiments on wave propagation in beds of randomly arranged grains 5 and ma- trices of specific packings 6 revealed propagation speeds that, as expected, seemed to scale with the elastic wave speed in the material of the particles   E /  where E is the Young’s modulus and  is the density and showed little dependence on particle size. The speed did seem to increase somewhat with the overall constraining pressure or overburden pressure p. An assumption of Hertzian contacts between the particles 7,8 led Duffy and Mindlin 6 to theorize a speed that in- creased like p 1/6 . The measurements follow this dependence at higher pressures but the behavior at lower confining pres- sures is more like p 1/4 . Goddard put forward two possible explanations for this stronger dependence at lower pressures: first, that conical asperities may dominate the interaction be- tween particles and, second, that the coordination number the number of contacts would increase as the pressure in- creased 9. These could explain the p 1/4 dependence at lower pressures. Makse et al. 10 conducted simulations that supported the coordination number explanation; the simula- tions also yielded wave speeds in good agreement with the experimental measurements. We note that Velický and Caroli 11 proposed another explanation for the p 1/4 dependence that identifies disorder-induced stress fluctuations as responsible. However, any review of the literature will clearly unveil a disturbingly broad range of measured values for the wave speed in granular beds particularly at lower pressures and it is difficult to correlate this range with the material properties involved. At low pressures the wave speeds range from 50 m/s 12 to 210 m/s 5 to 500 m/s 13 for glass or sand see Ref. 14 for details. Stresses in a static granular bed are carried by “force chains,” preferentially stressed chains of particles that are responsible for the nonisotropic distribution of stress in a granular material 15,16. Stress waves are therefore propagated primarily along these force chains. The complication is that force chains can be altered as a result of very small perturbations. This heterogeneous and ephemeral nature of a granular bed is one possible ex- planation for the above-described discrepancies. It leads to complicated and often elusive wave propagation characteris- tics that are the subject of this paper. Liu and Nagel investigated wave propagation in a bed of 5-mm glass beads using an accelerometer of a size compa- rable to an individual grain; thus they were detecting the wave transmitted through a single force chain. Though the waves seemed to be nondispersive in a general, average sense, the experiments yielded wave speeds that were highly susceptible to very small perturbations in the bed. This sug- gested an extreme fragility in the force chain microstructure; indeed the stress waves may themselves be altering that structure and therefore the local wave speed 17. Jia et al. further addressed these issues and distinguished between propagation at low frequencies where the wavelength is long compared with the particle size and waves in several force chains will retain coherence. In contrast, waves at high fre- quency where the wavelength is comparable with the particle size will lack coherence at any detector 13. Later we refer to this limit of Jia et al. which, according to their analysis, would occur around 50 kHz for 4-mm glass beads. The purpose of this paper is to delve further into these intricate complications of wave propagation in a granular material, utilizing wave propagation measurements and simulations in beds of particles of different size and material. In addition to the wave propagation speed, we also examine the attenuation which is of considerable technological inter- est since granular materials are often used for acoustic insu- lation. There are several mechanisms by which wave energy is dissipated in a granular medium. Energy is clearly dissi- *Currently at the Department of Mechanical and Aerospace Engi- neering, Case Western Reserve University. Electronic address: hostler@case.edu PHYSICAL REVIEW E 72, 031303 2005 1539-3755/2005/723/03130313/$23.00 ©2005 The American Physical Society 031303-1