Model for the propagation of sound in granular materials

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Abstract
This paper presents a simple model for the propagation of small-amplitude vibrations in a granular material. In this model, the grains are taken to be spherical balls that interact via linear springs. The positional disorder in the real system is ignored and the particles are placed on the vertices of a square lattice. The only disorder in the system comes from a random distribution of the spring constants. Despite its apparent simplicity, this model is able to reproduce the complex frequency response seen in measurements of sound propagation in a granular system. In order to understand this behavior, the role of the resonance modes of the system is investigated. Finally, this simple model is generalized to include relaxation behavior in the force network—a behavior which is also seen in real granular materials. This model gives quantitative agreement with experimental observations of relaxation.
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