Towards Stochastic and Deterministic Modeling of Mechanical Waves in Disordered Media

{What?} Disorder of size (polydispersity) and mass of discrete elements or particles in randomly structured media (e.g., granular matter such as soil) has numerous effects on materials’ sound propagation characteristics. The influence of disorder on energy and momentum transport during vibration propagation (mechanical/sound wave), the sound wave speed and its low-pass frequency-filtering characteristics is the subject of this study. {Why?} The goal is understanding the connection between the particle-microscale disorder and dynamics and the system-macroscale wave propagation, which can be applied to nondestructive testing, seismic exploration of buried objects (oil, mineral, etc.) or to study the internal structure of the Earth. {How?} The mechanical wave/vibration propagating through granular media exhibits a specific signature in time; a coherent pulse or wavefront arrives first with multiply scattered waves (coda) arriving later. The coherent pulse is of low frequency nature and micro-structure independent i.e. it depends only on the bulk properties of the disordered granular sample, the sound wave velocity and hence, bulk and shear moduli. The coda or the multiply scattered waves are of high frequency nature and are micro-structure dependent. Numerical and stochastic techniques for 1-D, 2-D discrete element systems and experiments with 1-D photoelastic particles constituting a granular chain have been employed to isolate and study different modes of the propagating waves (namely, P and S waves), disorder dependent dispersion relations, sound wave velocity and diffusive transport of spectral energy. {Results} Increase in mass disorder (where disorder has been defined such that it is independent of the shape of the probability distribution of masses) decreases the sound wave speed along a granular chain. Averaging over energies associated with the eigenmodes can be used to obtain better quality dispersion relations and can be formulated in a way to give a Master Equation of energy in terms of wavenumber or frequency which identifies the switching and cross-talk of energy between different frequency bands; these dispersion relations confirm the decrease in pass frequency and wave speed with increasing disorder acting opposite to the wave acceleration close to the source. Also, it is observed that an ordered granular chain exhibits ballistic propagation of energy whereas, a disordered granular chain exhibits more diffusive like propagation, which eventually becomes localized at long time periods