Reconstruction of waves traveling in a nonlinear granular chain from acoustic measurements

This paper presents a method for reconstructing transient waves traveling in a discrete granular chain from acoustic pressure measurements made in the resulting sound field. A transient vibro-acoustic model is developed, based on the acoustic impulse response of an accelerating sphere, which superposes acoustic radiation from each bead of a granular chain using a convolution integral. We assume straight-line radiation of the acoustic pressure and constant position of the granular chain during propagation of a primary traveling wave. The continuous convolution integral is discretized to obtain the transfer matrix of the acoustic field response that correlates to vibrational motion of a discrete granular chain. To overcome ill-posedness in inverse processing, Tikhonov regularization with appropriate parameter-choice methods is applied. Acoustic field pressures are sampled with a linear microphone array conformal to the geometry of a granular chain at various standoff distances. It is found that the amplitude and speed of the primary traveling wave in a discrete granular chain can be reconstructed from acoustic field responses. The accuracy of reconstruction and the parameter sensitivity are examined with finite element analysis results as benchmarks. For acoustic field pressures contaminated by white noise, root-mean-square error analysis demonstrates that reasonably good reconstruction accuracy can still be achieved when the ill-conditioning of the convolution matrix and the matching of acoustic field pressures are balanced well. This study provides a useful approach to measure and visualize pulse transmission through a discrete granular medium in a time-efficient and cost-effective manner.