Self-demodulation of elastic waves in a one-dimensional granular chain

The self-demodulation process in a nonlinear granular chain of identical beads is studied analytically and numerically. In such a medium, in accordance with the dispersion relation, longitudinal waves that have a frequency higher than the so-called cutoff frequency of the chain are evanescent. Here, we study the influence on the self-demodulation process of the transition from the propagative to the evanescent regime in pump wave propagation that takes place when the pump frequency increases. An analytical solution in discrete coordinates is derived for the case of two primary frequencies mixing into a single difference frequency. This solution is then numerically integrated in order to analyze the demodulation of the acoustic wave packet (i.e., of the harmonic acoustic wave modulated in a pulse mode). Temporal demodulated profiles can be strongly sensitive to the regime (propagative or evanescent) of primary wave transport. This model allows us to detect the cutoff frequency of longitudinal elastic waves in the chain, without receiving the primary waves, but receiving the low frequency nonlinearly radiated signal. The roles of frequency dependent attenuation, velocity dispersion, and observation distance are analyzed.