Sound propagation in impure granular columns

We present a detailed simulational study of the vertical propagation of weak and strong impulses in deep gravitationally compacted granular columns [see R. S. Sinkovits and S. Sen, Phys. Rev. Lett. 74, 2686 (1995)]. The intergrain potential is assumed to be $V\left(\delta \right)\sim {\delta }^n$, $n>~2$, where $\delta$ is the overlap between the grains. Due to gravitational compaction, the magnitude of the overlap between the grains increases progressively with increasing depth. Therefore the sound velocity increases as an impulse travels vertically downward into a granular column. For weak impulses, our large scale simulational studies show that the sound velocity $c_{\mathrm{weak}}\propto z^{\frac{[1-\frac{1}{\mathrm{}(n-1\mathrm{})\mathrm{}}]}{2}}$, where $z$ is the depth at which $c_{\mathrm{weak}}$ is measured. This result, which has been obtained from particle dynamical studies, is in perfect agreement with the predictions based upon elasticity theory. We then extend our analysis to show that (i) for columns with small void fractions, $\epsilon$, $c_{\mathrm{weak}}\propto (1-\epsilon )z^{\frac{[1-\frac{1}{\mathrm{}(n-1\mathrm{})\mathrm{}}]}{2}}$ and (ii) for large amplitude impulses, the velocity of the perturbation $c_{\mathrm{strong}}$ behaves very differently compared to $c_{\mathrm{weak}}$ at shallow depths with $c_{\mathrm{strong}}\to c_{\mathrm{weak}}$ as $z\to \infty$. We also present a detailed numerical study of the velocity power spectra of the individual grains as a function of depth $z$. We close with a discussion of the effects of both light and heavy impurities on the vertical sound and shock propagation.