Asymptotic solution for solitary waves in a chain of elastic spheres

Abstract

We consider a solitary wave in a chain of contacting, but initially unstressed, particles between which the compressive force F as a function of relative approach x is ${F=kx}^n.$ By “initially unstressed” we mean that there is zero contact force between neighboring particles that are infinitely far from the crest of the wave. For a chain of elastic spheres in Hertzian contact, $n=\frac{3}{2}.$ In this work, n is treated as “slightly” greater than 1, and an asymptotic solution for the solitary wave is developed in terms of the associated small parameter. The solution for the propagating velocity wave is found as a slightly perturbed Gaussian. Comparison with numerics shows that the asymptotic solution is very good even for the fairly large value of $n=\frac{3}{2}$ and is substantially more accurate than the presently available approximate solution given by Nesterenko.