Long wavelength solitary waves in Hertzian chains
Published 2017-05-18Version 1
Properties of solitary waves in weakly pre-compressed Hertzian chains are studied in the long wavelength regime using a well-known continuum model. Several main results are obtained by parameterizing solitary waves in terms of their impulse momentum and their asymptotic amplitude. First, solitary waves with a fixed impulse are shown to comprise a one-parameter family of solutions, and the shape of these solitary waves is shown to be highly sensitive to their wave speed. Second, an explicit approximate formula for their physical height and width is derived. This formula is used to compare the shape of the solitary waves to the shape of Nesterenko's compacton solution having the same impulse momentum. An important conclusion is that solitary waves have a noticeably different shape than the compacton if the speed ratio between the solitary waves and the compacton exceeds approximately 10%. Exact solitary wave solutions are used to illustrate all of these results.