Longitudinal-strain soliton focusing in a narrowing nonlinearly elastic rod

Abstract

The evolution of a longitudinal-strain solitary wave (a soliton) is studied theoretically and in experiments in a nonlinearly elastic tapered rod. Amplification (focusing) of the soliton is predicted and observed in the rod with decreasing cross section. An asymmetric soliton deformation when focused is observed. An approach is developed to obtain analytical relationships between longitudinal and shear nonlinear strains, and an asymptotic solution to the problem is found, accurately satisfying the boundary conditions on the lateral rod's surface. The explicit relationship is obtained for the soliton amplitude dependence upon the cross section radius' variations of the nonlinearly elastic rod. An allowed interval of soliton velocities is shown to exist that is dependent on elasticity. It was proved in experiments that the elastic strain soliton is not absorbed even at distances much greater than the typical linear dissipation length for linear waves in polystyrene.