Some remarks on one-dimensional models of wave motion in elastic rods

Abstract

An exact derivation from three-dimensional elasticity of a model equation for the longitudinal vibrations of a cylindrical elastic rod is presented, based on the results of [1]. Similarities and differences are discussed with the model of [2], whose study strongly motivated the work leading to [1] and opened the way to the present discussion. A difference is that the model of è2+ is not exact, being obtained through a line of reasoning that involves truncated expansions in the radius of the cross section; a similarity is that the resulting equations share the mathematically relevant properties, and describe the same physical phenomenology (in particular, they support traveling wave solutions of the solitary type).