Nonlinear Wave Equation for an Elastic String: Derivation, Symmetries, and Conserved Quantities

Abstract

The goal of this thesis is to study the exact, most general wave equation governing the large-amplitude motion of an elastic string. The linear one-dimensional wave equation, commonly presented in undergraduate textbooks to describe the motion of an elastic string, relies on simplifying assumptions such as Hooke's law, all of which fail for large-amplitude motion. We examine these assumptions and offer a brief survey of derivations in the literature that relax some of these assumptions while retaining Hooke's law. We then present two derivations of the exact wave equation from nonlinear elasticity and continuum mechanics, both consistent with each other, to describe large amplitude motion. Finally, we turn to symmetries and conservation laws, two important tools in the study of partial differential equations. We find and interpret the kinematic symmetries and conservation laws admitted by this wave equation. The conservation laws are obtained from the symmetries by applying Noether's theorem. We describe the theory behind both computations and perform them using Maple software.