Wave Solutions

TL;DR

This paper discusses the mathematical modeling of wave solutions in classical continuum physics, emphasizing linear and nonlinear partial differential equations, their properties, and the analytical techniques used to find such solutions.

Contribution

It provides an overview of the types of wave solutions in PDEs, highlighting the differences between linear and nonlinear cases and the challenges in solving complex equations.

Findings

Wave solutions arise from hyperbolic PDE properties.

Dispersion is explained through higher-order derivatives.

Nonlinear wave solutions require case-by-case analysis.

Abstract

In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic equations, there also exists a general theory for finding wave solutions. In addition, nonlinear parabolic partial differential equations are sometimes said to posses wave solutions, though they lack hyperbolicity, because it may be possible to find solutions that translate in space with time. Unfortunately, an all-encompassing methodology for solution of partial differential equations with any possible combination of nonlinearities does not exist. Thus, nonlinear wave solutions must be sought on a case-by-case basis depending on the governing equation.