Part two - Nonlinear Waves

Summary

So far we have studied only linear wave systems. With the exception of electromagnetic waves, we formulated the governing equations by looking at small amplitude disturbances of steady states – a string in equilibrium, a motionless ideal gas or elastic solid, the flat, undisturbed surface of a fluid or solid. If y₁ and y₂ are solutions of a linear system of equations, then a₁y₁ + a₂y₂ is also a solution for any constants a₁ and a₂. In particular, this means that separation of variables and integral transform methods allow us to determine the solution. In fact, these are the only techniques we have used. Compare what happens for nonlinear systems, for example, disturbances of a steady state that do not have a small amplitude. If y₁ and y₂ are solutions of a nonlinear system of equations, then, in general, neither y₁ + y₂ nor ky₁, with k a constant, are solutions. Our standard mathematical techniques fail, and we must think again.

We begin by introducing some of the techniques that can be used to study nonlinear systems of equations by looking at two specific examples in chapter 7. In section 7.1 we study in detail a simple model for the flow of traffic. The governing equation determines how the density of cars changes along a road with a single lane. In chapter 3 we studied small amplitude disturbances to a compressible gas. In section 7.2 we investigate finite amplitude disturbances.