In the hyperbolic case the solution can exhibit steep moving fronts which are difficult to resolve numerically. Waves of different wavelenths will travel at the same phase speed if the waves are nondispersive and travel with different phase speeds if the waves are dispersive. This means that dispersion tends to sort out the waves based on their phase speeds. If the nonlinearity tends to increase the phase speed with amplitude, then large waves tend to catch up with smaller waves. The result is a steepening of the waveform with time which could lead to a shock.
Waves traveling in shallow water also steepen and sometimes break.
There is more than one method to analyze nondispersive waves traveling in either a homogeneous or a heterogeneous media. By including viscous effects we come back to Burgers' equation. It serves as a demonstration for a nonlinear scalar hyberbolic equation. The web page includes a graph which allows the user to modify the curve which actually represents the density of the cars. The horizontal direction of the graph represents distance in terms of a one-way street with cars driving to the right.