In this chapter we shall study some problems of wave propagation in the nonlinear case. We shall consider models without Newtonian viscosity which give rise to hyperbolic waves. In general we deal with first order quasilinear systems and first order nonlinear systems which are not quasilinear. In §5.7 we showed how to reduce a nonlinear system of order N in independent variables (x,t) or (x,y) into a quasilinear system of order 3N by differentiating the nonlinear system once. A nonlinear system of order N in three independent variables can be reduced to a quasilinear system of order 4N, etc. An important point is that the principal part in such differentiated systems is only of order N. N characteristic directions are possible. We expect shocks to occur in quasilinear problems when characteristics intersect. In genuinely nonlinear problems the characteristics are not parallel, they intersect. At the point of intersection of the characteristics the solution may become multiple valued. This can be interpreted as the breaking or overturn of the wave (see Figure 20.2), a point of view developed in the fine book of Whitham  on nonlinear waves.
KeywordsNonlinear Wave Finite Time Jump Condition Maxwell Model Shock Speed
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