Second Order Hyperbolic Equations with Small Nonlinearities

A second order partial differential equation which describes the propagation of one-dimensional nonlinear waves in a bounded, inhomogeneous, dissipative medium is analyzed using the method of multiple scales. The conditions under which the oppositely traveling components of the nonlinear motion uncouple to first order are given. The nonlinear interaction occurs at second order. Two examples are given: the first is a nonlinear free vibration problem; the second is a problem of resonant forced oscillations induced by a time-periodic body force. The nonlinear resonant motion is always bounded, but may contain shocks.