Interactions of fast and slow waves in hyperbolic systems with two time scales

We consider certain symmetric hyperbolic systems of nonlinear partial differential equations whose solutions vary on two time scales. The large part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives is allowed. We show that if the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude O(?), then the perturbation in the solution will also be of amplitude O(?). Further, if the large part of the spatial operator is nonsingular, we show that the error introduced in the slow scale motion will be of amplitude O(?²), even though fast scale waves of amplitude O(?) will be present in the solution.