Interactions of fast and slow waves in problems with two time scales

Consideration is given to certain symmetric hyperbolic systems of nonlinear partial differential equations whose solutions vary on two time scales, a slow scale t and a fast scale t/epsilon. It is shown that if the initial data are not prepared correctly for the suppression of the fast scale motion and contain errors of amplitude O(epsilon to the mu), so that only mu time derivatives of the solution are bounded at t = 0, then fast waves of amplitude O(epsilon to the mu) will be present in the solution. The error introduced in the slow scale motion by nonlinear interactions of these waves, however, will only be of amplitude O(epsilon to the 2 mu) + O(epsilon to the mu + 1). Since this holds for any mu greater than 0, it extends the results given by Kreiss (1979, 1980). It is concluded that the effects of the fast waves can be controlled more easily than has been thought. There is thus partial justification for neglecting them in certain physical situations.