The method of multiple time scales is used to obtain a uniformly valid perturbation solution for initial value problems (periodic or isolated initial disturbance) arising in shallow water motion or in compressible gas flow. The results are used to modify the original equations so as to lead to a particularly convenient problem formulation. The time at which discontinuities first appear is found to be accurately predictable.Although the specific problem considered here does not seem to have been previously treated by this method, the basic ideas are familiar in related contexts. The paper is in part expository, and comparisons are made with other methods, including conventional perturbations, perturbation of characteristics, and direct numerical solutions.