MODULATED HIGH-FREQUENCY WAVES

We construct asymptotic approximations to solutions of nonlinear hyperbolic conservation laws, when the initial data is small-amplitude high-frequency waves with modulated wave number. We show that the nonlinear multiwave interaction terms approach zero in the asymptotic limit, so that the wave components satisfy decoupled Burgers equations, provided a certain nonresonance condition holds. This extends previous results on more strictly nonresonant or everywhere resonant waves, to permit modulated high frequencies to pass through resonant and nonresonant values. We show how these results apply to high-frequency wave propagation in a nonhomogeneous medium or on a nonuniform base state. We illustrate our conclusions with numerical examples and discuss a phenomenon of localized resonance.