7 - Two-and Three-Wave Resonance

Summary

INTRODUCTION

Resonance of linear and weakly nonlinear waves gives rise to a wealth of physical phenomena and many interesting mathematical problems. Resonant interactions are responsible for exchange of energy between existing wave modes, for preferential amplification from infinitesimal levels of previously undetected modes, for the disintegration of initially uniform wavetrains into more complex motion and for enhanced extraction of energy from primary shear flows. While simple linear models may provide acceptable theoretical descriptions of some aspects of these phenomena, others require more sophisticated theories such as that of inverse scattering and of strange attractors, or the brute force of modern computers. Sometimes, too, one is fortunate enough to find simple particular solutions of the nonlinear evolution equations.

Linear, or direct, resonance of two waves is considered in Section 2, and Sections 3-5 concern weakly nonlinear three-wave resonance. The aim has been to provide an overview of the range of phenomena and a review of the available mathematical solutions. Particular attention is paid in Section 5 to non-conservative systems, for which the mathematical theory is least developed. Special cases such as subharmonic resonance and further complications involving quadratic interaction of more than three waves are mentioned only briefly. Cubic and higher-order nonlinearities are not discussed, owing to lack of space. Likewise, the more general problem of weak interactions among Fourier components of a continuous wave spectrum is not confronted.

LINEAR RESONANCE

It is convenient, first, to consider the three-layer fluid flow of Figure 1. Incompressible inviscid fluid has density p(z) and unidirectional primary velocity [U(z),0,0], relative to Cartesian axes x,y,z, given by the piecewise-constant functions with p₁< p₂ < P₃. Here, z is measured upwards, gravitational acceleration is (0,0,-g) and there may be interfacial surface tensions y, y’ at the respective interfaces z=h and -h.

If Ui = U2 = U3 = 0, linear, irrotational, capillary-gravity waves may exist at either interface and the properties of waves on one interface are influenced by the proximity of the other. Waves with periodicity exp[i(kx-wt) ] are normally only weakly affected by the other interface when kh > 1.