Subharmonic resonance of nonlinear cross-waves: Theory and experiments.

Abstract

The generation and evolution of cross-waves in a channel are investigated analytically, numerically and experimentally. The derivation of the modulation equation governing the inviscid cross-wave amplitude yields the nonlinear Schrodinger equation with a homogeneous Robin boundary condition at the wavemaker. Either of two uniformly valid scalings--cross-wave amplitude of the same order as or much larger than the wavemaker amplitude--may be used in the derivations. The differences between the two scalings are discussed. The inviscid modulation equation is augmented by a linear damping term, the coefficient of which is determined empirically from the measured neutral stability curve. The viscous modulation equation is solved numerically. The theory is compared to experiments in a channel 30.9 cm wide, for mode n = 6, for frequencies close to the cutoff frequency 7.82 Hz. Measurements include the neutral stability curve, the onset of modulation, cross-wave phase along the channel, and cross-wave amplitude as functions of wavemaker amplitude, forcing frequency and distance from the wavemaker. These measurements are in good agreement with the numerical results. The results are also observed to be sensitive to viscous effects. Additionally, both numerical calculations and experiment reveal trapped and propagating modes. The trapped mode is most easily observed at positive detuning.