Steep gravity–capillary waves within the internal resonance regime

Abstract

Steep gravity–capillary waves are studied experimentally in a channel. The range of cyclic frequencies investigated is 6.94–9.80 Hz; namely, the high‐frequency portion of the regime of internal resonances according to the weakly nonlinear theory (Wilton’s ripples). These wave trains are stable according to the nonlinear Schrödinger equation. The experimental wave trains are generated by large, sinusoidal oscillations of the wavemaker. A comparison is made between the measured wave fields and the (symmetric) numerical solutions of Schwartz and Vanden‐Broeck [J. Fluid Mech. 95, 119 (1979)], Chen and Saffman [Stud. Appl. Math. 60, 183 (1979); 62, 95 (1980)], and Huh (Ph.D. dissertation, University of Michigan, 1991). The waves are shown to be of slightly varying asymmetry as they propagate downstream. Their symmetric parts, isolated by determining the phase which provides the smallest mean‐square antisymmetric part, compare favorably with the ‘‘gravity‐type’’ wave solutions determined by numerical computations. The antisymmetric part of the wave profile is always less than 30% of the peak‐to‐peak height of the symmetric part. As nonlinearity is increased, the amplitudes of the short‐wave undulations in the trough of the primary wave increase; however, there are no significant changes in these short‐wave frequencies. The lowest frequency primary‐wave experiments, which generate the highest frequency short‐wave undulations, exhibit more rapid viscous decay of these high‐frequency waves than do the higher‐frequency primary wave experiments.