ON PARASITIC CAPILLARY WAVES GENERATED BY STEEP GRAVITY-WAVES - AN EXPERIMENTAL INVESTIGATION WITH SPATIAL AND TEMPORAL MEASUREMENTS

An experimental investigation of steep, high-frequency gravity waves ( approximately 4 to 5 Hz) and the parasitic capillary waves they genera te is reported. Spatial, as well as temporal, non-intrusive surface me asurements are made using a new technique. This technique employs cyli ndrical lenses to magnify the vertical dimension in conjunction with a n intensified, high-speed imaging system, facilitating the measurement of the disparate scales with a vertical surface-elevation resolution on the order of 10 mum. Thus, high-frequency parasitic capillary waves and the underlying gravity wave are measured simultaneously and accur ately in space and time. Time series of spatial surface-elevation meas urements are presented. It is shown that the location of the capillary waves is quasi-stationary in a coordinate system moving with the phas e speed of the underlying gravity wave. Amplitudes and wavenumbers of the capillaries are modulated in space; however, they do not propagate with respect to the gravity wave. As capillary amplitudes are seen to decrease significantly and then increase again in a recurrence-like p henomenon, it is conjectured that resonance mechanisms are present. Me asured surface profiles are compared to the theories of Longuet-Higgin s (1963) and Crapper (1970) and the exact, two-dimensional numerical f ormulation of Schwartz & Vanden-Broeck (1979). Significant discrepanci es are found between experimental and theoretical wavetrains in both a mplitude and wavenumber. The theoretical predictions of the capillary wave amplitudes are much smaller than the measured amplitudes when the measured phase speed, amplitude, and wavelength of the gravity wave a re used in the Longuet-Higgins model. In addition, this theory predict s larger wavenumbers of the capillaries as compared to experiments. Th e Crapper model predicts the correct order-of-magnitude capillary wave amplitude on the forward face of the gravity wave, but predicts large r amplitudes on the leeward face in comparison to the experiments. Als o, it predicts larger capillary wavenumbers than are experimentally de termined. Comparison of the measured profiles to multiple solutions of the stationary, symmetric, periodic solutions determined using the Sc hwartz & Vanden-Broeck numerical formulation show similar discrepancie s. In particular, the assumed symmetry of the waveform about crest and trough in the numerical model precludes a positive comparison with th e experiments, whose underlying waves exhibit significantly larger cap illaries on their forward face than on their leeward face. Also, the a priori unknown multiplicity of numerical solutions for the same dimen sionless surface tension and steepness parameters complicates comparis on. Finally, using the temporal periodicity of the wave field, composi te images of several successive wavelengths are constructed from which potential energy and surface energy are calculated as a function of d istance downstream.