Model and experiments for resonant generation of second harmonic capillary–gravity waves

The generation and propagation of a second harmonic water wave have been investigated in the frequency range between 7 Hz and 60 Hz where the velocity vs. frequency curve attains its minimum value. A model is proposed by assuming that the second harmonic is locally generated by point sources on the wavefront of the fundamental wave, and that at any point along the propagation direction the second harmonic be given by the cumulative contribution from all the sources up to the considered point. In the frequency range examined the combined effects of gravity and capillarity yield the so called resonance condition where the fundamental and second harmonic waves share the very same phase velocity. In such a case, wave shape matching condition is maintained between the two waves along all the propagation directions, with the amplitude of the second harmonic only limited by the attenuation effect. Evidence is given experimentally of such effect through the wavenumbers mismatching produced by the model vs. frequency and the detection of the maximum distance of second harmonic amplitude from the wave source. Furthermore, it is found that the resonance condition is a threshold effect with respect to water depth.