The excitation of surface waves near a cut-off frequency

Abstract
This paper describes a study of surface waves in a uniform channel, where the waves are generated by a plane flap executing torsional oscillations about a vertical axis at a frequency near a cut-off value for a wave mode. Experiments indicate that, near a cut-off frequency, the wave response is relatively large, and indeed linear inviscid theory suggests that the wave amplitudes are infinitely large at the cut-off frequency itself. Here we present theories for the modification of this result by making allowance (separately) for nonlinear terms in the surface boundary condition and for viscous dissipation. In order to estimate the effectiveness of the wavemaker in forcing the motions, a separate calculation was made to apportion the driving condition into a part driving a parasitic non-propagating field and a part forcing the wave modes. Also described in the paper are experiments in which the wave response has been measured in a similar situation to that modelled by the analytic work, and one of the main purposes of this study is to try to ascertain how well the theoretical model describes the experimental situation. An important feature to emerge from the comparison is that, even though the observed wave amplitudes were rather large and the temporal decay rate of standing waves corresponding to the cut-off mode was quite small, the dissipative effect played a crucial role in determining the structure of the response. Because of this the theoretical response was determined by numerical computation. Some of the results show a similarity with the response of a nonlinear spring, but there are significant differences. The results indicate that the model gave a good qualitative description of the experiments, and accordingly our main conclusions to the study are: (i) the multiple-scale calculation, by which the nonlinear effects were estimated, appears to have given useful results in this particular case; (ii) the way in which the dissipative effects were modelled appears to have been satisfactory; (iii) the method of estimating the effective driving condition at the wavemaker seems to have worked very well.

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