Analysis of capillary-gravity waves using the discrete periodic inverse scattering transform

An online measurement technique based on capacitance measurements has been used to study surface water waves induced by harmonic forcing propagating in a laboratory flow cell with dimensions of the order of a few centimeters. It has been postulated theoretically that such nonlinear dispersive waves propagating on an interface of length just a few multiples of the fluid depth satisfy an evolution equation with steeper nonlinearity than the conventional Korteweg–de Vries (KdV) equation for shallow water theory. In this paper a nonlinear Fourier analysis is performed on time-series of surface displacement made at two locations downstream of an oscillating membrane. Results from a discrete periodic inverse scattering transform based on the KdV equation, which decomposes a signal into soliton, cnoidal and sinusoidal components, indicate that whilst the amplitudes of linear modes are conserved as the disturbance propagates between the two sensors, the amplitudes of the nonlinear modes increase. This suggests that the nonlinearity of such surface waves is indeed stronger than that predicted by the KdV equation.

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► Harmonically induced waves in ‘finite’ length wave tank more nonlinear than KdV.
► Displacement fluctuations calculated via calibration with variation in capacitance.
► Surface displacement analyzed by discrete periodic inverse scattering transform.