Experimental Realization of Periodic Deep-Water Wave Envelopes with and without Dissipation

International audienceThe Korteweg-de Vries equation, that describes surface gravity water wave dynamics in shallow water, is well-known to admit cnoidal wave solutions, i.e. periodic travelling waves with stationary wave shape. Such type of periodic wave patterns can be also found in the deep water waves with the envelopes that follow the dynamics of the nonlinear Schrödinger equation (NLS). A particular class of NLS periodic and stationary solutions are cnoidal (CN) and dnoidal (DN) envelopes. These one-parameter solutions have as limiting cases either the envelope soliton or a constant background. In this experimental study we discuss the physical features of such waves and emphasize the particular effect of dissipation, as observed in several hydrodynamic experiments that have been conducted in several water wave facilities with different dimensions. Experiments on such type of periodic wave envelopes in a large facility have been already reported in the early 90s. These studies demonstrated significant deviation of the DN-type envelopes from theory. Here, we show that these deviations are due to the effect of dissipation that can be qualitatively considered by adapting the NLS framework accordingly. Reducing the amplitude of the carrier wave makes the wave field susceptible to dissipation effects. Our experiments prove that the dissipation is indeed responsible for phase-shift pulsations for DN-type envelopes. 1 Introduction Simplified evolution equations are useful for the fundamental understanding of complex dynamics. One example for water waves in deep and finite water depth is the nonlinear Schrödinger equation (NLS), which is the simplest weakly nonlinear framework that allows the description of dispersive and weakly nonlinear hydrodynamics of wave packets in time and space. An essential feature of the NLS is its integrability. Using advanced integration techniques, a wide range of analytical stationary and pulsating envelope solutions can be derived. These allow for the numerical and experimental investigation of particular localized wave patterns as well as the controlled generation of waves in laboratory environments. The dynamics of such wave packets can be initiated using the exact initial or boundary conditions, depending on wether the evolution is in time or in space, as determined by the initial model