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The study analyzes the microwave imaging of shoaling nearshore surface gravity waves during the process of steepening and breaking over two beaches for a wide range of environmental conditions. Data are sourced from two coherent X-band radars, operating under low-grazing angle conditions. Using automatic wave tracking on the radar images, the evolution of individual waves is followed over hundreds of meters. The extracted backscatter intensity and Doppler speed form distinct patterns that reveal a nonnegligible dependence on environmental conditions. Statistical representations of the backscattered signal are presented by conditional Doppler speed–intensity histograms. These are composed of ensembles extracted at the radar-facing fronts. This technique helps to focus on the steepening of the wave and minimizes the impact of extremely low-grazing angle imaging mechanisms that are still not well understood. The combination of wind speed and direction, as well as initial wave steepness, local depth, and degree of nonlinearity, contributes to the shapes and centroid positions of the histograms. The backscatter signature exhibited by breaking waves remained consistent and similar throughout all datasets. The results are consistent across radars. Before consolidating the findings, it is imperative to conduct a further investigation of radar imaging of nonlinear wave dynamics in shallow water to eliminate any possible influence of imaging mechanisms.

Real-time prediction of free-surface elevation is necessary for a variety of applications. Assuming a Gaussian wave field, the wave spectrum can be used to calculate the statistically-optimal predictor, for a given prediction configuration (i.e. for a given combination of measurement instants and spatial locations, relative to the instants and locations where and when the wave is predicted). More specifically, the optimal predictor is linear, and its coefficients need only be updated as the wave condition evolves. This spectrum-based prediction (SBP) approach encompasses, in a unified theoretical framework, both “time-series” and “spatially-distributed” prediction configurations. In this paper, the validity of the SBP theoretical framework is tested against real-sea wave data, which originate from a measurement campaign using an Acoustic Doppler Current Profiler (ADCP), and consist of free-surface elevation time series, at the corners and centre of a 25m-by-25m square. The directional wave spectra, corresponding to the sea states where the time series are provided, have also been calculated. The empirical SBP accuracy, obtained by applying the SBP in the real-sea time series, is assessed in various sea conditions and prediction configurations, and compared with the theoretical SBP accuracy, evaluated based on the wave spectra. Although the ADCP measurement layout is clearly not ideal for the purpose of wave forecasting, empirical results are physically and statistically consistent, and show good agreement with theoretical results, thus supporting the relevance of the SBP framework.

Recent investigations in nonlinear fiber optics have shown a renewed interest in certain classes of analytical solutions of the Nonlinear Schrödinger equation which, although present in the mathematical literature for 25 years, have been largely overlooked in studies of nonlinear fiber propagation. In this paper we review recent experiments that have shown the power of this analytic approach.

The steady response of an infinite unbroken floating ice sheet to a moving load is considered. It is assumed that the ice sheet is supported below by water of finite uniform depth. For a concentrated line load, earlier studies based on the linearization of the problem have shown that there are two ‘critical’ load speeds near which the steady deflection is unbounded. These two speeds are the speed c0 of gravity waves on shallow water and the minimum phase speed cmin. Since deflections cannot become infinite as the load speed approaches a critical speed, Nevel (1970) suggested nonlinear effects, dissipation or inhomogeneity of the ice, as possible explanations. The present study is restricted to the effects of nonlinearity when the load speed is close to cmin. A weakly nonlinear analysis, based on dynamical systems theory and on normal forms, is performed. The difference between the critical speed cmin and the load speed U is taken as the bifurcation parameter. The resulting normal form reduces at leading order to a forced nonlinear Schrödinger equation, which can be integrated exactly. It is shown that the water depth plays a role in the effects of nonlinearity. For large enough water depths, ice deflections in the form of solitary waves exist for all speeds up to (and including) cmin. For small enough water depths, steady bounded deflections exist only for speeds up to U*, with U* < cmin. The weakly nonlinear results are validated by comparison with numerical results based on the full governing equations. The model is validated by comparison with experimental results in Antarctica (deep water) and in a lake in Japan (relatively shallow water). Finally, nonlinear effects are compared with dissipation effects. Our main conclusion is that nonlinear effects play a role in the response of a floating ice plate to a load moving at a speed slightly smaller than cmin. In deep water, they are a possible explanation for the persistence of bounded ice deflections for load speeds up to cmin. In shallow water, there seems to be an apparent contradiction, since bounded ice deflections have been observed for speeds up to cmin while the theoretical results predict bounded ice deflection only for speeds up to U* < cmin. But in practice the value of U* is so close to the value of cmin that it is difficult to distinguish between these two values.

In a two-fluid system where the upper surface of the upper fluid is free, there are two independent modes of oscillation about the state of equilibrium, an ‘internal’ mode and an ‘external’ mode, which are described by two distinct dispersion curves. An efficient numerical scheme based on Fourier series expansions is used to calculate periodic waves of permanent form and of finite amplitude. Three kinds of waves are calculated: combination waves resulting from the interaction between an ‘internal’ mode and an ‘external’ mode with the same phase speed but wavelengths in a ratio of 2 (1:2 resonance), combination waves resulting from the interaction between a long ‘internal’ mode and a short ‘external’ mode with the same phase speed, and pure ‘external’ waves. It is shown that the 1:2 resonance, which is well-known for capillary – gravity surface waves and can profoundly affect wave field evolution, can affect pure gravity waves in a two-fluid system, but not in oceanic conditions. On the other hand, it is shown that the long/short wave resonance can occur in ocean-type conditions. Finally it is confirmed that pure external waves of finite amplitude behave like surface waves.

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem. The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.