Experimental study of wave dispersion and stability in crossing seas and on sheared currents
Steer, James Nicholas
This thesis considers the effects of crossing seas and linearly sheared current on the dispersion and stability of surface gravity waves. Experimental data are compared against predictions by three different nonlinear Schr¨odinger equations (NLSE): the constant-vorticity (vor-NLSE), used to simulate wave evolution on a linearly sheared current; the coupled (CNLSE), which predicts the interaction between two crossing wave systems; and the two-dimensional (2D+1 NLSE), which allows an angle between the carrier wave and the packet of a single wave system. In chapter 2, the linearly sheared currents examined are one-dimensional, in accordance with wave propagation and consist of a velocity profile varying linearly with depth. Such currents have constant-vorticity and, although rotational, admit potential flow solutions. Both the linear evolution and the weakly nonlinear behaviour of waves on five constant vorticity sheared currents in the shear rate, Ω range, 0 s-1 ≤ Ω ≤ -0:87 s-1 are measured and compared to predictions by the vor-NLSE and vor-dispersion relation. It is found that the constant-vorticity equations agree extremely well with the experimental measurements in all cases. Significant differences between the vor-equations and uniform velocity equations are found at the strongest shear cases for both stability and linear dispersion experiments (-0:48-s-1 ≤ Ω ≤ -0:87 s-1). In chapter 3, the coupled nonlinear Schrodinger equation (CNLSE) is used to quantify the effect of a crossing angle between two weakly nonlinear coupled wave systems. Individually (when unidirectional) both systems show modulational instability. This is augmented by the addition of a crossing angle between the two wave systems. Linear stability analysis of the CNLSE indicates that wavetrains become increasingly stable as the crossing angle is increased, reaching stability at a critical angle of 35:26°. The experiments presented in this thesis measured the stability of crossing angles up to 88° for a coupled system showing clear instability when the wavetrains are unidirectional. Initially strong instabilities for the interacting unidirectional case are quickly stabilised as the crossing angle is increased. The system becomes entirely stabilised when the crossing angle is increased beyond the critical angle. In chapter 4, the two-dimensional nonlinear Schrodinger equation is used to impose a crossing angle between the carrier wave and continuous sidebands of a narrow-banded wave group. Measurements of a low-steepness wave group envelope showed normal dispersive behaviour when unidirectional. However, as the two-dimensional nonlinear Schrodinger equation predicts, at the critical angle of 35:26° it was found that the Gaussian wave group propagated with entirely unchanging form, displaying nondispersive behaviour. Similarly, when a medium-steepness Gaussian group was propagated at the critical angle, not only was the group nondispersive, the focusing present in its unidirectional propagation (due to nonlinear focusing) became negligible. Nonlinear effects were seen as the development of a double peaked wave envelope as larger waves travelled to the front of the group. These results show that wave groups are capable of travelling for extended periods of time with extreme waves at their centre. Two nondispersive crossed groups with crossing angles of ±35:26° were superimposed to create the first observations of a hydrodynamic X-wave. Such waves have previously been observed in optics, Bose-Einstein condensates, and plasmas. The X-wave has a large central amplitude where the two groups cross.