Progressive, Deep-Water Wavetrains with 1D and 2D Surface Patterns

Abstract: We report on experiments and analysis of deep-water waves with (i) 2D surface patterns using our 6-ft x 12-ft wavebasin and (ii) 1D surface patterns using our 43-ft wave channel. We generate 2D surface patterns by programming our 32-paddle wavemaker array to correspond to (i) a single carrier wave with a Jacobi elliptic sine function modulation in the transverse direction and (ii) two symmetric carrier waves interacting at an oblique angle. We conducted experiments using a range of two parameters: (i) a measure of two-dimensionality (aspect ratio) and (ii) a measure of nonlinearity (wave steepness). We catalogue a list of twelve features of the surface patterns. A theoretical explanation for some of these features is given by Fuhrman & Madsen and confirmed by our further experiments.

The persistence of the 2D wave patterns in the wavebasin is surprising in light of classic instability results (the Benjamin-Feir instability) for deep-water waves. To understand the persistence, we reconsidered the stability of a uniform wavetrain using the NLS equation modified to include linear damping. We proved that the presence of damping, no matter how small, stabilizes (with linear and nonlinear stability) the uniform wavetrain solution of the damped NLS equation. Here, we report on experiments using waves with 1D surface patterns that test the details of predicted evolution from the damped NLS equation. We show excellent agreement between experiments and predictions of nine independent parameters. We use the experiments to determine a measure of when the NLS-type equations are valid models for water waves by monitoring in the data the conserved quantities of these Hamiltonian systems.

Fuhrman & Madsen (2005) A numerical investigation of short-crested waves in deep water. submitted to J. Fluid Mech.

Hammack, Henderson, Segur (2005) Progressive waves with two-dimensional surface patterns in deep water. J. Fluid Mech. 532: 1-52.

Henderson, Patterson, Segur (2005) On the laboratory generation of deep-water progressive waves with two-dimensional surface patterns. (in preparation).

Segur, Henderson, Carter, Hammack, Li, Pheiff, Socha (2005) Stabilizing the Benjamin-Feir instability. J. Fluid Mech. 539:229-271.