Joe Hammack

TWO-DIMENSIONAL PERIODIC WAVES IN SHALLOW WATER

Experimental data are presented that demonstrate the existence of a family of gravitational water waves that propagate practically without change of form on the surface of shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional and fully periodic, i.e. they are periodic in two spatial directions and in time. The amplitudes of these waves need not be small; their form persists even up to breaking. The waves are easy to generate experimentally, and they are observed to propagate in a stable manner, even when perturbed significantly. The measured waves are described with reasonable accuracy by a family of exact solutions of the Kadomtsev-Petviashvili equation (KP solutions of genus 2) over the entire parameter range of the experiments, including waves well outside the putative range of validity of the KP equation. These genus-2 solutions of the KP equation may be viewed as two-dimensional generalizations of cnoidal waves.

Stabilizing the Benjamin–Feir instability

odinger equation is also well established as an approximate model based on the same assumptions as required for the derivation of the Benjamin–Feir theory: a narrow-banded spectrum of waves of moderate amplitude, propagating primarily in one direction in a dispersive medium with little or no dissipation. In this paper, we show that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation (of a certain type) stabilizes the instability. We arrive at this stability result first by proving it rigorously for a damped version of the nonlinear Schr¨ equation, and then by confirming our theoretical predictions with laboratory experiments on waves of moderate amplitude in deep water. The Benjamin–Feir instability is often cited as the first step in a nonlinear process that spreads energy from an initially narrow bandwidth to a broader bandwidth. In this process, sidebands grow exponentially until nonlinear interactions eventually bound their growth. In the presence of damping, this process might still occur, but our work identifies another possibility: damping can stop the growth of perturbations before nonlinear interactions become important. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.

Two-dimensional periodic waves in shallow water. Part 2. Asymmetric waves

We demonstrate experimentally the existence of a family of gravity-induced finiteamplitude water waves that propagate practically without change of form in shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional, and periodic. The basic template of a wave is hexagonal, but it need not be symmetric about the direction of propagation, as required in our previous studies (e.g. Hammack et al. 1989). Like the symmetric waves in earlier studies, the asymmetric waves studied here are easy to generate, they seem to be stable to perturbations, and their amplitudes need not be small. The Kadomtsev–Petviashvili (KP) equation is known to describe approximately the evolution of waves in shallow water, and an eight-parameter family of exact solutions of this equation ought to describe almost all spatially periodic waves of permanent form. We present an algorithm to obtain the eight parameters from wave-gauge measurements. The resulting KP solutions are observed to describe the measured waves with reasonable accuracy, even outside the putative range of validity of the KP model.

Long-time dynamics of the modulational instability of deep water waves

In this paper, we experimentally and theoretically examine the long-time evolution of modulated periodic 1D Stokes waves which are described, to leading-order, by the nonlinear Schrodinger (NLS) equation. The laboratory and numerical experiments indicate that under suitable conditions modulated periodic wave trains evolve chaotically. A Floquet spectral decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave states to left- and right-going modulated traveling waves. Numerical experiments using a higher-order nonlinear Schrodinger equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary conditions the long-time evolution of modulated wave trains is chaotic. Further, the numerical experiments indicate that the macroscopic features of the evolution can be described by the HONLS equation. Ultimately, these laboratory experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.

Experiments on ripple instabilities. Part 3. Resonant quartets of the Benjamin–Feir type

Instabilities and long-time evolution of gravity-capillary wavetrains (ripples) with moderate steepnesses (e f ≤ 25 Hz are generated mechanically in a channel containing clean, deep water; no artificial perturbations are introduced. Frequency spectra are obtained from in situ measurements; two-dimensional wavenumber spectra are obtained from remote sensing of the water surface using a high-speed imaging system. The analytical models are in viscid, uncoupled NLS (nonlinear Schrodinger) equations: one that describes the temporal evolution of longitudinal modulations and one that describes the spatial evolution of transverse modulations. The experiments show that the evolution of wavetrains with sensible amplitudes and frequencies exceeding 9.8 Hz is dominated by modulational instabilities, i.e. resonant quartet interactions of the Benjamin–Feir type. These quartet interactions remain dominant even for wavetrains that are unstable to resonant triad interactions ( f > 19.6 Hz) – if selective amplification does not occur (see Parts 1 and 2). The experiments further show that oblique perturbations with the same frequency as the underlying wavetrain, i.e. rhombus-quartet instabilities, amplify more rapidly and dominate all other modulational instabilities. The inviscid, uncoupled NLS equations predict the existence of modulational instabilities for wavetrains with frequencies exceeding 9.8 Hz, typically underpredict the bandwidth of unstable transverse modulations, typically overpredict the bandwidth of unstable longitudinal modulations, and do not predict the dominance of the rhombus-quartet instability. When the effects of weak viscosity are incorporated into the NLS models, the predicted bandwidths of unstable modulations are reduced, which is consistent with our measurements for longitudinal modulations, but not with our measurements for transverse modulations. Both the experiments and NLS equations indicate that wavetrains in the frequency range 6.4–9.8 Hz are stable to modulational instabilities. However, in these experiments, wavetrains with sensible amplitudes excite one of the members of the Wilton ripples family. When second-harmonic resonance occurs, both the first-and second-harmonic wavetrains undergo rhombus-quartet instabilities. When third-harmonic resonance occurs, only the third-harmonic wavetrain undergoes rhombus-quartet instabilities.

Experiments on ripple instabilities. Part 2 Selective amplification of resonant triads

Resonant three-wave interactions among capillary-gravity water waves are studied experimentally using a test wavetrain and smaller background waves (noise) generated mechanically in a channel. The spectrum of the background waves is varied from broad-banded to one with discrete components

A note on the generation and narrowness of periodic rip currents

Periodic rip currents on a wide planar beach are generated in the laboratory by shoaling water waves that are periodic in time and in two spatial directions: one normal (x direction) and one parallel (y direction) to the shoreline. These short-crested waves propagate in water of uniform depth with nearly permanent form. They are described analytically by a family of solutions of the Kadomtsev-Petviashvili (KP) equation (KP solutions of genus 2). During shoaling, genus 2 waves retain their spatial pattern past breaking, and they quickly generate periodic rip currents along the beach with a spacing of one-half the y wavelength of the incident waves. KP theory also provides a plausible explanation and prediction for the narrow widths, relative to their longshore spacing, of rip currents generated in this manner. An estimate of their widths is one-half the x wavelength of the incident waves.