Diane M. Henderson

Solitary water wave interactions

This article concerns the pairwise nonlinear interaction of solitary waves in the free surface of a body of water lying over a horizontal bottom. Unlike solitary waves in many completely integrable model systems, solitary waves for the full Euler equations do not collide elastically; after interactions, there is a nonzero residual wave that trails the post-collision solitary waves. In this report on new numerical and experimental studies of such solitary wave interactions, we verify that this is the case, both in head-on collisions (the counterpropagating case) and overtaking collisions (the copropagating case), quantifying the degree to which interactions are inelastic. In the situation in which two identical solitary waves undergo a head-on collision, we compare the asymptotic predictions of Su and Mirie [J. Fluid Mech. 98, 509 (1980)] and Byatt-Smith [J. Fluid Mech. 49, 625 (1971)], the wavetank experiments of Maxworthy [J. Fluid Mech. 76, 177 (1976)], and the numerical results of Cooker, Weidman, and ...

Surface-wave damping in a circular cylinder with a fixed contact line

The natural frequencies and damping ratios for surface waves in a circular cylinder are calculated on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. These theoretical predictions are compared with the measurements for the first six modes in a brimfull, sharp-edged cylinder of radius 2.77 cm and depth 3.80 cm. The differences between the predicted and observed frequencies were less than 0.5 % for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%; this difference was significantly larger for the higher modes with a clean surface and for all of the modes with a contaminated surface.

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.

On the pinch-off of a pendant drop of viscous fluid

The pinch-off of a drop of viscous fluid is observed using high-speed digital imaging. The behavior seen by previous authors is observed here; namely, the filament that attaches the drop to the orifice evolves into a primary thread attached to a much thinner, secondary thread by a slight bulge. Here, we observe that the lengths of the primary and secondary threads are reproducible among experiments to within 3% and 10%. The secondary thread becomes unstable as evidenced by wave-like disturbances. The actual pinch-off does not occur at the point of attachment between the secondary thread and the drop. Instead, it occurs between the disturbances on the secondary thread. After the initial pinch-off, additional breaks occur between the disturbances, resulting in several secondary satellite drops with a broad distribution of sizes. The pinch-off of the thread at the orifice is similar to that at the drop with one main difference: there is no distinct secondary thread. Instead, the primary thread necks down mon...

Single-mode Faraday waves in small cylinders

Experiments on single-mode Faraday waves in small rectangular and circular cylinders in which both capillary and viscous effects were significant aie reported. Measurements of threshold forcing (for neutral stability) and steady-state wave amplitudes are compared with theoretical predictions. Theoretical predictions of the resonant frequency of a single mode and of the threshold amplitude for its excitation on the hypothesis of linear boundary-layer damping agree well with the measured data. (The theory must use the measured damping rate to predict these quantities for waves in the rectangular cylinder.) Theoretical predictions of wave amplitudes are in reasonable agreement with those observed in the circular cylinder; however, the theory provides only qualitative predictions of amplitudes for waves in the rectangular cylinder. In experiments in which two modes are theoretically admissible, the one with the smaller damping rate is observed; however, a single-mode calculation proves inadequate for the prediction of the stability boundary.

Progressive waves with persistent two-dimensional surface patterns in deep water

Experiments are conducted to generate progressive wave fields in deep water with two-dimensional surface patterns for which two parameters are systematically varied: (i) the aspect ratio of the cells comprising the surface patterns and (ii) a measure of nonlinearity of the input wave field. The goal of these experiments is to determine whether these patterns persist, what their main features are, whether standard models of waves describe these features, and whether there are parameter regimes in which the patterns are stable. We find that in some parameter regimes, surface patterns in deep water do persist with little change of form during the time of the experiment. In other parameter regimes, particularly for large-amplitude experiments, the patterns evolve more significantly. We characterize the patterns and their evolutions with a list of observed features. To describe the patterns and features, we consider two models: (

Recovering the Water-Wave Profile from Pressure Measurements

a

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

) the standard (

Effects of surfactants on Faraday-wave dynamics

2+1

The motion of a falling liquid filament

) nonlinear Schrodinger equation and (