Particle trajectories in nonlinear capillary waves

1984 ◽  
Vol 143 ◽  
pp. 243-252 ◽  
Author(s):  
S. J. Hogan
Keyword(s):  
Drift Velocity ◽  
Phase Speed ◽  
Analytic Form ◽  
Capillary Waves ◽  
Particle Trajectories

The particle trajectories of nonlinear capillary waves are derived. The properties of the surface and subsurface particles are presented in exact analytic form, up to and including the highest wave. It is found that the orbits of the steeper waves are neither circular nor closed. For the highest wave, a particle moves through a distance [X] equal to 7.99556 λ in one orbit, where λ is the wavelength. It moves with an average horizontal drift velocity U equal to 0.88883c, where c is the phase speed of the wave. In addition, the subsurface particles (at depths nearly three-quarters that of the wavelength) move at speeds up to one-tenth that of surface particles.

scholarly journals The generation of gravity–capillary solitary waves by a pressure source moving at a trans-critical speed

2016 ◽  
Vol 810 ◽  
pp. 448-474 ◽  
Author(s):  
Naeem Masnadi ◽  
James H. Duncan
Keyword(s):  
Free Surface ◽  
Numerical Simulations ◽  
Model Equation ◽  
Surface Deformation ◽  
Phase Speed ◽  
Capillary Waves ◽  
Deformation Pattern ◽  
Pressure Source ◽  
Exponential Decay Law ◽  
Response State

The unsteady response of a water free surface to a localized pressure source moving at constant speed $U$ in the range $0.95c_{min}\lesssim U\leqslant 1.02c_{min}$, where $c_{min}$ is the minimum phase speed of linear gravity–capillary waves in deep water, is investigated through experiments and numerical simulations. This unsteady response state, which consists of a V-shaped pattern behind the source, and features periodic shedding of pairs of depressions from the tips of the V, was first observed qualitatively by Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) and called state III. In the present investigation, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free-surface deformation pattern downstream of the source as it moves along a towing tank, while numerical simulations of the model equation described by Cho et al. (J. Fluid Mech., vol. 672, 2011, pp. 288–306) are used to extend the experimental results over longer times than are possible in the experiments. From the experiments, it is found that the speed–amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity–capillary lumps of inviscid potential theory. The decay rate of the depressions is measured from their height–time characteristics, which are well fitted by an exponential decay law with an order one decay constant. It is found that the shedding period of the depression pairs decreases with increasing source strength and speed. As the source speed approaches $c_{min}$, this period tends to approximately 1 s for all source magnitudes. At the low-speed boundary of state III, a new response with unsteady asymmetric shedding of depressions is found. This response is also predicted by the model equation.

scholarly journals Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments

2011 ◽  
Vol 672 ◽  
pp. 268-287 ◽  
Author(s):  
JAMES D. DIORIO ◽  
YEUNWOO CHO ◽  
JAMES H. DUNCAN ◽  
T. R. AKYLAS
Keyword(s):  
Surface Pressure ◽  
Deep Water ◽  
Measurement Techniques ◽  
Phase Speed ◽  
Wave Pattern ◽  
Capillary Waves ◽  
Pressure Amplitude ◽  
Pressure Source ◽  
Nonlinear Features ◽  
Pressure Magnitude

The wave pattern generated by a pressure source moving over the free surface of deep water at speeds, U, below the minimum phase speed for linear gravity–capillary waves, cmin, was investigated experimentally using a combination of photographic measurement techniques. In similar experiments, using a single pressure amplitude, Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) pointed out that the resulting surface response pattern exhibits remarkable nonlinear features as U approaches cmin, and three distinct response states were identified. Here, we present a set of measurements for four surface-pressure amplitudes and provide a detailed quantitative examination of the various behaviours. At low speeds, the pattern resembles the stationary state (U = 0), essentially a circular dimple located directly under the pressure source (called a state I response). At a critical speed, but still below cmin, there is an abrupt transition to a wave-like state (state II) that features a marked increase in the response amplitude and the formation of a localized solitary depression downstream of the pressure source. This solitary depression is steady, elongated in the cross-stream relative to the streamwise direction, and resembles freely propagating gravity–capillary ‘lump’ solutions of potential flow theory on deep water. Detailed measurements of the shape of this depression are presented and compared with computed lump profiles from the literature. The amplitude of the solitary depression decreases with increasing U (another known feature of lumps) and is independent of the surface pressure magnitude. The speed at which the transition from states I to II occurs decreases with increasing surface pressure. For speeds very close to the transition point, time-dependent oscillations are observed and their dependence on speed and pressure magnitude are reported. As the speed approaches cmin, a second transition is observed. Here, the steady solitary depression gives way to an unsteady state (state III), characterized by periodic shedding of lump-like disturbances from the tails of a V-shaped pattern.

On the changes in phase speed of one train of water waves in the presence of another

1988 ◽  
Vol 192 ◽  
pp. 97-114 ◽  
Author(s):  
S. J. Hogan ◽  
Idith Gruman ◽  
M. Stiassnie
Keyword(s):  
Integral Equation ◽  
Perturbation Method ◽  
Gravity Waves ◽  
Water Waves ◽  
Phase Speed ◽  
Capillary Waves ◽  
The Other ◽  
Wide Range ◽  
Correct Form ◽  
General Method

We present calculations of the change in phase speed of one train of water waves in the presence of another. We use a general method, based on Zakharov's (1968) integral equation. It is shown that the change in phase speed of each wavetrain is directly proportional to the square of the amplitude of the other. This generalizes the work of Longuet-Higgins & Phillips (1962) who considered gravity waves only.In the important case of gravity-capillary waves, we present the correct form of the Zakharov kernel. This is used to find the expressions for the changes in phase speed. These results are then checked using a perturbation method based on that of Longuet-Higgins & Phillips (1962). Agreement to 6 significant digits has been obtained between the calculations based on these two distinct methods. Full numerical results in the form of polar diagrams over a wide range of wavelengths, away from conditions of triad resonance, are provided.

Particle trajectories in linear periodic capillary and capillary–gravity water waves

2007 ◽  
Vol 365 (1858) ◽  
pp. 2241-2251 ◽  
Author(s):  
David Henry
Keyword(s):  
Surface Tension ◽  
Linear Theory ◽  
Significant Role ◽  
Water Waves ◽  
Small Amplitude ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Particle Trajectories ◽  
Particle Paths

Surface tension plays a significant role as a restoration force in the setting of small-amplitude waves, leading to pure capillary and gravity-capillary waves. We show that within the framework of linear theory, the particle paths in a periodic gravity–capillary or pure capillary wave propagating at the surface of water over a flat bed are not closed.

scholarly journals Particle Motion in Longitudinal Waves. I Subluminal Waves

1992 ◽  
Vol 45 (1) ◽  
pp. 1 ◽  
Author(s):  
ET Rowe
Keyword(s):  
Electric Field ◽  
Longitudinal Wave ◽  
Drift Velocity ◽  
Particle Motion ◽  
Equations Of Motion ◽  
Phase Speed ◽  
Relativistic Energy ◽  
Longitudinal Waves ◽  
Arbitrary Phase ◽  
Particle Orbit

The classical equations of motion for a particle moving in a parallel longitudinal wave of arbitrary phase speed are discussed and the case of subluminal waves is considered in detail. Motion of both trapped and untrapped particles is explored with particular reference to the ability of a wave to accelerate particles to relativistic energy. The particle orbit is found in both closed and expanded forms, taking the electric field into account exactly. Expressions are also found for the 'drift velocity' of a particle, which is an important quantity because it is a constant of the particle motion that describes the motion of the centre of oscillation.

Fractal Particle Trajectories in Capillary Waves: Imprint of Wavelength

1997 ◽  
Vol 79 (10) ◽  
pp. 1845-1848 ◽  
Author(s):  
Adam E. Hansen ◽  
Elsebeth Schröder ◽  
Preben Alstrøm ◽  
Jacob Sparre Andersen ◽  
Mogens T. Levinsen
Keyword(s):  
Capillary Waves ◽  
Particle Trajectories ◽  
Fractal Particle
Sign in / Sign up

Export Citation Format

Share Document