Pure Capillary Waves Research Articles

Current modified higher-order Schrödinger equation of broader bandwidth capillary-gravity waves

A higher-order nonlinear Schrödinger equation of capillary-gravity waves for broader bandwidth on infinite depth of water including the effect of depth uniform current is established. The derivation is made from Zakharov's integral equation by extending the narrow bandwidth restriction to make it more suited for application to problems involving actual sea waves. On the basis of this equation, a stability analysis is made for uniform Stokes waves. After obtaining an instability condition, instability regions in the perturbed wave number space are displayed that are in good agreement with the exact numerical findings. It is found that the modifications in the stability characteristics at the fourth-order term are due to the interaction between the frequency-dispersion term and the mean flow. It is seen that the growth rate of sideband instability decreases due to the effects of both surface tension and depth uniform following currents. Significant deviations of the instability regions are observed between narrow-banded and broader-banded results. In addition, we have depicted the instability growth rate for the case of pure capillary waves.

A modification to the Schrödinger equation for broader bandwidth gravity-capillary waves on deep water with depth-uniform current

We derive a nonlinear Schrödinger equation for the propagation of the three-dimensional broader bandwidth gravity-capillary waves including the effect of depth-uniform current. In this derivation, the restriction of narrow bandwidth constraint is extended, so that this equation will be more appropriate for application to a realistic sea wave spectrum. From this equation, an instability condition is obtained and then instability regions in the perturbed wavenumber space for a uniform wave train are drawn, which are in good agreement with the exact numerical results. As it turns out, the corrections to the stability properties that occur at the fourth-order term arise from an interaction between the mean flow and the frequency-dispersion term. Since the frequency-dispersion term, in the absence of depth-uniform current, for pure capillary waves is of opposite sign for pure gravity waves, so too are the corrections to the instability properties. doi: 10.1017/S1446181123000020

Exact solutions for steadily travelling water waves with submerged point vortices

This paper presents a novel theoretical framework, based on the concept of the Schwarz function of a wave, for understanding water waves with vorticity in the absence of gravity and capillarity. The framework leads naturally to a taxonomy of three subcases, herein referred to as cases 1, 2 and 3, into which fall three existing studies of water waves incorporating uniform vorticity and submerged point vortices. This provides a theoretical unification of several seemingly unrelated results in the literature. It also provides a route to finding new exact solutions with this paper focussing on new solutions falling within the case 2 category. Among several presented here are a submerged point vortex pair cotravelling with a solitary deep-water wave, von Kármán point vortex streets cotravelling with a periodic deep-water wave and a point vortex row cotravelling with a wave in water of finite depth. Some other more exotic waveforms are also constructed. All these new solutions generalize those of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014) who found steady waves in deep water cotravelling with a submerged point vortex row for which the free surface shapes turn out to coincide with those of pure capillary waves on deep water found by Crapper (J. Fluid Mech., vol. 2, 1957). The new exact solutions are likely to provide a useful basis for asymptotic or numerical studies when additional effects such as gravity and capillarity are incorporated.

A MODIFICATION TO THE SCHRÖDINGER EQUATION FOR BROADER BANDWIDTH GRAVITY-CAPILLARY WAVES ON DEEP WATER WITH DEPTH-UNIFORM CURRENT

AbstractWe derive a nonlinear Schrödinger equation for the propagation of the three-dimensional broader bandwidth gravity-capillary waves including the effect of depth-uniform current. In this derivation, the restriction of narrow bandwidth constraint is extended, so that this equation will be more appropriate for application to a realistic sea wave spectrum. From this equation, an instability condition is obtained and then instability regions in the perturbed wavenumber space for a uniform wave train are drawn, which are in good agreement with the exact numerical results. As it turns out, the corrections to the stability properties that occur at the fourth-order term arise from an interaction between the mean flow and the frequency-dispersion term. Since the frequency-dispersion term, in the absence of depth-uniform current, for pure capillary waves is of opposite sign for pure gravity waves, so too are the corrections to the instability properties.

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

AbstractWe investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

No solitary waves in 2D gravity and capillary waves in deep water

A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves.

Stationary water waves on rotational flows of two vortical layers

Stationary waves of constant shape and constant propagation speed on rotational flows of two layers are computed numerically. Two layers are assumed to be of distinct constant vorticity distributions. Three different kinds of waves of finite depth are considered: pure capillary, capillary-gravity, and gravity waves. The problem is formulated as a bifurcation problem, which involves many parameters and produces a complicated structure of solutions. We adopted a numerical method by which waves with stagnation points can be computed, and obtained variety of new solutions. It is also reported that the locations of the stagnation points vary curiously with the prescribed parameters and that they offer an interesting problem.

Numerical simulation of the wave turbulence on the surface of a ferrofluid in a horizontal magnetic field

The turbulence of capillary waves on the surface of a ferrofluid with a high permeability in a horizontal magnetic field is considered in the framework of a one-dimensional weakly nonlinear model. In the limit of a strong magnetic field, the surface waves under study can propagate without distortions along or against the direction of external field, i.e., similar to Alfvén waves in a perfectly conducting fluid. The interaction of counter-propagating nonlinear waves leads to the development of wave turbulence on the surface of the liquid. The computational data show that the spectrum of turbulence is divided into two parts: a low-frequency dispersionless region, where the magnetic forces dominate and a high-frequency dispersive one, in which the influence of capillary forces becomes significant. In the first region, the spectrum of the surface elevation has the same exponent in k and ω domains and its value is close to -3.5, what is in a good agreement with the estimation obtained from the dimensional analysis of the weak turbulence spectra. At the high frequencies, the computed spatial spectrum of the surface waves is close to k-5/2 which corresponds to ω-5/3 in terms of the frequency. This spectrum does not coincide with the Zakharov-Filonenko spectrum obtained for pure capillary waves. A possible explanation of this fact is in the influence of coherent structures (like shock waves) usually arising in media with weak dispersion.

Gravity-Capillary Wave Spectral Modulation by Gravity Waves

In order to more fully understand the specific hydrodynamic relationship between young wind-generated gravity-capillary waves and longer gravity waves, a laboratory experiment was devised to observe changes in short wave spectral behavior over the phase of a long wave. This paper endeavors to expand on the body of laboratory wave modulation data and extend the investigation in support of the radar remote sensing of ocean surface waves. Measurements were made in the University of Miami’s surge–structure–atmosphere interaction facility in the air–sea interaction saltwater tank wind-wave tank, with 10 m referenced wind speeds ranging between 5 and 23 m/s and paddle-generated wave steepnesses “ak” varying between $0.05 . A polarimetric camera was used to capture high sampling frequency maps of wave slope, yielding spatiotemporal information about short wind-wave behavior [provided as temporal variations in the wavenumber spectrum, where $k~\approx \mathscr {O}$ (100–1000) rad/m]. The simultaneous and colocated long wave phase was measured via a side-looking camera. Hydrodynamic modulation transfer function (MTF) phases are found to be in general agreement with established values (between 2 and 10 radians) at the given wind speeds. The positive phase of the modulation places it immediately downwind of the long wave crest, with MTF magnitudes strongest for high wavenumbers at the lowest wind speeds. The results are also presented to show the modulation of gravity-capillary and pure capillary waves as variations in mean square slope over the long wave phase, with peak roughness enhancement found to move upwind of the long wave crest with increasing wind forcing.

Numerical simulation of surface waves instability on a homogeneous grid

We performed full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence.Instability of both propagating and standing waves was studied. We separately studied pure capillary wave, that was unstable due to three-wave interactions and pure gravity waves, that were unstable due to four-wave interactions. The theoretical description of instabilities in all cases is included in the article. The numerical algorithm used in these and many other previous simulations performed by the authors is described in detail.

Global bifurcation theory for periodic traveling interfacial gravity–capillary waves

We consider the global bifurcation problem for spatially periodic traveling waves for two-dimensional gravity–capillary vortex sheets. The two fluids have arbitrary constant, non-negative densities (not both zero), the gravity parameter can be positive, negative, or zero, and the surface tension parameter is positive. Thus, included in the parameter set are the cases of pure capillary water waves and gravity–capillary water waves. Our choice of coordinates allows for the possibility that the fluid interface is not a graph over the horizontal. We use a technical reformulation which converts the traveling wave equations into a system of the form “identity plus compact.” Rabinowitz' global bifurcation theorem is applied and the final conclusion is the existence of either a closed loop of solutions, or an unbounded set of nontrivial traveling wave solutions which contains waves which may move arbitrarily fast, become arbitrarily long, form singularities in the vorticity or curvature, or whose interfaces self-intersect.

Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

We consider nonlinear gravity-capillary waves with the nonlinearity parameter ɛ ∼ 0.1–0.25. For this nonlinearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits—pure capillary and pure gravity waves on a fluid surface—coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov’s equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.

Gravity perturbed Crapper waves

Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity–capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity–capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of single-valued and multi-valued travelling waves in the gravity–amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.

ENERGY SPECTRUM OF THE ENSEMBLE OF WEAKLY NONLINEAR GRAVITY--CAPILLARY WAVES ON A FLUID SURFACE

In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 ÷0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in \emph{Kartashova, \emph{EPL} \textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.

Capillary-wave scattering from an infinitesimal barrier and dissipation at dynamic contact lines

AbstractThe interaction of pure capillary waves with boundaries that constrain the contact line are of interest for problems involving liquids contained by minimal solid contact for applications in low gravity and at small scales in normal gravity. Time-harmonic capillary waves on a liquid cylinder axially incident on and scattered by an infinitesimal concentric barrier are considered theoretically in the inviscid limit. The barrier is taken to be infinitesimally small in the sense that its immersed depth is of the order of the amplitude of contact-line motion. Edge conditions on the barrier that are investigated include a pinned contact line and a moving contact line by an effective-slip model, assuming that contact-line velocity is proportional to the deviation of the contact angle from equilibrium multiplied by a slip coefficient. The incident waves are taken to be those with wavelengths short enough to be stable on the liquid cylinder. Scattering and dissipation by the contact line are determined as a function of wavenumber and slip coefficient. Zero transmission is approached in the long-wave limit. The short-wave limit agrees with established results for the scattering of planar gravity–capillary waves on a deep liquid by a surface-piercing vertical barrier in the limit of zero barrier depth and zero gravity. We find that contact-line dissipation at the barrier is a maximum for incident waves whose phase speed is of the order of the slip coefficient, which is interpreted as an effect of impedance matching. Transmission past an infinitesimal barrier is found to be low over all parameter space, illustrating the importance of contact-line constraints.

Laboratory experiments on wave turbulence

This review paper is devoted to a presentation of recent progress in wave turbulence. I first present the context and state of the art of this field of research both experimentally and theoretically. I then focus on the case of wave turbulence on the surface of a fluid, and I discuss the main results obtained by our group: caracterization of the gravity and capillary wave turbulence regimes, the first observation of intermittency in wave turbulence, the occurrence of strong fluctuations of injected power in the fluid, the observation of a pure capillary wave turbulence in low gravity environment and the observation of magnetic wave turbulence on the surface of a ferrofluid. Finally, open questions in wave turbulence are discussed.

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

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.

Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves

We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.

Efficient Atomization Using MHz MEMS-Based Ultrasonic Nozzles

AbstractThis paper reports on the preliminary results of atomization using micro-electro-mechanical system (MEMS)-based 3-Fourier horn 0.5 MHz silicon ultrasonic nozzles. The droplets produced are monodisperse with 7.1 μm drop diameter in good agreement with that predicted by pure capillary wave atomization mechanism. This drop diameter is much smaller than that obtainable using conventional ultrasonic nebulizers operating at three times higher frequency.

Uniqueness Issues on Permanent Progressive Water-Waves

We consider two-dimensional water-waves of permanent shape with a constant propagation speed. Two theorems concerning the uniqueness of certain solutions are reported. Uniqueness of Crapper’s pure capillary waves is proved under a positivity assumption. The proof is based on the theory of positive operators. Also proved is the uniqueness of the positive gravity waves of infinite depth with moderately large amplitude. This is accomplished by a combination of new inequalities and a numerical verification algorithm. Possibilities and impossibilities of other uniqueness theorems are discussed.