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.

Numerical calculations of steady gravity-capillary waves using an integro-differential formulation

1979 ◽  
Vol 94 (4) ◽  
pp. 777-793 ◽  
Author(s):  
James W. Rottman ◽  
D. B. Olfe
Keyword(s):  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Solutions ◽  
Capillary Wave ◽  
Capillary Waves ◽  
The Other ◽  
Wave Number ◽  
Infinite Depth ◽  
Free Surface Waves ◽  
Differential Formulation

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

Some effects of surface tension on steep water waves. Part 3

1981 ◽  
Vol 110 ◽  
pp. 381-410 ◽  
Author(s):  
S. J. Hogan
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Original Work ◽  
Fourier Coefficients ◽  
Solution Procedure ◽  
Capillary Waves ◽  
Wave Profile ◽  
Final Part ◽  
Wave Properties ◽  
Good Agreement

This is the final part in the work on steady gravity–capillary waves by the author. On extending the work of Pierson & Fife (1961), the phenomenon of Wilton's ripples is resolved. These singularities of the traditional solution procedure are merely consequences of the non-uniformity in the ordering of the Fourier coefficients of the wave profile. Results presented here include wave properties and profiles. Near-resonant waves are also considered. Good agreement is found between this work and previous papers in the series as well as with other authors. An appendix contains all the results in numerical form. Minor algebraic errors in Wilton's original work are corrected.

scholarly journals Capillary-gravity water waves with vorticity : steady wind-driven waves and waves with a submerged dipole

2019 ◽  
Author(s):  
◽  
Hung Duc Le
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Elliptic Equations ◽  
Small Amplitude ◽  
Finite Depth ◽  
Classical Solutions ◽  
Implicit Function ◽  
Mathematical Problems ◽  
Orbital Instability ◽  
Incompressible Euler

In this thesis, we study two mathematical problems on water waves in the setting of the incompressible Euler equations with vorticity, gravity, and surface tension. We investigate the existence of small-amplitude steady wind-driven water waves in finite depth, using the Crandall Rabinowitz theorem. As part of the result, elliptic equations with transmission and Wentzell boundary conditons are also examined, and Schauder type estimates on classical solutions are established. The second chapter considers the existence and instability of solitary water waves with a nite dipole in in nite depth. We construct waves of this type using an Implicit Function Theorem argument. Then we establish orbital instability. This is proved using a modi cation of the classical Grillaks Shatash Strauss method.

scholarly journals A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

2018 ◽  
Vol 854 ◽  
pp. 146-163 ◽  
Author(s):  
H. C. Hsu ◽  
C. Kharif ◽  
M. Abid ◽  
Y. Y. Chen
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Modulational Instability ◽  
Capillary Waves ◽  
Rate Of Growth ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Wave Trains

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

scholarly journals Registration of high-frequency waves on the surface by the interference methods

2019 ◽  
Vol 213 ◽  
pp. 02075
Author(s):  
Anastasia Shmyrova ◽  
Andrey Shmyrov ◽  
Irina Mizeva ◽  
Alexey Mizev
Keyword(s):  
Surface Tension ◽  
Capillary Wave ◽  
Surface Profile ◽  
Wave Generation ◽  
Cylindrical Wave ◽  
Capillary Waves ◽  
Practical Implementation ◽  
Excitation Techniques ◽  
High Frequency Waves ◽  
Profile Reconstruction

Capillary waves are frequently used to measure the surface tension of liquids. However, this approach has not found wide application in the manufacture of modern commercial tensiometers because of the limitations imposed by capillary wave excitation techniques and the labor input associated with its practical implementation. In this paper we introduce a modified version of the capillary wave method which allows one to avoid the existing limitations and disadvantages. The distinguishing features of the proposed technique are as follows: acoustic wave generation and application of an interferometry technique for 3D surface profile reconstruction. A dynamic speaker with controlled vibration frequency and amplitude is used to produce acoustic vibrations. Application of a conventional Fizeau interferometer and the spatial phase shifting method makes it possible to perform surface form measurements with a high accuracy. For calculating wavelengths and the damping co-efficient, the surface profile is fitted with a decaying cylindrical wave equation. The accuracy of surface tension measurement by the modified capillary wave technique is 0.3 %. Owing to the non-contact way of wave generation and the small amounts of the examined fluid, the proposed method can be used in different studies.

Some effects of surface tension on steep water waves. Part 2

1980 ◽  
Vol 96 (3) ◽  
pp. 417-445 ◽  
Author(s):  
S. J. Hogan
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Expansion Method ◽  
Capillary Waves ◽  
Maximum Wave Height ◽  
Pure Gravity ◽  
Wave Profiles ◽  
The Waves ◽  
Wave Properties

This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

Sign in / Sign up

Export Citation Format

Share Document