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

2020 ◽  
Author(s):  
Sunao Murashige ◽  
Wooyoung Choi
Keyword(s):  
Linear Stability ◽  
Deep Water ◽  
Finite Amplitude ◽  
Capillary Waves

Linear stability of finite-amplitude capillary waves on water of infinite depth

2012 ◽  
Vol 696 ◽  
pp. 402-422 ◽  
Author(s):  
Roxana Tiron ◽  
Wooyoung Choi
Keyword(s):  
Linear Stability ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
One Dimensional ◽  
Highly Nonlinear ◽  
Non Local ◽  
Two Bubbles

AbstractWe study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

Effects of plane progressive irrotational waves on thermal boundary layers

1971 ◽  
Vol 50 (2) ◽  
pp. 321-334 ◽  
Author(s):  
James Witting
Keyword(s):  
Boundary Layers ◽  
Deep Water ◽  
Maximum Amplitude ◽  
Capillary Waves ◽  
Temperature Gradients ◽  
Two Dimensional ◽  
Inviscid Incompressible Fluid ◽  
Mean Temperature ◽  
The Waves ◽  
Thermal Boundary Layers

The average changes in the structure of thermal boundary layers at the surface of bodies of water produced by various types of surface waves are computed. the waves are two-dimensional plane progressive irrotational waves of unchanging shape. they include deep-water linear waves, deep-water capillary waves of arbitrary amplitude, stokes waves, and the deep-water gravity wave of maximum amplitude.The results indicate that capillary waves can decrease mean temperature gradients by factors of as much as 9·0, if the average heat flux at the air-water interface is independent of the presence of the waves. Irrotational gravity waves can decrease the mean temperature gradients by factors no more than 1·381.Of possible pedagogical interest is the simplicity of the heat conduction equation for two-dimensional steady irrotational flows in an inviscid incompressible fluid if the velocity potential and the stream function are taken to be the independent variables.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

A new family of capillary waves

1980 ◽  
Vol 98 (1) ◽  
pp. 161-169 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Joseph B. Keller
Keyword(s):  
Integral Equation ◽  
Nonlinear Integral Equation ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Wave Steepness ◽  
Algebraic Equations ◽  
Infinite Depth ◽  
New Family ◽  
Nonlinear Algebraic Equations ◽  
Finite Set

A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

The blockage of gravity and capillary waves by longer waves and currents

1990 ◽  
Vol 217 ◽  
pp. 115-141 ◽  
Author(s):  
Jinn-Hwa Shyu ◽  
O. M. Phillips
Keyword(s):  
Multiple Scale ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Reflected Waves ◽  
Dominant Wave ◽  
Constant Vorticity ◽  
Waves And Currents ◽  
Uniform Solution ◽  
Group Velocities

Surface waves superimposed upon a larger-scale flow are blocked at the points where the group velocities balance the convection by the larger-scale flow. Two types of blockage, capillary and gravity, are investigated by using a new multiple-scale technique, in which the short waves are treated linearly and the underlying larger-scale flows are assumed steady but can have a considerably curved surface and uniform vorticity. The technique first provides a uniformly valid second-order ordinary differential equation, from which a consistent uniform asymptotic solution can readily be obtained by using a treatment suggested by the result of Smith (1975) who described the phenomenon of gravity blockage in an unsteady current with finite depth.The corresponding WKBJ solution is also derived as a consistent asymptotic expansion of the uniform solution, which is valid at points away from the blockage point. This solution is obviously represented by a linear combination of the incident and reflected waves, and their amplitudes take explicit forms so that it can be shown that even with a significantly varied effective gravity g’ and constant vorticity, wave action will remain conserved for each wave. Furthermore, from the relative amplitudes of the incident and reflected waves, we clearly demonstrate that the action fluxes carried by the two waves towards and away from the blockage point are equal within the present approximation.The blockage of gravity–capillary waves can occur at the forward slopes of a finite-amplitude dominant wave as suggested by Phillips (1981). The results show that the blocked waves will be reflected as extremely short capillaries and then dissipated rapidly by viscosity. Therefore, for a fixed dominant wave, all wavelets shorter than a limiting wavelength will be suppressed by this process. The minimum wavelengths coexisting with the long waves of various wavelengths and slopes are estimated.

Finite-amplitude deep-water waves on currents

1979 ◽  
Vol 292 (1392) ◽  
pp. 371-390 ◽  
Keyword(s):  
Wave Propagation ◽  
Deep Water ◽  
Water Waves ◽  
Large Scale ◽  
Finite Amplitude ◽  
The Other ◽  
Wave Trains ◽  
Deep Water Waves

Accurate integral properties of plane periodic deep-water waves of amplitudes up to the steepest are tabulated by Longuet-Higgins (1975). These are used to define an averaged Lagrangian which, following Whitham, is used to describe the properties of slowly varying wave trains. Two examples of waves on large-scale currents are examined in detail. One flow is that of a shearing current, V ( x ) j , which causes waves to be refracted. The other flow, U ( x ) i , varies in the direction of wave propagation and causes waves to either steepen or become more gentle. Some surprising features are found.

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