Capillary wave - Citizendia

This article is about ripples on fluid interfaces. For for ripples in electricity, see ripple (physics). For other uses, see ripple.

Ripples on Lifjord in Øksnes, Norway. Øksnes is a municipality in the county of Nordland, Norway Øksnes was established as a municipality January 1 1838 (see Formannskapsdistrikt) Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional

Capillary waves produced by droplet impacts. A drop or droplet is a small volume of Liquid, bounded completely or almost completely by Free surfaces Surface tension

A capillary wave is a wave travelling along the interface between two fluids, whose dynamics are dominated by the effects of surface tension. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to Capillary waves are common in nature and home, and are often referred to as ripple. Nature, in the broadest sense is equivalent to the natural world, physical universe, material world or material universe. The wavelength of capillary waves is typically less than a few centimeters. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

A gravity–capillary wave on a fluid interface is influenced by both the effects of surface tension and gravity, as well as by the fluid inertia. Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving

1 Capillary waves, proper
2 Gravity–capillary waves
3 Gallery
4 References
5 External links
6 See also

Capillary waves, proper

The dispersion relation for capillary waves is

$\omega^2=\frac{\sigma}{\rho+\rho'}\, |k|^3,$

where ω is the angular frequency, σ the surface tension, ρ the density of the heavier fluid, ρ' the density of the lighter fluid and k the wavenumber. Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters The wavelength is $\lambda=\frac{2 \pi}{k}.$

Gravity–capillary waves

Dispersion of gravity–capillary waves on the surface of deep water (zero mass density of upper layer, ρ′ = 0). In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Phase and group velocity divided by $\scriptstyle \sqrt[4]{g\sigma/\rho}$ as a function of inverse relative wavelength $\scriptstyle \frac{1}{\lambda}\sqrt{\sigma/(\rho g)}$ .
Blue lines (A): phase velocity, Red lines (B): group velocity.
Drawn lines: dispersion relation for gravity–capillary waves.
Dashed lines: dispersion relation for deep-water gravity waves.
Dash-dot lines: dispersion relation valid for deep–water capillary waves.

In general, waves are also affected by gravity and are then called gravity–capillary waves. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth^[1]^[2]:

$\omega^2=|k|\left( \frac{\rho-\rho'}{\rho+\rho'}g+\frac{\sigma}{\rho+\rho'}k^2\right),$

where g is the acceleration due to gravity, ρ and ρ‘ are the mass density of the two fluids (ρ > ρ‘). Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

Gravity wave regime

For large wavelengths (small k = 2π/λ), only the first term is relevant and one has gravity waves. In this limit, the waves have a group velocity half the phase velocity: following a single wave's crest in a group one can see the wave appearing at the back of the group, growing and finally disappearing at the front of the group. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space

Capillary wave regime

Shorter (large k) waves (e. g. 2 mm), which are proper capillary waves, do the opposite: an individual wave appears at the front of the group, grows when moving towards the group center and finally disappears at the back of the group. Phase velocity is two thirds of group velocity in this limit.

Phase velocity minimum

Between these two limits, an interesting and common situation occurs when the dispersion caused by gravity cancels out the dispersion due to the capillary effect. At a certain wavelength, the group velocity equals the phase velocity, and there is no dispersion. At precisely this same wavelength, the phase velocity of gravity-capillary waves as a function of wavelength (or wave number) has a minimum. Waves with wavelengths much smaller than this critical wavelength λ_c are dominated by surface tension, and much above by gravity. The value of this wavelength is^[1]:

$\lambda_c = 2 \pi \sqrt{ \frac{\sigma}{(\rho-\rho') g}}.$

For the air–water interface, λ_c is found to be 1. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five Water is a common Chemical substance that is essential for the survival of all known forms of Life. 7 cm^[1]. A centimetre ( American spelling: centimeter, symbol cm) is a unit of Length in the Metric system, equal to one hundredth

Derivation

As Richard Feynman put it, ". Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum . . [water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses. . . are the worst possible example. . . they have all the complications that waves can have". ^[3] The derivation of the general dispersion relation is therefore quite involved (see e. g. Ref. ^[4] for a more detailed description. )

Therefore, first the assumptions involved are pointed out. There are three contributions to the energy, due to gravity, to surface tension, and to hydrodynamics. For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of g and σ. For gravity, an assumption is made of the density of the fluids being constant (i. e. , incompressibility), and likewise g (waves are not high for gravitation to change appreciably). For surface tension, the deviations from planarity (as measured by derivatives of the surface) are supposed to be small. Both approximations are excellent for common waves.

The last contribution involves the kinetic energies of the fluids, and is the most involved. The kinetic energy of an object is the extra Energy which it possesses due to its motion One must use a hydrodynamic framework to tackle this problem. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Incompressibility is again involved (which is satisfied if the speed of the waves is much less than the speed of sound in the media), together with the flow being irrotational — the flow is then potential; again, these are typically good approximations for common situations. In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. In Fluid dynamics, a potential flow is a Velocity field which is described as the Gradient of a scalar function the velocity potential The resulting equation for the potential (which is Laplace equation) can be solved with the proper boundary conditions. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties On one hand, the velocity must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more involved result is obtained, see Ocean surface waves. Ocean surface waves are Surface waves that occur on the Free surface of the Ocean. ) On the other, its vertical component must match the motion of the surface. This contribution ends up being responsible for the extra k outside the parenthesis, which causes all regimes to be dispersive, both at low values of k, and high ones (except around the one value at which the two dispersions cancel out. )

Gallery

Ripples on water

Ripples on water created by water striders

Ripples of tapwater over a plughole

References

^ ^a ^b ^c Lamb, H. (1994). The family Gerridae contains insects commonly known as water striders, water bugs, magic bugs, pond skaters, skaters, skimmers Sir Horace Lamb FRS ( 29 November 1849 – 4 December 1934) was a British Applied mathematician and author of several influential Hydrodynamics, 6^th edition, Cambridge University Press. ISBN 9780521458689. §267, page 458–460.
^ Dingemans, M. W. (1997). Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering 13. World Scientific, Singapore, 2 Parts, 967 pages. ISBN 981-02-0427-2. Section 2. 1. 1, p. 45.
Phillips, O. M. (1977). The dynamics of the upper ocean, 2^nd edition, Cambridge University Press. ISBN 0-521-29801-6. Section 3. 2, p. 37.
^ R.P. Feynman, R. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4.
^ Samuel Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994.
^ Lamb (1994), §174 and §230.
^ ^a ^b ^c ^d ^e Lamb (1994), §266.
^ ^a ^b Lamb (1994), §61.
^ Lamb (1994), §20
^ Lamb (1994), §230.
^ ^a ^b Whitham, G. B. (1974). Linear and nonlinear waves. Wiley-Interscience. ISBN 0 471 94090 9. See section 11. 7.
^ Lord Rayleigh (J. W. Strutt) (1877). John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered "On progressive waves". Proceedings of the London Mathematical Society 9: 21–26. Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.

External links

Capillary waves entry at sklogwiki

Contents