8.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Capillary wave | 1/2 | https://en.wikipedia.org/wiki/Capillary_wave | reference | science, encyclopedia | 2026-05-05T07:34:33.305688+00:00 | kb-cron |
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension. Capillary waves are common in nature, and are often referred to as ripples. The wavelength of capillary waves on water is typically less than a few centimeters, with a phase speed in excess of 0.2–0.3 meter/second. A longer wavelength on a fluid interface will result in gravity–capillary waves which are influenced by both the effects of surface tension and gravity, as well as by fluid inertia. Ordinary gravity waves have a still longer wavelength. Light breezes upon the surface of water which stir up such small ripples are also sometimes referred to as 'cat's paws'. On the open ocean, much larger ocean surface waves (seas and swells) may result from coalescence of smaller wind-caused ripple-waves.
== Dispersion relation == The dispersion relation describes the relationship between wavelength and frequency in waves. Distinction can be made between pure capillary waves – fully dominated by the effects of surface tension – and gravity–capillary waves which are also affected by gravity.
=== Capillary waves, proper === The dispersion relation for capillary waves is
ω
2
=
σ
ρ
+
ρ
′
|
k
|
3
,
{\displaystyle \omega ^{2}={\frac {\sigma }{\rho +\rho '}}\,|k|^{3},}
where
ω
{\displaystyle \omega }
is the angular frequency,
σ
{\displaystyle \sigma }
the surface tension,
ρ
{\displaystyle \rho }
the density of the heavier fluid,
ρ
′
{\displaystyle \rho '}
the density of the lighter fluid and
k
{\displaystyle k}
the wavenumber. The wavelength is
λ
=
2
π
k
.
{\displaystyle \lambda ={\frac {2\pi }{k}}.}
For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
ω
2
=
σ
ρ
|
k
|
3
.
{\displaystyle \omega ^{2}={\frac {\sigma }{\rho }}\,|k|^{3}.}
=== Gravity–capillary waves ===
When capillary waves are also affected substantially by gravity, they are called gravity–capillary waves. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:
ω
2
=
|
k
|
(
ρ
−
ρ
′
ρ
+
ρ
′
g
+
σ
ρ
+
ρ
′
k
2
)
,
{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\sigma }{\rho +\rho '}}k^{2}\right),}
where
g
{\displaystyle g}
is the acceleration due to gravity,
ρ
{\displaystyle \rho }
and
ρ
′
{\displaystyle \rho '}
are the densities of the two fluids
(
ρ
>
ρ
′
)
{\displaystyle (\rho >\rho ')}
. The factor
(
ρ
−
ρ
′
)
/
(
ρ
+
ρ
′
)
{\displaystyle (\rho -\rho ')/(\rho +\rho ')}
in the first term is the Atwood number.
==== Gravity wave regime ====
For large wavelengths (small
k
=
2
π
/
λ
{\displaystyle k=2\pi /\lambda }
), 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.
==== Capillary wave regime ==== Shorter (large
k
{\displaystyle k}
) waves (e.g. 2 mm for the water–air interface), 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 is a point at which 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
m
{\displaystyle \lambda _{m}}
are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed
c
m
{\displaystyle c_{m}}
are:
λ
m
=
2
π
σ
(
ρ
−
ρ
′
)
g
and
c
m
=
2
(
ρ
−
ρ
′
)
g
σ
ρ
+
ρ
′
.
{\displaystyle \lambda _{m}=2\pi {\sqrt {\frac {\sigma }{(\rho -\rho ')g}}}\quad {\text{and}}\quad c_{m}={\sqrt {\frac {2{\sqrt {(\rho -\rho ')g\sigma }}}{\rho +\rho '}}}.}
For the air–water interface,
λ
m
{\displaystyle \lambda _{m}}
is found to be 1.7 cm (0.67 in), and
c
m
{\displaystyle c_{m}}
is 0.23 m/s (0.75 ft/s). If one drops a small stone or droplet into liquid, the waves then propagate outside an expanding circle of fluid at rest; this circle is a caustic which corresponds to the minimal group velocity.