Capillary wave

Capillary waves on the water

Capillary waves are transverse waves on a liquid surface , the properties of which, including the speed of propagation, depend mainly on the surface tension of the liquid. This is the case up to a wavelength of about one centimeter.

With increasing wavelength, capillary waves change into gravity waves , in which the influence of gravity predominates.

Physical description

According to the Young-Laplace equation, the capillary pressure acts at the highest point of a wave crest ( ) ${\displaystyle y=h}$

${\displaystyle p_{\mathrm {kap} }=\sigma {\Bigl (}{\frac {1}{r}}+{\frac {1}{r^{\prime }}}{\Bigr )}={\frac {\sigma }{r}}}$

With

• the surface tension in N / m ${\displaystyle \sigma }$
• the radii of curvature of the surface in the direction of propagation and along the back of the wave. The following applies here , where the function is the shape of the surface according to the wave equation ${\displaystyle r}$ ${\displaystyle r^{\prime }=\infty }$ ${\displaystyle r\approx 1/y^{\prime \prime }}$ ${\displaystyle y(x)}$

${\displaystyle y(x)=h\cdot \sin(2\pi \cdot x/\lambda )}$

indicating with

• the vertical coordinate ${\displaystyle y}$
• the horizontal coordinate ${\displaystyle x}$
• the amplitude ${\displaystyle h.}$

Thus the capillary pressure is through on the wave crest

${\displaystyle p_{\mathrm {kap} }=\sigma {\frac {4\pi ^{2}}{\lambda ^{2}}}h}$

given and for the trough with a correspondingly changed sign.

The speed of the liquid particles on the wave crest is lower than in the wave trough: For an observer who follows the wave, the particles have the speed in the former (in terms of magnitude) and the speed in the latter . Here is the speed of propagation of the wave and (half) the speed difference. For an observer at rest with the liquid, the particles do not move (to a good approximation). For him, the propagation of the wave through the liquid corresponds to a circular motion of the individual particles with radius and radial velocity . As usual , the angular velocity is linked to the wavelength : ${\displaystyle c_{\mathrm {kap} }-v}$ ${\displaystyle c_{\mathrm {kap} }+v}$ ${\displaystyle c_{\mathrm {kap} }}$ ${\displaystyle v}$ ${\displaystyle h}$ ${\displaystyle v}$ ${\displaystyle \omega }$ ${\displaystyle \lambda }$

${\displaystyle \omega ={\frac {v}{h}}=2\pi ~{\frac {c_{\mathrm {kap} }}{\lambda }}}$.

The difference in kinetic energy per volume (with liquid density ) ${\displaystyle \rho }$

${\displaystyle {\frac {\Delta E_{\mathrm {kin} }}{V}}={\frac {1}{2}}~\rho ~(c_{\mathrm {kap} }+v)^{2}-{\frac {1}{2}}~\rho ~(c_{\mathrm {kap} }-v)^{2}=2~\rho ~c_{\mathrm {kap} }v={\frac {4\pi ~h~\rho ~c_{\mathrm {kap} }^{2}}{\lambda }}}$

between mountain and valley corresponds to a pressure ( Bernoulli formula ) which counteracts the capillary pressure .

From the condition that this dynamic pressure difference between wave crest and wave trough is equal to the capillary pressure difference (this corresponds to twice the amount given above ) between these two regions, it follows for the propagation speed ${\displaystyle ~p_{\mathrm {kap} }}$

${\displaystyle c_{\mathrm {kap} }={\sqrt {\frac {2\pi \cdot \sigma }{\rho \cdot \lambda }}}}$.

This means that capillary waves have anomalous dispersion ; H. their speed of propagation decreases with increasing wavelength . ${\displaystyle \lambda }$

See also

• Capillary wave spectroscopy
• Water wave
• Ultrasonic nebulizer

Individual evidence

1. ↑ Joachim Grehn (Ed.): Metzler Physik . 2nd Edition. Schroedel Schulbuchverlag GmbH, Hanover 2005, ISBN 3-507-05209-1 , p. 124 .
2. ↑ Dieter Meschede : Gerthsen Physik (=  Springer textbook ). Springer Berlin Heidelberg, Berlin, Heidelberg 2015, ISBN 978-3-662-45976-8 , pp. 204–205 , doi : 10.1007 / 978-3-662-45977-5 ( springer.com [accessed February 12, 2020]).

literature

• Erich Truckenbrodt: Elementary flow processes of density variable fluids as well as potential and boundary layer flows . In: Fluid Mechanics . 4th edition. tape 2 . Springer, 2008, ISBN 3-540-79023-3 .
• Dieter Meschede : Gerthsen Physics . 23rd edition. Springer, Berlin / Heidelberg / New York 2006, ISBN 3-540-25421-8 .
• Joachim Grehn (Ed.): Metzler Physik . 2nd Edition. Schroedel Schulbuchverlag GmbH, Hanover 2005, ISBN 3-507-05209-1 .

Web links

• Lecture notes on hydraulics and water waves (PDF; 179 kB)
• Entry about capillary waves in the Sklogwiki