Capillary wave
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 ( )
With
- the surface tension in N / m
- 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
- indicating with
- the vertical coordinate
- the horizontal coordinate
- the amplitude
Thus the capillary pressure is through on the wave crest
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 :
- .
The difference in kinetic energy per volume (with liquid density )
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
- .
This means that capillary waves have anomalous dispersion ; H. their speed of propagation decreases with increasing wavelength .
See also
Individual evidence
- ↑ Joachim Grehn (Ed.): Metzler Physik . 2nd Edition. Schroedel Schulbuchverlag GmbH, Hanover 2005, ISBN 3-507-05209-1 , p. 124 .
- ↑ 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 .