Capillary Wave’s Depth Decay

Author(s)
Kern E. Kenyon

ABSTRACT

Depth decay rates for pressure and velocity variations of a propagating capillary wave are found to be significantly different from each other, and neither one is expected to have the classical exponential character. To obtain these results Bernoulli’s equation along streamlines in the steady reference frame is combined with the force balance on fluid particles in the cross-stream direction: a pressure gradient offsets the centrifugal force on particles moving along a curved path. The two starting equations for pressure and velocity are nonlinear, but two linear first order ordinary differential equations are produced from them, one for each variable, and they can be integrated immediately. A full solution awaits further information on the non-constant coefficient, the radius of curvature function for the streamlines, either from observations or another theory.

Depth decay rates for pressure and velocity variations of a propagating capillary wave are found to be significantly different from each other, and neither one is expected to have the classical exponential character. To obtain these results Bernoulli’s equation along streamlines in the steady reference frame is combined with the force balance on fluid particles in the cross-stream direction: a pressure gradient offsets the centrifugal force on particles moving along a curved path. The two starting equations for pressure and velocity are nonlinear, but two linear first order ordinary differential equations are produced from them, one for each variable, and they can be integrated immediately. A full solution awaits further information on the non-constant coefficient, the radius of curvature function for the streamlines, either from observations or another theory.

Cite this paper

Kenyon, K. (2014) Capillary Wave’s Depth Decay.*Natural Science*, **6**, 1241-1243. doi: 10.4236/ns.2014.616112.

Kenyon, K. (2014) Capillary Wave’s Depth Decay.

References

[1] Milne-Thomson, L.M. (1953) Theoretical Hydrodynamics. 3rd Edition, MacMillan, New York, 394-397.

[2] Lamb, H. (1945) Hydrodynamics. 6th Edition, Dover, New York, 455-458.

[3] Kundu, P.K. (1990) Fluid Mechanics. Academic Press, San Diego, 205-207.

[4] Faber, T.E. (1995) Fluid Dynamics for Physicists. Cambridge University Press, Cambridge, 173-174.

http://dx.doi.org/10.1017/CBO9780511806735

[5] Kenyon, K.E. (2010) Vena Contracta and Surface Tension. Physics Essays, 23, 579.

http://dx.doi.org/10.4006/1.3486756

[6] Kenyon, K.E. (2013) Depth Decay Rate of Surface Gravity Wave Pressure and Velocity. Natural Science, 5, 44.

http://dx.doi.org/10.4236/ns.2013.51007

[1] Milne-Thomson, L.M. (1953) Theoretical Hydrodynamics. 3rd Edition, MacMillan, New York, 394-397.

[2] Lamb, H. (1945) Hydrodynamics. 6th Edition, Dover, New York, 455-458.

[3] Kundu, P.K. (1990) Fluid Mechanics. Academic Press, San Diego, 205-207.

[4] Faber, T.E. (1995) Fluid Dynamics for Physicists. Cambridge University Press, Cambridge, 173-174.

http://dx.doi.org/10.1017/CBO9780511806735

[5] Kenyon, K.E. (2010) Vena Contracta and Surface Tension. Physics Essays, 23, 579.

http://dx.doi.org/10.4006/1.3486756

[6] Kenyon, K.E. (2013) Depth Decay Rate of Surface Gravity Wave Pressure and Velocity. Natural Science, 5, 44.

http://dx.doi.org/10.4236/ns.2013.51007