Capillary gravity waves caused by a moving disturbance: Wave resistance

E. Raphaël; P.-G. de Gennes

doi:10.1103/PhysRevE.53.3448

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Capillary gravity waves caused by a moving disturbance: Wave resistance

E. Raphaël and P.-G. de Gennes

Laboratoire de Physique de la Matière Condensée, URA CNRS 792, Collège de France, 75231 Paris Cedex 05, France

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Phys. Rev. E 53, 3448 – Published 1 April, 1996

DOI: https://doi.org/10.1103/PhysRevE.53.3448

Abstract

The dispersive property of capillary gravity waves is responsible for the complicated wave pattern generated at the free surface of a calm liquid by a disturbance moving with a velocity V greater than the minimum phase speed $c^{\min}$ =(4gγ/ρ $)^{1 / 4}$ (ρ is the liquid density, γ is the liquid-air surface tension, and g is the acceleration due to gravity). The disturbance may be produced by a small object immersed in the liquid or by the application of an external surface pressure distribution. The waves generated by the moving disturbance continually remove energy to infinity, and, consequently, the disturbance experiences a drag called the wave resistance. The wave resistance corresponding to a surface pressure distribution symmetrical about a point was analyzed by Havelock in the particular case of pure gravity waves (i.e., γ=0) for which the minimum phase speed reduces to zero. Here, we investigate the more general case of capillary gravity waves using a linearized theory. We also analyze the integral depression of the liquid, the momentum carried by the liquid, and the effective mass of the disturbance for velocities V smaller than $c^{\min}$ . These results may possibly lead to a new method of probing soft surfaces. © 1996 The American Physical Society.

References (15)

E. Raphaël and P.-G. de Gennes, C. R. Acad. Sci. Ser. II 317, 153 (1993).
E. Raphaël and P.-G. de Gennes, Europhys. Lett. 31, 293 (1995).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, New York, 1987).
D. J. Acheson, Elementary Fluid Dynamics (Clarendon, Oxford, 1990).
J. Lighthill, Waves in Fluids, 6th ed. (Cambridge University Press, Cambridge, 1979).
H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1993).
Lord Rayleigh, Proc. London Math. Soc. 15, 69 (1883).
Omitted end note.
T. H. Havelock, Proc. R. Soc. A 95, 354 (1918).
See, for instance, G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).
In order to properly calculate Ω, one has first to calculate ∫dxt $ζ_{ε}$ (x) and then to take the limit ε→0.
See, for instance, Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1972).
Omitted end note.
The following analysis is reminiscent of the development given in Landau and Lifshitz concerning the drag force in the potential flow past a body, see Ref. 3 , pp. 26–31.
Note that Q′(V≪ $c^{\min}$ ) is not simply proportional to the bump volume:mQ′(V≪ $c^{\min}$ )≠ρΩ.

[c1] E. Raphaël and P.-G. de Gennes, C. R. Acad. Sci. Ser. II 317, 153 (1993).

[c2] E. Raphaël and P.-G. de Gennes, Europhys. Lett. 31, 293 (1995).

[c3] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, New York, 1987).

[c4] D. J. Acheson, Elementary Fluid Dynamics (Clarendon, Oxford, 1990).

[c5] J. Lighthill, Waves in Fluids, 6th ed. (Cambridge University Press, Cambridge, 1979).

[c6] H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1993).

[c7] Lord Rayleigh, Proc. London Math. Soc. 15, 69 (1883).

[c8] Omitted end note.

[c9] T. H. Havelock, Proc. R. Soc. A 95, 354 (1918).

[c10] See, for instance, G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

[c11] In order to properly calculate Ω, one has first to calculate ∫dxt $ζ_{ε}$ (x) and then to take the limit ε→0.

[c12] See, for instance, Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1972).

[c13] Omitted end note.

[c14] The following analysis is reminiscent of the development given in Landau and Lifshitz concerning the drag force in the potential flow past a body, see Ref. 3 , pp. 26–31.

[c15] Note that Q′(V≪ $c^{\min}$ ) is not simply proportional to the bump volume:mQ′(V≪ $c^{\min}$ )≠ρΩ.

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Capillary gravity waves caused by a moving disturbance: Wave resistance

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