Capillary Waves Generated by a Rock in a Stream

Authors

  • Takeo Nakagawa Satellite College Hakusan
  • Ai Nakagawa Satellite College Hakusan

Keywords:

Capillary Wave, Gravity Wave, Surface Tension, Kelvin Wave, Group Velocity

Abstract

This paper is concerned with capillary waves generated by a rock, or fishing line in a stream.  It is found mathematicaly that the angle between the capillary wave envelope and the direction of a point source is dependent of the velocity, surface tension, and density of the fluid:

This angle increases with increasing the surface tension, but decreases with increasing the square of velocity and the density of fluid.   These theoretical outcomes are consistent with the detailed behaviors of capillary waves observed in the natural running streams. To enhance the mathematical analyses on capillary waves generated by a fishing line with constant speed, the relevant non-dimensional parameters 1=Ut/r and

2=r3 /(t2 ) have derived based on the Buckingham -Theorem, where U the relative stream velocity, t the time, r the distance from the point source of wave generation,  the surface tension, and  the density of the fluid. 

It has been confirmed by the present proposed approach that the angle between the gravity wave envelope and the direction of a duck or ship moving on the calm water surface is constant of 19.28 which agrees with the result obtained by “Thomson, W., 1887 On ship waves, Institution of Mechanical Engineers, Minutes of Proceedings, 409-434” and “Adam, J.A., 2003 Mathematics in nature: modelling patterns in the natural world, Princeton University Press, Princeton, 161-172”.

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Author Biographies

Takeo Nakagawa, Satellite College Hakusan

Academy of Hakusan,

Ai Nakagawa, Satellite College Hakusan

Academy of Hakusan,

References

Adam, J. A.(2003) Mathematics in Nature: Modeloing Patterns in the natural World. Princeton University Press, Princeton, pp.161-172.

Adam, J.A.(2009) A Mathematical Nature Walk. Princeton University Press, Princeton, pp. 140-157 https://doi.org/10.1515/9781400832903

Lamb, H.(1895) Hydrodynamics. Cambridge

Lamb, H.(1932) Hydrodynamics. 6th ed. Dover, New York. pp.433-5.

Rayleigh Lord(1883) The form of standing waves on the surface of running water. Scientic Papers, 2, 258-267.

Nakagawa, T., Chanson, H.(2006) Fluid Mechanics for Ecologists. Applied Edition, IPC, Tokyo.(in Japanese)

Reynolds, O.(1877) On the rate of progression of groups of waves and the rate which energy is transmitted by waves. Papers on Mechanical and Physical Subjects. 1, 170-82.

Russell, J.S.(1943) Notice of a report of the committee on the form of ships. British Association for the Advancement of Science, Report.

Stokes, G.G.(1876) Smith prize examination papers for 2 February 1976. Mathematical Physical Papers, 5, pp.362.

Thomson, W. (1871) Letter to Tait. Mathematical and Physical Papers, 4, 79-80.

Thomson, W. (1887) On ship waves. Institution of Mechanical Engineers, Minutes of Proceedings, 409-34. https://doi.org/10.1243/PIME_PROC_1887_038_028_02

Thomson, W. (1906) Deep sea ship-waves. Philosophical Magazine, 4, 394-418.

Zirep, J., Nakagawa, T.(1996) Similarity Laws and Modelling. IPC, Tokyo pp.219.(in Japanese).

Published

2020-04-30

How to Cite

Takeo Nakagawa, & Ai Nakagawa. (2020). Capillary Waves Generated by a Rock in a Stream. To Physics Journal, 5, 123-137. Retrieved from http://purkh.com/index.php/tophy/article/view/723

Issue

Section

Research Articles