Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T20:50:34.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Free gravitational oscillations in rotating rectangular basins

Published online by Cambridge University Press:  28 March 2006

Desiraju B. Rao
Desiraju B. Rao
Affiliation:
The University of Chicago
This paper is a condensed version of a Doctoral Dissertation (Rao 1965).
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of free oscillations of a homogeneous liquid under gravity in rotating rectangular basins of uniform depth is undertaken from both theoretical and experimental considerations.

The theoretical study is in the framework of the quasistatic equations. It is also assumed that the curvature of the free surface can be ignored. Numerical computations for the frequencies and modal structures were carried out for several of the slowest antisymmetric and symmetric modes in a square basin and in a rectangular basin of two-to-one dimension ratio, without any restriction on the angular speed of rotation of the basin. These computations are in agreement with a numerical value obtained many years ago by Taylor, and also with several values found by Corkan & Doodson. They exhibit the typical frequency-splitting associated with certain multiplets in the zero-rotation spectrum. Further, the theoretical calculations indicate that the slopes of the curves of frequency versus speed of rotation change sign for some of the modes in rectangular geometry. Such behaviour is not present in circular basins. Negative modes are found to be ‘unstable’ in the sense of Corkan & Doodson; that is, they are transformed into positive modes for sufficiently high rotation. Calculations were also made for the slowest longitudinal oscillations in highly elongated basins to demonstrate the decreasing importance of rotation on the frequencies of these modes.

Experimental work was carried out in a flat-bottomed square tank for the slowest positively and negatively propagating antisymmetric modes and the slowest positively propagating symmetric mode. Good agreement was found between theory and experiment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 25 , Issue 3 , July 1966 , pp. 523 - 555
Copyright
© 1966 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Present affiiliation: National Center for Atmospheric Research, Boulder, Colorado.

References

Corkan, R. H. & Doodson, A. T. 1952 Free tidal oscillations in a rotating square sea. Proc. Roy. Soc., A 215, 147.Google Scholar
Defant, A. 1925 Gezeitenprobleme des Meeres in Landnähe. (Problem der Kosmischen Physik, 6.) Hamburg: Henri Grand.
Defant, A. 1961 Physical Oceanography, vol. II. New York: Pergamon Press.
Fultz, D. 1962 An experimental view of some atmospheric and oceanic behavioral problems. Trans. N.Y. Acad. Sci. 24, 421.Google Scholar
Goldsbrough, G. R. 1931 The tidal oscillations in rectangular basins. Proc. Roy. Soc., A 132, 689.Google Scholar
Jeffreys, H. 1925 The free oscillations of water in an elliptical lake. Proc. Lond. Math. Soc. 23, 455.Google Scholar
Lamb, H. 1924 Hydrodynamics, 5th ed. Cambridge University Press.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Miles, J. W. 1964 Free-surface oscillations in a slowly rotating liquid. J. Fluid Mech. 18, 187.Google Scholar
Platzman, G. W. & Rao, D. B. 1964 The free oscillations of Lake Erie. Studies on Oceanography (Hidaka volume) (K. Yoshida, editor), pp. 359–382. University of Washington. Press.Google Scholar
Proudman, J. 1916 On the dynamical theory of tides. Part II. Flat seas. Proc. Lond. Math. Soc. (2nd Series), 18, 21.Google Scholar
Proudman, J. 1933 Note on the free tidal oscillations of a sea with slow rotation. Proc. Lond. Math. Soc. (2nd series), 35, 75.Google Scholar
Proudman, J. 1953 Dynamical Oceanography. London: Methuen.
Rao, D. B. 1965 Free gravitational oscillations in rotating rectangular basins. Ph.D. Thesis, Department of the Geophysical Sciences, The University of Chicago.
Rayleigh, Lord 1903 On the vibrations of a rectangular sheet of rotating liquid. Phil. Mag. 5, 297.Google Scholar
Rayleigh, Lord 1909 Notes concerning tidal oscillations upon a rotating globe. Proc. Roy. Soc., A 82, 448.Google Scholar
Taylor, G. I. 1922 Tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc. 20, 148.Google Scholar
Van Dantzig, D. & Lauwerier, H. A. 1960 The North Sea Problem. IV. Free oscillations of a rotating rectangular sea. Proc. K. ned. Akad. Wet. (Series A), 63, 339.Google Scholar
Veltkamp, G. W. 1960 Spectral properties of Hilbert Space operators associated with tidal motions. Ph.D. Thesis, University of Utrecht.