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Gravity waves in a stratified fluid

Published online by Cambridge University Press:  28 March 2006

Chia-Shun Yih
Chia-Shun Yih
Affiliation:
Department of Engineering Mechanics, University of Michigan
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Abstract

A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturm's comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturm's oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified.

In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 8 , Issue 4 , August 1960 , pp. 481 - 508
Copyright
© 1960 Cambridge University Press

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References

Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. Roy. Soc. A, 225, 505–15.Google Scholar
Benton, G. S. 1953 A General Solution for the Celerity of Long Gravitational Waves in a Stratified Fluid, pp. 149–62. Fluid Models in Geophysics, U.S. Government Printing Office.
Courant, R. 1926 Über die Anwendung der Variationsrechnung. Acta Math. vol. 49.Google Scholar
Courant, R. & Hilbert, D. 1931 Methoden der Mathematischen Physik, Vol. I. Berlin: Julius Springer.
Fjeldstad, J. E. 1933 Interne Wellen. Geof. Publ. Vol. X, no. 6, Oslo.Google Scholar
Fjeldstad, J. E. 1952 Article in Gravity Waves, National Bureau of Standards Circular, 521, 39–45.
Görtler, H. 1954 Neuere Anwendungen der Charakteristikentheorie in der Hydrodynamik, Colloque Julius Massau des Mémoires des Sciences, Tom XXVIII.
Greenhill, A. G. 1887 Wave motion in hydrodynamics. J. Math., 9, 62–112.Google Scholar
Ince, E. L. 1944 Ordinary Differential Equations. New York: Dover Publications.
Lamb, H. 1945 Hydrodynamics. New York: Dover Publications.
McLachlan, N. W. 1947 Theory and Applications of Mathieu Functions. Oxford.
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. Roy. Soc. A, 142, 621–8.Google Scholar
Sz.-Nagy, B. 1947 Vibrations d'une corde non homogène. Bull. Soc. Math. France, vol. 75, pp. 193–208. See also, Riesz, F and Sz.-Nagy, B. 1956 Functional Analysis, §99. London and Glasgow: Blackie and Son.Google Scholar
Webb, R. R. 1884 Math. Tripos Papers, Cambridge.
Yih, C.-S. 1959a Effects of gravitational or electromagnetic fields on fluid motion. Quart. Appl. Math. 16, 409–15.Google Scholar
Yih, C.-S. 1959b Effects of density variation on fluid flow. J. Geophys. Res. 64, 2219–23.Google Scholar