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Steady wave patterns on a non-uniform steady fluid flow

Published online by Cambridge University Press:  28 March 2006

F. Ursell
F. Ursell
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge University
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Abstract

A steady slightly non-uniform flow with a free surface is subject to a concentrated surface pressure which gives rise to a pattern of surface waves. (For gravity waves on deep water this is the well-known Kelvin ship-wave pattern.) The motion is assumed inviscid, and the waves are assumed small. A theory is developed for the wave pattern, based on the following assumptions:

  1. The stream velocity component normal to a wave crest is equal to the phase velocity based on the local wavelength;

  2. the separation between consecutive crests is equal to the local wave-length. These assumptions are expressed in mathematical form, and the existence of a set of characteristic curves (associated with the group velocity) is deduced from them. These characteristics are not identical with the crests. Let the additional assumption be made that

  3. the characteristics all pass through the point disturbance; the characteristics are then completely defined and may be constructed by a step-by-step process starting at the point disturbance. The same construction gives the direction of the wave crests at all points. The wave crests can then be deduced.

Assumptions of the same type as (1) and (2) have long been familiar in various applications of ray tracing. For uniform flows the present theory gives the same pattern as the method of stationary phase.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 9 , Issue 3 , November 1960 , pp. 333 - 346
Copyright
© 1960 Cambridge University Press

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References

Arthur, R. S., Munk, W. H. & Isaacs, J. D. 1952 The direct construction of wave rays. Trans. Amer. Geophys. Un. 33, 855–65.Google Scholar
Courant, R. & Hilbert, D. 1937 Methoden der Mathematischen Physik, Vol. 2. Berlin: Springer.
Havelock, T. H. 1908 The propagation of groups of waves in dispersive media, with application to waves on water produced by a travelling disturbance. Proc. Roy. Soc. A, 81, 398–430.Google Scholar
Lamb, H. 1916 On wave patterns due to a travelling disturbance. Phil. Mag. (6) 31, 539–48.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Longuet-Higgins, M. S. 1957 On the transformation of a continuous spectrum by refraction. Proc. Camb. Phil. Soc. 53, 226–9.Google Scholar
Peters, A. S. & Stoker, J. J. 1957 The motion of a ship, as a floating rigid body, in a seaway. Comm. Pure Appl. Math. 10, 399–490.Google Scholar
Stoker, J. J. 1957 Water Waves. New York: Interscience.
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. II. Waves on fluid sheets. Proc. Roy. Soc. A, 253, 289–321.Google Scholar
Ursell, F. 1960 On Kelvin's ship-wave pattern. J. Fluid Mech., 8, 418–31.Google Scholar
Whitham, G. B. 1960 A note on group velocity. J. Fluid Mech., 9, 347–52.Google Scholar