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Gravity effects in the water entry problem

Published online by Cambridge University Press:  09 April 2009

A. G. Mackie
A. G. Mackie
Affiliation:
Victoria University of Wellington, Wellington, New Zealand.
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The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 5 , Issue 4 , November 1965 , pp. 427 - 433
Copyright
Copyright © Australian Mathematical Society 1965

References

[1]Mackie, A. G., A linearized theory of the water entry problem, Quart. J. Mech. Appl. Math. 15 (1962), 137–151.CrossRefGoogle Scholar
[2]Mackie, A. G., Initial value problems in water wave theory, J. Austral. Math. Soc. 3 (1963), 340–350.CrossRefGoogle Scholar
[3]Stoker, J. J., Water Waves, (Interscience, New York, 1957).Google Scholar
[4]Terazawa, K., On deep-sea water waves caused by a local disturbance on or beneath the surface, Proc. Roy. Soc. A 92 (1916), 57–81.Google Scholar
[5]Wagner, H., Über Stoss- und Gleitvorgänge an den Oberflächen von Flüssigkeiten, Z. Angew. Math. Mech. 12 (1932), 193–215.CrossRefGoogle Scholar