Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T19:08:02.420Z Has data issue: false hasContentIssue false

Periodic Waves in a Running Stream

Published online by Cambridge University Press:  20 November 2018

Marvin Shinbrot*
Affiliation:
Centre de Recherche de Mathématiques Appliquées, Université de Montréal, Montréal, Québec H3C 3J7
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we discuss questions of the existence and calculation of periodic, steady flows over periodic streambeds. There are some surprises.

Problems such as this, of flows in running streams, are free-surface problems, and part of the difficulty is that the domain occupied by the fluid is not completely known a priori.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Gerber, Robert, Sur les solutions exactes des équations du mouvement avec surface libre d'un liquide pesant. J. de Math. 34 (1962) 185-299.Google Scholar
2. Beckert, Herbert, Existenzbeweise in der Théorie permanenter Schwerewellen. Arch. Rat. Mech. Anal. 9 (1962) 379-394.Google Scholar
3. Hewgill, Denton E., Reeder, John, and Shinbrot, Marvin, Some exact solutions of the nonlinear problem of water waves. To appear in Pacific J. Math. Google Scholar
4. Hilbig, H., Existenzbeweis für Potentialströmungen längs eines Kanals mit welliger Kanalsohle unter Einfluss der Schwerkraft und der Oberflâchenspannung. Arch. Rat Mech. Anal. 51 (1962) 295-303.Google Scholar
5. Joseph, Daniel D., Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rat. Mech. Anal. 51 (1962) 295-303.Google Scholar
6. Krasnosel'skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York, 1964.Google Scholar
7. Krasovskii, Yu. P., The theory of steady state waves of large amplitude. Translated in: Soviet Physics Dokl. 5 (1962) 62-65.Google Scholar
8. Lamb, Sir Horace, Hydrodynamics, 6th ed. Cambridge, at the University Press, 1932.Google Scholar
9. Maxwell, James Clerk, Capillary action. Encyclopedia Britannica, 11th ed. Cambridge, 1911.Google Scholar
10. Moiseev, N. N., On the flow of a heavy fluid over a wavy bottom. Prikl. Mat. Mekh. (Russian). 21 (1962) 15-20.Google Scholar
11. Reeder, John and Shinbrot, Marvin, Three-dimensional, nonlinear wave interaction in water of constant depth. To appear.Google Scholar
12. Reeder, John and Shinbrot, Marvin, The initial value problem for surface waves under gravity, II and III. Indiana Univ. Math. J. 25 (1962) 1049-1071, and J. Math. Anal, and Appl. 67 (1962) 340-391.Google Scholar
13. Sattinger, D., On the free surface of a viscous fluid motion. Proc. Roy. Soc. London A 349 (1962) 183-204.Google Scholar
14. Shinbrot, Marvin, Lectures on Fluid Mechanics. Gordon and breach, New York, London, Paris, 1973.Google Scholar
15. Shinbrot, Marvin, Water waves over periodic bottoms in three dimensions. To appear in J. of the Inst, of Maths, and Applications. Google Scholar
16. Shinbrot, Troy, On salient phenomena of stationary waves in water. Undergraduate thesis, Reed College, Oregon, 1978.Google Scholar
17. Stoker, J. J., Water Waves. Interscience Publishers, New York, 1957.Google Scholar
18. Wehausen, John V. and Laitone, Edmond V., Surface waves. In: Handbuch der Physik, Fliigge, S., ed., v. IX, pp. 446-778. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1960.Google Scholar
19. Zeidler, Eberhard, Beiträge zur Théorie und Praxis Freier Randwertaufgaben. Akademie- Verlag, Berlin, 1971.Google Scholar
20. Zeidler, Eberhard, Topologischer Existenzbeweis für Kapillar-Schwerewellen. Math. Nachr. 49 (1962) 85-99.Google Scholar