Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T20:29:20.518Z Has data issue: false hasContentIssue false

The reflexion of long waves at a step

Published online by Cambridge University Press:  24 October 2008

E. F. Bartholomeusz
Affiliation:
Faculty of EngineeringUniversity of CeylonColombo, Ceylon

Abstract

A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1958

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.)

References

REFERENCES

(1)Kreisel, G.Quart. Appl. Math. 7 (1949), 21.Google Scholar
(2)Lamb, H.Hydrodynamics, 6th ed. (Cambridge, 1932).Google Scholar
(3)Havelock, T. H.Phil. Mag. (7), 8 (1929), 569.CrossRefGoogle Scholar
(4)Schmidt, E.Math. Ann. 64 (1907), 161.CrossRefGoogle Scholar
(5)Whittaker, E. T. and Watson, G. N.Modern analysis, 4th ed. (Cambridge, 1940).Google Scholar
(6)Titchmarsh, E. C.Theory of functions, 2nd ed. (Oxford, 1939).Google Scholar
(7)Muskhelishvili, N. I.Singular integral equations (Groningen, 1953).Google Scholar