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16.—Waves over Obstacles on a Shallow Seabed

Published online by Cambridge University Press:  14 February 2012

Alan Jeffrey  and
Saw Tin
Alan Jeffrey
Affiliation:
Department of Engineering Mathematics, University of Newcastle upon Tyne
Saw Tin
Affiliation:
Department of Engineering Mathematics, University of Newcastle upon Tyne
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Synopsis

In this paper we consider the effect of the passage of shallow water waves over vertical walled objects on a flat seabed. The effect of reflections at the successive walls is taken into account when determining the place of breaking of the wave. The results are obtained by the introduction of special reflection and transmission coefficients at each wall and recurrence relations are formulated connecting their values at successive walls. The effect of this is to reduce the calculation of the place of breaking of the wave over a stepped seabed to an equivalent one for a wave over a flat seabed, the result of which is well known.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 71 , Issue 3 , 1973 , pp. 181 - 192
Copyright
Copyright © Royal Society of Edinburgh 1973

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References

References to Literature

[1]Bartholomeusz, E. F., 1958. The reflection of long waves at a step. Proc. Camb. Phil. Soc. Math. Phys. Sci., 54, 106–118.CrossRefGoogle Scholar
[2]Bürger, W., 1967. A note on the breaking of waves on non-uniformly sloping beaches. J. Math. Mech., 16, 1131–1142.Google Scholar
[3]Carrier, G. F. and Greenspan, H. P., 1958. Water waves of finite amplitude on a sloping beach. J. Fluid Mech., 4, 97–109.CrossRefGoogle Scholar
[4]Courant, R. and Hilbert, D., 1962. Methods of Mathematical Physics, II. New York: Inter-science.Google Scholar
[5]Greenspan, H. P., 1958. On the breaking of water waves of finite amplitude on a sloping beach. J. Fluid Mech., 4, 330–334.CrossRefGoogle Scholar
[6]Jeffrey, A., 1964. The breaking of waves on a sloping beach. Z. Angew. Math. Phys., 15, 97–106. See also Addendum, Z. Angew. Math. Phys., 16, 712, 1965.CrossRefGoogle Scholar
[7]Jeffrey, A., 1967. On a class of non-breaking finite amplitude water waves. Z. Angew. Math. Phys., 18, 57–65. See also Addendum, Z. Angew. Math. Phys., 18, 918, 1967.CrossRefGoogle Scholar
[8]Jeffrey, A., 1973. The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients. Part I—Fundamental theory. J. Appl. Anal., 3, 79–100.CrossRefGoogle Scholar
[9]Jeffrey, A., 1973. The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients. Part II—Special cases and application. J. Appl. Anal, (in press).CrossRefGoogle Scholar
[10]Lax, P. D., 1954. The initial value problem for nonlinear hyperbolic equations in two independent variables. Ann. Math. Stud., 33, 211–229.Google Scholar
[11]Newman, J. N., 1965. Propagation of water waves past long two-dimensional obstacles. J. Fluid Mech., 23, 23–29.CrossRefGoogle Scholar
[12]Stoker, J. J., 1957. Water Waves. New York: Interscience.Google Scholar