Hostname: page-component-88dd8db54-7p5v5 Total loading time: 0 Render date: 2024-03-05T08:47:17.776Z Has data issue: false hasContentIssue false

On the solution of the problem of scattering of surface water waves by a sharp discontinuity in the surface boundary conditions

Published online by Cambridge University Press:  17 February 2009

A. Chakrabarti
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India; e-mail: alok@math.iisc.ernet.in
Rights & Permissions [Opens in a new window]

Abstract

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.

Closed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Evans, D. V., “The solution of a class of boundary value problems with smoothly varying boundary conditions”, Q. J. Mech. Appl. Math. 38 (1994) 521536.Google Scholar
[2]Evans, D. V. and Linton, C. M., “On the step approximation for water wave problems”, J. Fluid Mech. 278 (1994) 229249.Google Scholar
[3]Fox, C. and Squire, V. A., “On the oblique reflection and transmission of ocean waves at shore fast sea ice”, Philos. Trans. Roy. Soc. London, Ser. A. 347 (1994) 185218.Google Scholar
[4]Gabov, S. A., Sveshnikov, A. G. and Shatov, A. K., “Dispersion of internal waves by an obstacle floating on the boundary separating two liquids”, Prikl. Mat. Mech. 53 (1989) 727730, (Russian).Google Scholar
[5]Gakhov, F. D., Boundary value problems (Pergamon, Oxford, 1966).Google Scholar
[6]Gradshteyn, I. M. and Rhyzik, I. S., Tables of integrals, infinite series and products (Academic Press, 1980).Google Scholar
[7]Keller, J. B. and Weitz, M., “Reflection and transmission coefficients of water waves entering or leaving an ice-field”, Comm. Pure Appl. Math. 6 (1953) 415417.Google Scholar
[8]Mitra, R. and Lee, S. W., Analytical techniques in the theory of guided wave (Mac-Millan, 1971).Google Scholar
[9]Muskhelishvili, N. I., Singular integral equations (Gröningen, Holland, Nordhoff, 1953).Google Scholar
[10]Newman, J. N., “Propagation of water waves over an infinite step”, J. Fluid Mech. 23 (1965) 399415.Google Scholar
[11]Peters, A. S., “The effect of a floating mat on water waves”, Comm. Pure Appl. Math. 3 (1950) 319354.Google Scholar
[12]Sneddon, I. N., The use of integral transforms (Tata McGraw Hill, New Delhi, 1974).Google Scholar
[13]Spence, D. A.,“The lift coefficient of a thin jet flapped wing, II: A solution of the integro-differential equation for the slope of the jet”, Proc. R. Soc., Ser. A 261 (1961) 97118.Google Scholar
[14]Stoker, J. J., Water waves (Wiley Interscience, 1957).Google Scholar
[15]Ursell, F., “The effect of a fixed vertical barrier on surface water waves in deep water”, Proc. Cambridge Philos. Soc. 43 (1947) 374382.Google Scholar
[16]Varley, E. and Walker, J. D. A., “A method for solving singular integro-differential equations”, IMA J. of Appl. Math. 43 (1989) 1115.Google Scholar
[17]Weitz, M. and Keller, J. B., “Reflection of water waves from floating ice in water of finite depth”, Comm. Pure Appl. Math. 3 (1950) 305318.Google Scholar