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MEMBRANE-COUPLED GRAVITY WAVE SCATTERING BY A VERTICAL BARRIER WITH A GAP

Published online by Cambridge University Press:  05 June 2014

S. R. MANAM*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India email manam@iitm.ac.in
R. B. KALIGATLA
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India email manam@iitm.ac.in
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Abstract

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We study the reflection of membrane-coupled gravity waves in deep water against a vertical barrier with a gap. A floating membrane is attached on both sides of the barrier. The associated mixed boundary value problem, which is not particularly well posed, is analysed. We utilize an orthogonal mode-coupling relation to reduce the problem to solving a set of dual integral equations with trigonometric kernel. We solve these by using a weakly singular integral equation. The reflection coefficient is determined explicitly, while having freedom to clamp the membrane with a spring of a certain stiffness on only one side of the vertical barrier. The physical problem is of capillary–gravity wave scattering by a vertical barrier with a gap, when the membrane density is neglected. In this case, the reflection coefficient is known up to an undetermined edge slope on either side of the barrier. The scattering quantity is computed and presented graphically against a wave parameter for different values of nondimensional parameters pertaining to the structures involved in the problem.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Chakrabarti, A., Benerjea, S., Mandal, B. N. and Sahoo, T., “A unified approach to problems of scattering of surface water waves by vertical barriers”, J. Aust. Math. Soc. (Series B) 39 (1997) 93103; doi:10.1017/S0334270000009231.CrossRefGoogle Scholar
Chakrabarti, A. and Manam, S. R., “Solution of the logarithmic singular integral equation”, Appl. Math. Lett. 16 (2003) 369373; doi:10.1016/S0893-9659(03)80059-9.CrossRefGoogle Scholar
Chakrabarti, A., Manam, S. R. and Benerjea, S., “Scattering of surface water waves involving a vertical barrier with a gap”, J. Engrg. Math. 45 (2003) 183194.CrossRefGoogle Scholar
Chakrabarti, A. and Vijaya Bharati, L., “Transmission of water waves through a gap in a submerged vertical barrier”, Indian J. Pure Appl. Math. 22 (1991) 491512.Google Scholar
Cho, I. H. and Kim, M. H., “Interactions of a horizontal flexible membrane with oblique incident waves”, J. Fluid Mech. 367 (1998) 139161; doi:10.1017/S0022112098001499.CrossRefGoogle Scholar
Evans, D. V., “The influence of surface tension on the reflection of water waves by a plane vertical barrier”, Proc. Camb. Philos. Soc. 64 (1968) 795810; doi:10.1017/S0305004100043504.CrossRefGoogle Scholar
Evans, D. V., “Diffraction of water waves by a submerged vertical plate”, J. Fluid Mech. 40 (1970) 433451; doi:10.1017/S0022112070000253.CrossRefGoogle Scholar
Hocking, L. M., “Waves produced by a vertically oscillating plate”, J. Fluid Mech. 179 (1987) 267281; doi:10.1017/S0022112087001526.CrossRefGoogle Scholar
Karmakar, D. and Sahoo, T., “Gravity wave interaction with floating membrane due to abrupt change in water depth”, Ocean Engrg. 35 (2008) 598615; doi:10.1016/j.oceaneng.2008.01.009.CrossRefGoogle Scholar
Kim, M. H. and Kee, S. T., “Flexible membrane wave barrier. I: analytic and numerical solutions”, ASCE J. Waterway Port Coastal Ocean Engrg. 122 (1996) 4653 ;doi:10.1061/(ASCE)0733-950X(1996)122:1(46).CrossRefGoogle Scholar
Manam, S. R., “Scattering of membrane coupled gravity waves by partial vertical barriers”, ANZIAM J. 51 (2009) 241260; doi:10.1017/S1446181110000064.CrossRefGoogle Scholar
Manam, S. R., “A dual integral equation method for capillary–gravity wave scattering”, J. Integral Equations Appl. 24 (2012) 81110; doi:10.1216/JIE-2012-24-1-81.CrossRefGoogle Scholar
Manam, S. R., Bhattacharjee, J. and Sahoo, T., “Expansion formulae in wave structure interaction problems”, Proc. R. Soc. Lond. Ser. A 462 (2006) 263287; doi:10.1098/rspa.2005.1562.Google Scholar
Mandal, B. N. and Chakrabarti, A., Water wave scattering by barriers (WIT Press, Southampton, 2000).Google Scholar
Mei, C. C., “Radiation and scattering of transient gravity waves by vertical plates”, Quart. J. Mech. Appl. Math. 19 (1966) 417440; doi:10.1093/qjmam/19.4.417.CrossRefGoogle Scholar
Porter, D., “The transmission of surface waves through a gap in a vertical barrier”, Proc. Camb. Philos. Soc. 71 (1972) 411421; doi:10.1017/S0305004100050647.CrossRefGoogle Scholar
Rhodes-Robinson, P. F., “The effect of surface tension on the progressive waves due to incomplete vertical wave-makers in water of infinite depth”, Proc. R. Soc. Lond. Ser. A 435 (1991) 293319 ;doi:10.1098/rspa.1991.0146.Google Scholar
Tuck, E. O., “Transmission of water waves through small apertures”, J. Fluid Mech. 49 (1971) 6574; doi:10.1017/S0022112071001939.CrossRefGoogle Scholar
Ursell, F., “The effect of a fixed vertical barrier on surface waves in deep water”, Proc. Camb. Philos. Soc. 43 (1947) 374382; doi:10.1017/S0305004100023604.CrossRefGoogle Scholar
Williams, W. E., “Note on the scattering of water waves by a vertical barrier”, Proc. Camb. Philos. Soc. 62 (1966) 507509; doi:10.1017/S0305004100040135.CrossRefGoogle Scholar
Yip, T. L., Sahoo, T. and Chwang, A. T., “Wave scattering by multiple floating membranes”, in: Proc. 11th Int. Offshore and Polar Engineering Conf. (Stavanger, Norway), Volume 3 (International Society of Offshore and Polar Engineers, Cupertino, CA, 2001), 379384.Google Scholar