Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T07:10:58.719Z Has data issue: false hasContentIssue false
  • English
  • Français

SCATTERING OF MEMBRANE COUPLED GRAVITY WAVES BY PARTIAL VERTICAL BARRIERS

Part of: Waves

Published online by Cambridge University Press:  08 June 2010

S. R. MANAM
S. R. MANAM*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India (email: [email protected])
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.

Scattering of membrane coupled gravity waves in deep water by partial vertical barriers is investigated by the recently developed expansion formulae for wave structure interaction problems. The horizontal thin membrane is considered to be under uniform tension and is covering the free surface. The analysis is based on the linearized theory of water waves, and by combining the kinematic and dynamic conditions at the membrane covered surface, one may derive a not so well-posed mixed boundary value problem for Laplace’s equation with third-order boundary condition. The flexible membrane is attached by a spring to the surface piercing barrier, giving suitable edge conditions for the unique solution. The boundary value problem has been converted into dual integral equations with kernels composed of trigonometric functions, which are then solved analytically. The important physical quantities such as reflection and transmission coefficients for both cases of submerged and surface piercing barriers are obtained analytically in terms of modified Bessel functions. It is found that complete reflection or transmission is possible at certain resonant frequencies for the incident membrane coupled waves. Numerical results are plotted and discussed for different values of the nondimensional membrane tension parameter.

Keywords

mixed boundary value problemlogarithmic kernelsreflection coefficientmode-coupling relation

MSC classification

Secondary: 74J05: Linear waves
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 2 , October 2009 , pp. 241 - 260
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Chakrabarti, A. and Manam, S. R., “Solution of the logarithmic singular integral equation”, Appl. Math. Lett. 16 (2003) 369–373.CrossRefGoogle Scholar
[2]Cho, I. H. and Kim, M. H., “Interaction of a horizontal flexible membrane with oblique incident waves”, J. Fluid Mech. 367 (1998) 139–161.CrossRefGoogle Scholar
[3]Cho, I. H. and Kim, M. H., “Wave deformation by a submerged circular disk”, Appl. Ocean Res. 21 (1999) 263–280.CrossRefGoogle Scholar
[4]Chwang, A. T. and Dong, Z. N., “Wave trapping due to a porous plate”, Proceedings of the 15th ONR Symposium on Naval Hydrodynamics, (National Academy Press, Washington, DC, 1984) 407–414.Google Scholar
[5]Estrada, R. and Kanwal, R. P., “Integral equations with logarithmic kernels”, IMA J. Appl. Math. 43 (1989) 133–155.CrossRefGoogle Scholar
[6]Evans, D. V., “The influence of surface tension on the reflection of water waves by a plane vertical barrier”, Proc. Cambridge Philos. Soc. 64 (1968) 795–810.CrossRefGoogle Scholar
[7]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products (Academic Press, London, 1980).Google Scholar
[8]Kim, M. H. and Kee, S. T., “Flexible membrane wave barrier. I: Analytic and numerical solutions”, ASCE J. Waterway, Port, Coastal Ocean Engng 122 (1996) 46–53.CrossRefGoogle Scholar
[9]Manam, S. R., Bhattacharjee, J. and Sahoo, T., “Expansion formulae in wave structure interaction problems”, Proc. R. Soc. Lond. A 462 (2006) 263–287.Google Scholar
[10]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. A 435 (1991) 293–319.Google Scholar
[11]Sahoo, T., Yip, T. L. and Chwang, A. T., “Wave interaction with a semi-infinite horizontal membrane”, Proceedings of XXIX IAHR Congress, Theme E-Hydraulics for Maritime Engineering, Beijing, China, 2001, (Tsinghua University Press, Beijing, 2001) 109–117.Google Scholar
[12]Sawaragi, T., Coastal engineering—waves, beaches, wave–structure interactions (Elsevier, Tokyo, 1995).Google Scholar
[13]Stoker, J. J., “Surface waves in water of variable depth”, Quart. Appl. Math. 5 (1947) 1–54.CrossRefGoogle Scholar
[14]Ursell, F., “The effect of a fixed vertical barrier on surface waves in deep water”, Proc. Cambridge Philos. Soc. 43 (1947) 374–382.CrossRefGoogle Scholar
[15]Williams, W. E., “A note on scattering of water waves by a vertical barriers”, Proc. Cambridge Philos. Soc. 62 (1966) 507–509.CrossRefGoogle Scholar
[16]Yip, T. L., Sahoo, T. and Chwang, A. T., “Wave scattering by multiple floating membranes”, Proceedings of the 11th International Offshore and Polar Engineering Conference, Stavangar, Norway, 2001, (International Society of Offshore and Polar Engineers, Cupertino, CA, 2001) 3, 379–384.Google Scholar
[17]Yip, T. L., Sahoo, T. and Chwang, A. T., “Trapping of surface waves by porous and flexible structures”, Wave Motion 35 (2002) 41–54.CrossRefGoogle Scholar