Abstract
The present study deals with the trapping of oblique surface gravity waves by a vertical submerged flexible porous plate located near a rigid wall in water of finite as well as infinite depths. The physical problem is based on the assumption of small amplitude water wave theory and structural response. The flexible plate is assumed to be thin and is modeled based on the Euler–Bernoulli beam equation. Using the Green’s function technique to the plate equation and associated boundary conditions, an integral equation is derived which relates the normal velocity on the plate to the difference in velocity potentials across the plate involving the porous-effect parameter and structural rigidity. Further, applying Green’s second identity to the free-surface Green’s function and the scattered velocity potentials on the two sides of the plate, a system of three more integral equations is derived involving the velocity potentials and their normal derivatives across the plate boundary along with the velocity potential on the rigid wall. The system of integral equations is converted into a set of algebraic equations using appropriate Gauss quadrature formula which in turn solved to obtain various quantities of physical interest. Utilizing Green’s identity, explicit expressions for the reflection coefficients are derived in terms of the velocity potentials and their normal derivatives across the plate. Energy balance relations are derived and used to check the accuracy of the computational results. As special cases of the submerged plate, wave trapping by the bottom-standing as well as surface-piercing plates is analyzed. Effects of various wave and structural parameters in trapping of surface waves are studied from the computational results by analyzing the reflection coefficients, wave forces exerted on the plate and the rigid wall, flow velocity, plate deflections and surface elevations. It is observed that surface-piercing plate is more effective for trapping of water waves compared to the bottom-standing and submerged plates. Further, irrespective of plate configurations, full reflection occurs for the same values of the distance between the plate and the rigid wall. Similar phenomenon is observed in case of angle of incidence. Irrespective of plate configurations, in the very long wave regime, full reflection occurs in case of partial plate of any length due to the occurrence of the wave diffraction through the gap region while zero reflection occurs in case of fully extended plate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Zhu S., Chwang A.T.: Experimental studies on caisson-type porous seawalls. Exp. Fluids 33, 512–515 (2002)
Williams A.N., Wang K.H.: Flexible porous wave barrier for enhanced wetlands habitat restoration. J. Eng. Mech. 129, 1–8 (2003)
Dean W.R.: On the reflection of surface waves by submerged plane barrier. Proc. Camb. Philos. Soc. 44, 483–491 (1945)
Ursell F.: The effect of a fixed vertical barrier on surface waves in deep water. Proc. Camb. Philos. Soc. 43, 374–382 (1947)
Mandal B.N., Chakrabarti A.: Water Wave Scattering by Barriers. WIT Press, Southampton (2000)
Linton C.M., McIver P.: Handbook of Mathematical Techniques for Wave/Structure Interactions. Chapman and Hall/CRC, New York (2001)
Sahoo T.: Mathematical Techniques for Wave Interaction with Flexible Structures. Chapman and Hall/CRC Press, London (2012)
Meylan M.H.: A flexible vertical sheet in waves. Int. J. Offshore Polar Eng. 5(2), 105–110 (1995)
Chakraborty R., Mandal B.N.: Water wave scattering by an elastic thin vertical plate submerged in finite depth water. J. Mar. Sci. Appl. 12, 393–399 (2013)
Chakraborty R., Mandal B.N.: Scattering of water waves by a submerged thin vertical elastic plate. Arch. Appl. Mech. 84, 207–217 (2014)
Koley, S., Kaligatla, R.B., Sahoo, T.: Oblique wave scattering by a vertical flexible porous plate. Stud. Appl. Math. 1–34 (2015). doi:10.1111/sapm.12076
Stokes, G.G.: Report on recent researches in hydrodynamics. Report to 16th Meeting of the British Association for the Advancement of Science, Southampton, Murrey, London 1–20 (1846)
Ursell F.: Trapping modes in the theory of surface waves. Proc. Camb. Philos. Soc. 47, 347–358 (1951)
Jones D.S.: The eigen values of \({\nabla^2u-\lambda u=0}\) when the boundary conditions are given on semi-infinite domains. Proc. Camb. Philos. Soc. 49, 668–684 (1953)
Greenspan H.P.: A note on edge waves in a stratified fluid. Stud. Appl. Math. 49(4), 381–388 (1970)
Leblond P.H., Mysak L.A.: Waves in the Ocean. Elsevier, Amsterdam (1978)
Linton C.M., Evans D.V.: Trapped modes above a submerged horizontal plate. Q. J. Mech. Appl. Math. 44(3), 487–506 (1991)
Linton C.M., McIver M.: The interaction of waves with horizontal cylinders in two-layer fluids. J. Fluid. Mech. 304, 213–229 (1995)
Kuznetsov N., Maz’ya V., Vainberg V.: Linear Water Waves: A Mathematical Approach. Cambridge University Press, UK (2004)
Sahoo T., Lee M.M., Chwang A.T.: Trapping and generation of waves by vertical porous structures. J. Eng. Mech. 126, 1074–1082 (2000)
Chwang, A.T., Dong, Z.: Wave-trapping due to a porous plate. In: Proceedings of the 15th Symposium on Naval Hydrodynamics, pp. 407–417. National Academy Press, Washington, DC (1984)
Wang K.H., Ren X.: An effective wave-trapping system. Ocean Eng. 21(2), 155–178 (1994)
Yip T.L., Sahoo T., Chwang A.T.: Trapping of waves by porous and flexible structures. Wave Motion 35, 41–54 (2002)
Behera H., Mandal S., Sahoo T.: Oblique wave trapping by porous and flexible structures in a two-layer fluid. Phys. Fluids 25, 1–23 (2013)
Koley S., Behera H., Sahoo T.: Oblique wave trapping by porous structures near a wall. J. Eng. Mech. ASCE 141(3), 1–15 (2015)
Behera H., Sahoo T.: Gravity wave interaction with porous structures in two-layer fluid. J. Eng. Math. 87, 73–97 (2014)
Clough R.B., Penzien J.: Dynamics of Structures. McGraw-Hill, New York (1975)
Williams A.N., Geiger P.T., McDogal W.G.: Flexible floating breakwater. J. Waterw. Port Coast. Ocean Eng. ASCE 117(5), 429–450 (1991)
Yu X., Chwang A.T.: Wave induced oscillation in harbor with porous breakwaters. J. Waterw. Port Coast. Ocean Eng. ASCE 120(2), 125–144 (1994)
Rhee J.P.: A note on the diffraction of obliquely incident water waves by a stepwise obstacle. Appl. Ocean Res. 23, 299–304 (2001)
Levine H.: Scattering of surface waves by a submerged circular cylinder. J. Math. Phys. 6, 1231–1234 (1965)
Tuck E.O.: Transmission of water waves through small apertures. J. Fluid Mech. 49, 65–74 (1971)
Ralston A.: A First Course in Numerical Analysis. McGraw Hill, New York (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaligatla, R.B., Koley, S. & Sahoo, T. Trapping of surface gravity waves by a vertical flexible porous plate near a wall. Z. Angew. Math. Phys. 66, 2677–2702 (2015). https://doi.org/10.1007/s00033-015-0521-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-015-0521-2