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Water waves over arrays of horizontal cylinders: band gaps and Bragg resonance

Published online by Cambridge University Press:  25 January 2011

C. M. LINTON
C. M. LINTON*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK
*
†Email address for correspondence: c.m.linton@lboro.ac.uk
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Abstract

The existence of a band-gap structure associated with water waves propagating over infinite periodic arrays of submerged horizontal circular cylinders in deep water is established. Waves propagating at right angles to the cylinder axes and at an oblique angle are both considered. In each case an exact linear analysis is presented with numerical results obtained by solving truncated systems of equations. Calculations for large finite arrays are also presented, which show the effect of an incident wave having a frequency within a band gap – with the amount of energy transmitted across the array tending to zero as the size of the array is increased. The location of the band gaps is not as predicted by Bragg's law, but we show that an approximate determination of their position can be made very simply if the phase of the transmission coefficient for a single cylinder is known.

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Papers
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Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 504 - 526
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
An, Z. & Ye, Z. 2004 Band gaps and localization of water waves over one-dimensional topographical bottoms. Appl. Phys. Lett. 84 (15), 2952–2954.CrossRefGoogle Scholar
Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2009 The interaction of flexural-gravity waves with periodic geometries. Wave Motion 46 (1), 57–73.CrossRefGoogle Scholar
Chamberlain, P. G. & Porter, D. 1995 Decomposition methods for wave scattering by topography with application to ripple beds. Wave Motion 22, 201–214.CrossRefGoogle Scholar
Chen, L.-S., Kuo, C.-H., Ye, Z. & Sun, X. 2004 Band gaps in the propagation and scattering of surface water waves over cylindrical steps. Phys. Rev. E 69, 066308.CrossRefGoogle ScholarPubMed
Chou, T. 1998 Band structure of surface flexural-gravity waves along periodic interfaces. J. Fluid Mech. 369, 333–350.CrossRefGoogle Scholar
Davies, A. G. 1982 The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans 6, 207–232.CrossRefGoogle Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.CrossRefGoogle Scholar
Dean, W. R. 1948 On the reflexion of surface waves by a submerged circular cylinder. Proc. Camb. Phil. Soc. 44, 483–491.CrossRefGoogle Scholar
Devillard, P., Dunlop, F. & Souillard, B. 1988 Localization of gravity waves on a channel with random bottom. J. Fluid Mech. 186, 521–538.CrossRefGoogle Scholar
Evans, D. V. 1990 The wide spacing approximation applied to multiple scattering and sloshing problems. J. Fluid Mech. 210, 647–658.CrossRefGoogle Scholar
Garnaud, X. & Mei, C. C. 2010 Bragg scattering and wave-power extraction by an array of small buoys. Proc. R. Soc. Lond. A 466 (2113), 79–106.Google Scholar
Hunter, G. K. 2004 Light is a Messenger: The Life and Science of William Lawrence Bragg. Oxford University Press.CrossRefGoogle Scholar
Levine, H. 1965 Scattering of surface waves by a submerged circular cylinder. J. Math. Phys. 6 (8), 1231–1243.CrossRefGoogle Scholar
Linton, C. M. & McIver, P. 2001 Handbook of Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.CrossRefGoogle Scholar
Markoš, P. & Soukoulis, C. M. 2008 Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press.CrossRefGoogle Scholar
McIver, P. 1990 The scattering of long water waves by a group of submerged, horizontal cylinders. Q. J. Mech. Appl. Math. 43 (4), 499–515.CrossRefGoogle Scholar
McIver, P. 2000 Water-wave propagation through an infinite array of cylindrical structures. J. Fluid Mech. 424, 101–125.CrossRefGoogle Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.CrossRefGoogle Scholar
Mei, C. C., Hara, T. & Naciri, M. 1988 Note on Bragg scattering of water waves by parallel bars on the seabed. J. Fluid Mech. 186, 147–162.CrossRefGoogle Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves. Part 1. Linear Aspects. World Scientific.Google Scholar
Newman, J. N. 1975 Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems. J. Fluid Mech. 71, 273–282.CrossRefGoogle Scholar
O'Leary, M. 1985 Radiation and scattering of surface waves by a group of submerged, horizontal, circular cylinders. Appl. Ocean Res. 7, 51–57.CrossRefGoogle Scholar
Porter, R. & Evans, D. V. 1998 The trapping of surface waves by multiple submerged horizontal cylinders. J. Engng Math. 34, 417–433.CrossRefGoogle Scholar
Porter, R. & Evans, D. V. 2006 Scattering of flexural waves by multiple narrow cracks in ice sheets floating on water. Wave Motion 43, 425–443.CrossRefGoogle Scholar
Porter, R. & Porter, D. 2003 Scattered and free waves over periodic beds. J. Fluid Mech. 483, 129–163.CrossRefGoogle Scholar
Schnute, J. T. 1967 Scattering of surface waves by submerged circular cylinders. Part II. Scattering by an infinite array of cylinders. Tech. Rep. 11. Department of Mathematics, Stanford University.Google Scholar
Schnute, J. T. 1971 The scattering of surface waves by two submerged cylinders. Proc. Camb. Phil. Soc. 69, 201–215.CrossRefGoogle Scholar
Shen, Y. & Zheng, Y. 2007 Interaction of oblique waves with an array of long horizontal circular cylinders. Sci. China E 50 (4), 490–509.CrossRefGoogle Scholar
Thorne, R. C. 1953 Multipole expansions in the theory of surface waves. Proc. Camb. Phil. Soc. 49, 707–716.CrossRefGoogle Scholar
Ursell, F. 1950 Surface waves on deep water in the presence of a submerged circular cylinder. Part I. Proc. Camb. Phil. Soc. 46, 141–152.CrossRefGoogle Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.CrossRefGoogle Scholar
Yang, S., Wu, F., Zhong, H. & Zhong, L. 2006 Large band gaps of water waves through two-dimensional periodic topography. Phys. Lett. A 352, 426–430.CrossRefGoogle Scholar