Skip to main content Accessibility help

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

  • C. M. LINTON (a1)


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.


Corresponding author

Email address for correspondence:


Hide All
Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
An, Z. & Ye, Z. 2004 Band gaps and localization of water waves over one-dimensional topographical bottoms. Appl. Phys. Lett. 84 (15), 29522954.
Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2009 The interaction of flexural-gravity waves with periodic geometries. Wave Motion 46 (1), 5773.
Chamberlain, P. G. & Porter, D. 1995 Decomposition methods for wave scattering by topography with application to ripple beds. Wave Motion 22, 201214.
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.
Chou, T. 1998 Band structure of surface flexural-gravity waves along periodic interfaces. J. Fluid Mech. 369, 333350.
Davies, A. G. 1982 The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans 6, 207232.
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419443.
Dean, W. R. 1948 On the reflexion of surface waves by a submerged circular cylinder. Proc. Camb. Phil. Soc. 44, 483491.
Devillard, P., Dunlop, F. & Souillard, B. 1988 Localization of gravity waves on a channel with random bottom. J. Fluid Mech. 186, 521538.
Evans, D. V. 1990 The wide spacing approximation applied to multiple scattering and sloshing problems. J. Fluid Mech. 210, 647658.
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), 79106.
Hunter, G. K. 2004 Light is a Messenger: The Life and Science of William Lawrence Bragg. Oxford University Press.
Levine, H. 1965 Scattering of surface waves by a submerged circular cylinder. J. Math. Phys. 6 (8), 12311243.
Linton, C. M. & McIver, P. 2001 Handbook of Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.
Markoš, P. & Soukoulis, C. M. 2008 Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press.
McIver, P. 1990 The scattering of long water waves by a group of submerged, horizontal cylinders. Q. J. Mech. Appl. Math. 43 (4), 499515.
McIver, P. 2000 Water-wave propagation through an infinite array of cylindrical structures. J. Fluid Mech. 424, 101125.
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315335.
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, 147162.
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves. Part 1. Linear Aspects. World Scientific.
Newman, J. N. 1975 Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems. J. Fluid Mech. 71, 273282.
O'Leary, M. 1985 Radiation and scattering of surface waves by a group of submerged, horizontal, circular cylinders. Appl. Ocean Res. 7, 5157.
Porter, R. & Evans, D. V. 1998 The trapping of surface waves by multiple submerged horizontal cylinders. J. Engng Math. 34, 417433.
Porter, R. & Evans, D. V. 2006 Scattering of flexural waves by multiple narrow cracks in ice sheets floating on water. Wave Motion 43, 425443.
Porter, R. & Porter, D. 2003 Scattered and free waves over periodic beds. J. Fluid Mech. 483, 129163.
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.
Schnute, J. T. 1971 The scattering of surface waves by two submerged cylinders. Proc. Camb. Phil. Soc. 69, 201215.
Shen, Y. & Zheng, Y. 2007 Interaction of oblique waves with an array of long horizontal circular cylinders. Sci. China E 50 (4), 490509.
Thorne, R. C. 1953 Multipole expansions in the theory of surface waves. Proc. Camb. Phil. Soc. 49, 707716.
Ursell, F. 1950 Surface waves on deep water in the presence of a submerged circular cylinder. Part I. Proc. Camb. Phil. Soc. 46, 141152.
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347358.
Yang, S., Wu, F., Zhong, H. & Zhong, L. 2006 Large band gaps of water waves through two-dimensional periodic topography. Phys. Lett. A 352, 426430.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

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

  • C. M. LINTON (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed