Skip to main content Accessibility help

Localisation of Rayleigh–Bloch waves and damping of resonant loads on arrays of vertical cylinders

  • Luke G. Bennetts (a1), Malte A. Peter (a2) (a3) and Fabien Montiel (a4)


Linear potential-flow theory is used to study loads imposed on finite line arrays of rigid, bottom-mounted, surface-piercing, vertical cylinders by surface water waves. Perturbations in the cylinder locations are shown to damp the resonant loads experienced by the unperturbed array. A relationship is established between the damping and the phenomenon of Anderson localisation. Specifically, the Rayleigh–Bloch waves responsible for the resonant loads are shown to attenuate along the array when perturbations are introduced, resulting in localisation when the attenuation rate is sufficiently large with respect to the array length. Further, an efficient solution method for line arrays is introduced that captures the Rayleigh–Bloch wave modes supported by unperturbed arrays from the scattering characteristics of an individual cylinder.


Corresponding author

Email address for correspondence:


Hide All
Bennetts, L. G. 2011 Wave attenuation through multiple rows of scatterers with differing periodicities. SIAM J. Appl. Maths 71 (2), 540558.
Bennetts, L. G. & Peter, M. A. 2013 Spectral analysis of wave propagation through rows of scatterers via random sampling and a coherent potential approximation. SIAM J. Appl. Maths 73 (4), 16131633.
Bennetts, L. G. & Squire, V. A. 2009 Wave scattering by multiple rows of circular ice floes. J. Fluid Mech. 639, 213238.
Berry, M. V. & Klein, S. 1997 Transparent mirrors: rays, waves and localization. Eur. J. Phys. 18, 222228.
Botten, L. C., White, T. P., Asatryan, A. A., Langtry, T. N., de Sterke, C. M. & McPhedran, R. C. 2004 Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory. Phys. Rev. E 70, 056606.
Callan, M., Linton, C. M. & Evans, D. V. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 5164.
Clemmow, P. C. 1966 The Plane Wave Spectrum Representation of Electromagnetic Fields. IEEE Press.
Colquitt, D. J., Craster, R. V., Antonakakis, T. & Guenneau, S. 2015 Rayleigh–Bloch waves along elastic diffraction gratings. Proc. R. Soc. Lond. A 471, 20140465.
Evans, D. V. & Porter, R. 1997 Trapped modes about multiple cylinders in a channel. J. Fluid Mech. 339, 331356.
Evans, D. V & Porter, R. 1999 Trapping and near-trapping by arrays of cylinders in waves. J. Engng Maths 35, 149179.
Hodges, C. H. & Woodhouse, J. 1983 Vibration isolation from irregularity in a nearly periodic structure: theory and measurements. J. Acoust. Soc. Am. 74 (3), 894905.
Kagemoto, H., Murai, M., Saito, M., Molin, B. & Malenica, Š. 2002 Experimental and theoretical analysis of the wave decay along a long array of vertical cylinders. J. Fluid Mech. 456, 113135.
Linton, C. M. & McIver, M. 2002 The existence of Rayleigh–Bloch surface waves. J. Fluid Mech. 470 (1994), 15.
Linton, C. M. & McIver, P. 2001 Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.
Linton, C. M., Porter, R. & Thompson, I. 2007 Scattering by a semi-infinite periodic array and the excitation of surface waves. SIAM J. Appl. Maths 67 (5), 12331258.
Maniar, H. D. & Newman, J. N. 1997 Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309330.
Martin, P. A. 2006 Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press.
Montiel, F., Squire, V. A. & Bennetts, L. G. 2015 Evolution of directional wave spectra through finite regular and randomly-perturbed arrays of scatterers. SIAM J. Appl. Maths 75, 630651.
Montiel, F., Squire, V. A. & Bennetts, L. G. 2016 Attenuation and directional spreading of ocean wave spectra in the marginal ice zone. J. Fluid Mech. 790, 492522.
Peter, M. A. & Meylan, M. H. 2007 Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473494.
Peter, M. A. & Meylan, M. H. 2009 Water-wave scattering by vast field of bodies. SIAM J. Appl. Maths 70 (5), 15671586.
Porter, R. & Evans, D. V. 1999 Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386, 233258.
Sheng, P. 2006 Introduction to Wave Scattering, Localisation and Mesoscopic Phenomena, 2nd edn. Springer.
Sommerfeld, A. 1949 Partial Differential Equations in Physics, vol. 1. Academic.
Thompson, I., Linton, C. M. & Porter, R. 2008 A new approximation method for scattering by long finite arrays. Q. J. Mech. Appl. Maths 61 (3), 333352.
Weaver, R. L. 1990 Anderson localization of ultrasound. Wave Motion 12 (2), 129142.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Localisation of Rayleigh–Bloch waves and damping of resonant loads on arrays of vertical cylinders

  • Luke G. Bennetts (a1), Malte A. Peter (a2) (a3) and Fabien Montiel (a4)


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.