Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T13:57:47.642Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 January 2017

Luke G. Bennetts Open the ORCID record for Luke G. Bennetts [Opens in a new window] ,
Malte A. Peter  and
Fabien Montiel
Luke G. Bennetts*
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
Malte A. Peter
Affiliation:
Institute of Mathematics, University of Augsburg, 86135 Augsburg, Germany Augsburg Centre for Innovative Technologies, University of Augsburg, 86135 Augsburg, Germany
Fabien Montiel
Affiliation:
Department of Mathematics and Statistics, University of Otago, Dunedin 9054, New Zealand
*
†Email address for correspondence: luke.bennetts@adelaide.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering Acoustics: Waves in random media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 813 , 25 February 2017 , pp. 508 - 527
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bennetts, L. G. 2011 Wave attenuation through multiple rows of scatterers with differing periodicities. SIAM J. Appl. Maths 71 (2), 540–558.Google Scholar
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), 1613–1633.Google Scholar
Bennetts, L. G. & Squire, V. A. 2009 Wave scattering by multiple rows of circular ice floes. J. Fluid Mech. 639, 213–238.CrossRefGoogle Scholar
Berry, M. V. & Klein, S. 1997 Transparent mirrors: rays, waves and localization. Eur. J. Phys. 18, 222–228.Google Scholar
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.Google Scholar
Callan, M., Linton, C. M. & Evans, D. V. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 51–64.Google Scholar
Clemmow, P. C. 1966 The Plane Wave Spectrum Representation of Electromagnetic Fields. IEEE Press.Google Scholar
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.Google Scholar
Evans, D. V. & Porter, R. 1997 Trapped modes about multiple cylinders in a channel. J. Fluid Mech. 339, 331–356.CrossRefGoogle Scholar
Evans, D. V & Porter, R. 1999 Trapping and near-trapping by arrays of cylinders in waves. J. Engng Maths 35, 149–179.Google Scholar
Hodges, C. H. & Woodhouse, J. 1983 Vibration isolation from irregularity in a nearly periodic structure: theory and measurements. J. Acoust. Soc. Am. 74 (3), 894–905.Google Scholar
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, 113–135.Google Scholar
Linton, C. M. & McIver, M. 2002 The existence of Rayleigh–Bloch surface waves. J. Fluid Mech. 470 (1994), 1–5.Google Scholar
Linton, C. M. & McIver, P. 2001 Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.Google Scholar
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), 1233–1258.Google Scholar
Maniar, H. D. & Newman, J. N. 1997 Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309–330.Google Scholar
Martin, P. A. 2006 Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press.Google Scholar
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, 630–651.Google Scholar
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, 492–522.Google Scholar
Peter, M. A. & Meylan, M. H. 2007 Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473–494.CrossRefGoogle Scholar
Peter, M. A. & Meylan, M. H. 2009 Water-wave scattering by vast field of bodies. SIAM J. Appl. Maths 70 (5), 1567–1586.Google Scholar
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, 233–258.Google Scholar
Sheng, P. 2006 Introduction to Wave Scattering, Localisation and Mesoscopic Phenomena, 2nd edn. Springer.Google Scholar
Sommerfeld, A. 1949 Partial Differential Equations in Physics, vol. 1. Academic.Google Scholar
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), 333–352.Google Scholar
Weaver, R. L. 1990 Anderson localization of ultrasound. Wave Motion 12 (2), 129–142.Google Scholar