Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-27T15:36:17.663Z Has data issue: false hasContentIssue false

Water-wave scattering by a semi-infinite periodic array of arbitrary bodies

Published online by Cambridge University Press:  07 March 2007

MALTE A. PETER
Affiliation:
Centre for Industrial Mathematics, FB3, University of Bremen, Germanympeter@math.uni-bremen.de
MICHAEL H. MEYLAN
Affiliation:
Department of Mathematics, University of Auckland, New Zealandmeylan@math.auckland.ac.nz

Abstract

We consider the scattering by a semi-infinite array of bodies of arbitrary geometry excited by an incident wave in the linear water-wave formulation (which reduces to the simpler case of Helmholtz scattering if the depth dependence can be removed). The theory presented here is extremely general, and we present example calculations for an array of floating elastic plates (a highly non-trivial scatterer). The solution method follows closely from the solution for point scatterers in a medium governed by Helmholtz's equation. We have made several extensions to this theory, considering water-wave scattering, allowing for bodies of arbitrary scattering geometry and showing how to include the effects of bound waves (called Rayleigh–Bloch waves in the water-wave context) in the formulation. We present results for scattering by arrays of cylinders that show the convergence of our methods and also some results for the case of scattering by floating elastic plates and fixed docks.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Abramowitz, M. & Stegun, I. A., (Eds.) 1970 Handbook of Mathematical Functions. Dover.Google Scholar
Hills, N. L. & Karp, S. N. 1965 Semi-infinite diffraction gratings I. Comm. Pure Appl. Maths 18, 203233.CrossRefGoogle Scholar
vonIgnatowsky, W. Ignatowsky, W. 1914 Zur Theorie der Gitter. Ann. Phys. 44, 369436.Google Scholar
Kagemoto, H. & Yue, D. K. P. 1986 Interactions among multiple three-dimensional bodies in water waves: an exact algebraic method. J. Fluid Mech. 166, 189209.CrossRefGoogle Scholar
Linton, C. M. 1998 The Green's function for the two-dimensional Helmholtz equation in periodic domains. J. Engng Maths 33, 377402.CrossRefGoogle Scholar
Linton, C. M. 2006 Schlömilch series that arise in diffraction theory and their efficient computation. J. Phys. A: Math. Gen. 39, 33253339.Google Scholar
Linton, C. M. & Evans, D. V. 1993 The interaction of waves with a row of circular cylinders. J. Fluid Mech. 251, 687708.CrossRefGoogle Scholar
Linton, C. M. & Martin, P. A. 2004 Semi-infinite arrays of isotropic point scatterers. A unified approach. SIAM J. Appl. Maths 64 (3), 10351056.CrossRefGoogle Scholar
Linton, C. M. & McIver, M. 2002 The existence of Rayleigh-Bloch surface waves. J. Fluid Mech. 470, 8590.CrossRefGoogle Scholar
McIver, P. 2002 Wave interaction with arrays of structures. Appl. Ocean Res. 24, 121126.CrossRefGoogle Scholar
Meylan, M. H. 2002 Wave response of ice floes of arbitrary geometry. J. Geophys. Res. – Oceans 107 (C1), doi: 10.1029/2000JC000713.CrossRefGoogle Scholar
Peter, M. A. & Meylan, M. H. 2004 Infinite-depth interaction theory for arbitrary floating bodies applied to wave forcing of ice floes. J. Fluid Mech. 500, 145167.CrossRefGoogle Scholar
Peter, M. A., Meylan, M. H. & Chung, H. 2004 Wave scattering by a circular elastic plate in water of finite depth: a closed form solution. Intl J. Offshore Polar Engng 14 (2), 8185.Google Scholar
Peter, M. A., Meylan, M. H. & Linton, C. M. 2006 Water-wave scattering by a periodic array of arbitrary bodies. J. Fluid Mech. 548, 237256.CrossRefGoogle 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, 233258.CrossRefGoogle Scholar
Porter, R. & Evans, D. V. 2005 Embedded Rayleigh-Bloch surface waves along periodic rectangular arrays. Wave Motion 43, 2950.CrossRefGoogle Scholar
Thompson, I. & Linton, C. M. 2006 Resonant effects in scattering by periodic arrays. In Proc. 21st Intl Workshop on Water Waves and Floating Bodies, pp. 173176.Google Scholar
Twersky, V. 1962 On scattering of waves by the infinite grating of circular cylinders. IRE Trans. Antennas and Propagation 10, 737765.CrossRefGoogle Scholar