The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on March 27, 2008
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbn006
A NEW APPROXIMATION METHOD FOR SCATTERING BY LONG FINITE ARRAYS
( Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU )
( Department of Mathematics, University of Bristol, Bristol BS8 1TW )
i.thompson{at}lboro.ac.uk
Received 6 September 2007. Revise 25 January 2008. Accepted 4 February 2008.
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Abstract |
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The scattering of water waves by a long array of evenly spaced, rigid, vertical circular cylinders is analysed under the usual assumptions of linear theory. These assumptions permit the reduction of the problem to that of solving the Helmholtz equation in two dimensions, with appropriate circular boundaries. Our primary goal is to show how solutions obtained for semi-infinite arrays can be combined to provide accurate and numerically efficient solutions to problems involving long, but finite, arrays. The particular diffraction problem considered here has been chosen both for its theoretical interest and for its applicability. The design of offshore structures supported by cylindrical columns is commonplace and understanding how the multiple interactions between the waves and the supports affect the field is clearly important. The theoretical interest comes from the fact that, for wavelengths greater than twice the geometric periodicity, the associated infinite array can support Rayleigh–Bloch surface waves that propagate along the array without attenuation. For a long finite array, we expect to see these surface waves travelling back and forth along the array and interacting with the ends. For particular sets of parameters, near-trapping has previously been observed and we provide a quantitative explanation of this phenomenon based on the excitation and reflection of surface waves by the ends of the finite array.