A new approximation method for scattering by long finite arrays

The scattering of water waves by a long array of rigid vertical circular cylinders is analysed under the usual assumptions of linear theory. 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 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, for the first time, a quantitative explanation of this phenomenon based on the excitation and reflection of surface waves by the ends of the finite array.