Bragg scattering by a line array of small cylinders in a waveguide. Part 1. Linear aspects

Abstract

Motivated by potential applications for offshore airports supported on vertical piles, we report a theory of wave diffraction by a periodic array of circular cylinders. The simple case of normal incidence on a rectangular array is studied here, which is equivalent to a line array along the centre of a long channel. An asymptotic theory is developed for cylinders much smaller than the incident wavelength, which is comparable to the cylinder spacing. Focus is on Bragg resonance near which scattering is strong. A combination of the method of multiple scales and the Bloch theorem leads to simple evolution equations coupling the wave envelopes. Dispersion of transient wave envelopes is investigated. Scattering of detuned waves by a large but finite number of cylinders is investigated for frequencies in and outside the band gap. Quantitative accuracy is assessed by comparisons with numerical computations via finite elements. The analytical theory prepares the ground for nonlinear studies and may facilitate future inclusion of real-fluid effects such as vortex shedding.