Scattering by a semi-infinite lattice and the excitation of Bloch waves

The interaction of a time-harmonic plane wave with a semi-infinite lattice of identical circular cylinders is considered. No assumptions about the radius of the cylinders, or their scattering properties, are made. Multipole expansions and Graf’s addition theorem are used to reduce the boundary value problem to an infinite linear system of equations. Applying the z transform and disregarding interaction effects due to certain strongly damped modes then leads to a matrix Wiener–Hopf equation with rational elements. This is solved by a straightforward method that does not require matrix factorisation. Implementation of the method requires that the zeros of the matrix determinant be located numerically, and once this is achieved, all far field quantities can be calculated. Numerical results that show the proportion of energy reflected back from the edge are presented for several different lattice geometries. 1