The two-dimensional problem of acoustic scattering of an incident plane wave by a semi-infinite array of either rigid or soft circular scatterers is solved. Solutions to the corresponding infinite array problems are used, together with a novel filtering approach, to enable accurate solutions to be computed efficiently. Particular attention is focussed on the accurate determination of the amplitude of the Rayleigh-Bloch waves that can be excited along the array. The far-field away from the array is shown to be the sum of a finite number of plane waves propagating in different directions (the number depending on the observation angle) and a circular wave emanating from the edge of the array. A uniform asymptotic expansion is derived which varies continuously across the shadow boundaries that exist. We consider resonant (when one wave propagates along or directly away from the array) and non-resonant cases.