The behavior of surface waves in truncated periodic arrays is examined through analysis of the currents. The surface waves to be studied are guided by the perfectly conducting elements of the array itself and are to be distinguished from the dielectric slab-guided surface waves encountered elsewhere in the literature. The conditions under which surface waves may arise are given. The surface wave currents are extracted from the method of moments solution for the finite by an infinite array using a least squares algorithm. Surface wave excitation and reflection coefficients are then be determined from the data and compared with the semi-infinite array solution.