Scattering-matrix analysis of linear periodic arrays

A spherical-wave source scattering-matrix description of acoustic radiators, along with reciprocity and power conservation, is applied to analyze infinite and finite linear periodic arrays that support traveling waves. We prove that for a general linear periodic array of small radiators, a traveling wave must be a slow wave with a propagation constant /spl beta/ greater than the propagation constant k of the medium in which the array is located. For an infinite periodic linear array of small isotropic radiators, the scattering-matrix analysis leads to a closed-form expression for the propagation constant of the traveling wave in terms of the normalized separation distance kd and the phase of the effective scattering coefficient of the array elements. These two parameters are the only critical variables in the N/spl times/N matrix equation, for N radiation coefficients, that is derived for a finite linear array of N elements. Resonances in the curves of total power radiated versus kd for a finite array excited with one feed element demonstrate the existence of the traveling wave predicted for the corresponding infinite array. The computed power curves, as well as directivity patterns, illustrate that the finite array becomes a more efficient endfire radiator as /spl beta/ approaches the value of k. The maximum attainable endfire directivity of a finite array with a single feed element is a monotonically increasing function of the phase velocity of the traveling wave, and this function is practically independent of the parameters of the array used to obtain this phase velocity. The basic formulation applies to any array composed of linear, reciprocal, lossless array elements, such as small linear periodic antennas.