The sound pressure and the sound velocity in the near field of a linear loudspeaker array is proposed and investigated from a mathematical point of view. Four different piston models, employing circular, elliptical and (two different models of) rectangular pistons, are explicitly calculated and closed-form solutions using higher transcendental functions are provided. As a representative overview of results, near field directional diagrams for different frequencies, different distances of the field point, as well as for varying phases of signal, are shown. Based on these results some optimization methods for phases of signal, loudspeaker geometry and array geometry are developed.
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