Many scenarios in underwater acoustics involve radiation from, or scattering by, configurations with periodic or quasiperiodic features. Depending on the operating conditions, the acoustic field generated by these processes carries the gross imprint of periodicity or quasiperiodicity, without or with effects of truncation, regardless of the detailed structure in each unit cell. To understand and quantify the phenomenology under time harmonic and pulsed excitation, especially in the mid- to high-frequency regime where the width of each cell can cover many wavelengths, it is instructive to explore alternative parametrizations that emphasize, respectively, the nondispersive direct radiation from each cell and the dispersive collective treatment of strict or weakly perturbed periodicity, without and with truncations. The prototype configuration for this two-dimensional study is a finite periodic array of linearly phased parallel filamentary elements. The formulation makes use of Poisson summation and subsequent asymptotics applied to the finite array, and is parametrized in the configuration-spectrum phase space; it highlights the connection between local and global phenomena in both the space-time and wave-number-frequency domains, with a view toward phase-space data processing. Formal aspects and general principles are presented in this paper and are applied to infinite and truncated periodic arrays to illustrate how known results obtained by other methods are recovered with the Poisson-based algorithm. The outcome is a new frequency and time domain Bragg-modulated ray acoustic model that generalizes the nonuniform and uniform ray fields of the geometrical theory of diffraction, so as to include effects of periodic dispersion, with truncations. Under transient conditions, such dispersion gives rise to new time domain Bragg modes. © 1994, Acoustical Society of America. All rights reserved.