This chapter presents a number of analytic solutions to the problem of sound field synthesis in three and 2.5 dimensions, whereby continuous distributions of secondary sources are assumed. A focus lies on the explicit solution of the synthesis equation, which provides a perfect solution for enclosing secondary source distributions. The explicit solution is derived for spherical, circular, planar, and linear geometries. It is then shown that the well-known Near-field Compensated Higher Order Ambisonics approach is equivalent to the explicit solution for spherical secondary source distributions. The recently proposed Spectral Division Methods is identified as the extension of Near-field Compensated Higher Order Ambisonics to planar and linear secondary source distributions. Apart from the explicit solution, an implicit solution exists, which has become known as Wave Field Synthesis. The latter is derived from the Rayleigh Integral and its modern formulation for arbitrary complex secondary source distributions is outlined.
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