We derive a rigorous method for finding the best-connected orthogonal communication channels, modes, or degrees of freedom, for scalar waves between two volumes of arbitrary shape and position, explicitly without assuming planar surfaces or paraxial approximations. The communication channels are the solutions of two eigenvalue problems, and are identical to the cavity modes of a double phase-conjugate resonator. We also derive a sum rule for the connection strengths, the sum being a simple volume integral. We use these results to analyze rectangular prism volumes, small volumes, thin volumes in different relative orientations, and arbitrary near-field volumes, all situations where previous planar approaches fail for one or more reasons. We reproduce previous planar results explicitly, extending these to finite depth. Depth is shown not to increase the number of communications modes unless the volumes are close compared to their depth. We discuss how to estimate the connection strengths in some cases without a full solution of the eigenvalue problem, so estimates of the number of usable communications modes can be made from the sum rule. In general, the approach gives a rigorous basis for handling problems related to volume sources and receivers. It may be applicable especially in near-field problems, and in situations where volume is an intrinsic part of the problem.

Full text available for download