We derive the coupled mode equations governing the Thompson–Quate experiment, that is, the parametric interaction of two counterpropagating bulk acoustic waves and a spatially uniform electric field oscillating at the second harmonic of the acoustic frequency. The derivation applies to propagation of any acoustic mode type in any direction in any piezoelectric or pyroelectric crystal. A new result of the derivation is a general but explicit expression for the material interaction coefficient that governs the strength of the process. We also find that the equations differ from the generic equations assumed by Thompson and Quate by the replacement of a phase velocity with the component of the group velocity normal to the crystal surface. As an aid to this derivation we also derive the coupled mode equations governing the parametric interaction of three acoustic waves. This derivation applies to the propagation of any mode types in any directions (consistent with being close to or at phase matching) in any dielectric, piezoelectric, or pyroelectric crystal.