The acoustoelectric effect, besides producing linear gain or loss of a single acoustic wave, couples together different acoustic waves. In this paper we first derive expressions for the second-order field acting on a given wave due to acoustoelectric interaction with other waves, either collinear or phase matched (and therefore noncollinear). These fields are obtained also for the case that trapped as well as free carriers are involved in acoustoelectric interaction. The wave equations are then set up for three coupled waves and solved for downconversion and upconversion, with pump depletion due to the interaction with other waves generally, but not always, neglected. In the absence of linear gain or loss, the coupled equations and solutions are quite similar to those of nonlinear optics, with displacements playing the role of electric fields. Many of the results derived in nonlinear optics are therefore easily adapted to the acoustoelectric case. The presence of sizable linear gains or losses, generally different for the different frequencies, changes the form of the solution, and also has the consequence that phase matching loses much of its importance. The dependence of gain rates for downconversion and upconversion on the frequencies of the waves involved, on the applied dc field, on the conductivity of the material, and on the trapping parameters is investigated. Detailed plots are given for some particular cases for which there are experimental data. Finally, the theory is compared with two sets of experimental data: (i) data of Zemon and Zucker, who studied the generation of subharmonics, second harmonics, and sum frequencies by a 1-GHz pump in CdS and (ii) data from many sources on the evolution of the noise frequency spectrum in moving domains in CdS and GaAs. It is found that the theory gives a good qualitative account of many of the observed phenomena.