Three-dimensionsal (two spatial dimension and time) partial differential equations of mass, momentum, and energy conservation have been integrated numerically by finite difference techniques over a sufficiently long time interval to allow convective motion to develop from computer round-off error. Adiabatic motion is not assumed. Except where specifically mentioned, attention is confined to linear results, by which is meant that the process that limit convection are negligible. The effects of several assumptions are discussed, and the method of solution is applied to a particular situation in which most of the assumptions are well satisfied: a shallow convective envelope with extensive convective motion only in the second helium ionization zone. The size of the convective cells, the evidence for the existence of only one growth rate of convective motion, the pressure perturbations, and the effects of nonadiabaticity are discussed. A comparison between the convective flux predicted by the computations and that from the standard mixing length theory is made, in which it is found that the mixing length theory predicts too little convective flux near the upper boundary of the convective region. Finally, the effects of nonlinear terms are discussed, and it is found that these terms tend to drive the convectivemore » motion toward a steady state even though the particle and radiative viscous dissipation is negligible on this scale of motion. (AIP)« less