Although eastward propagation has long been considered one of the essential features of the Madden-Julian waves, recent observations have revealed a stationary or quasi-stationary component in the oscillations, particularly in measures of the diabatic heating rate. Wave-CISK theories of the low-frequency oscillations have struggled to explain the observed period and vertical structure of the waves. On the other hand, theoretical and numerical studies have shown that low-frequency waves strongly resembling the observed oscillations can be excited by specified low-frequency oscillations of the convective heating. A problem with the latter set of theories is that the cause of the oscillatory heating has not been satisfactorily explained. It is proposed here that the observed low-frequency wave motions are the response to forcing by an essentially stationary, self-excited oscillating heat source that is produced by nonlinear interactions among radiation, cumulus convection, and the surface fluxes of sensible heat and moisture. Feedback of the large-scale motions on the latent heating is not required. Results from two very different one-dimensional models are presented to support this hypothesis. The physical processes included in the models are essentially the same, that is, radiation, cumulus convection, and the surface fluxes of sensible heat and moisture; the first model is highly simplified, however, while the second includes relatively sophisticated parameterizations of all the relevant physical processes. Results from both models show low-frequency oscillations of the latent heating, temperature, and moisture. Experiments show that the oscillations are favored by a warm sea surface and weak surface wind speeds, consistent with the observed conditions over the Indian Ocean and the tropical western Pacific Ocean.