It is shown in this work that two prominent intraseasonal oscillations of the tropical atmosphere-namely, the 3-4-day westward propagating wave observed over the equatorial Atlantic and Pacific Oceans and the quasi-biweekly (or 10-20 day) oscillation observed over the Indian Ocean summer monsoon region-can be understood as arising from selective excitation of the tropical normal modes in the presence of moist feedbacks and a moisture relaxation timescale. Unlike in earlier studies of a similar nature, the central theme of the present work is that the dynamics of the moisture variable is governed by a moisture relaxation timescale tau. In other words, the large-scale flow is not in exact quasi equilibrium with the precipitational heating. The implications of this hypothesis are investigated by using a shallow-water model of the tropical atmosphere on an equatorial beta plane, with a fixed vertical structure. The two major findings of the present work are: 1) both the 3-4-day wave and the 10-20-day wave can be understood as intrinsic modes of the tropical atmosphere, excited by the same basic mechanism-namely, moist feedbacks in the presence of a moisture relaxation timescale, and 2) while the 3-4-day wave is represented by a maximally growing mixed Rossby gravity wave, driven selectively unstable by moist feedbacks, the 10-20-day wave represents a new mode of the tropical atmosphere that is excited by the moist feedbacks in the presence of mean westerlies. For both of these waves, the agreement between observed structure and theoretical predictions is excellent. Thus, the present work presents a mechanism that explains two major low-frequency tropical oscillations as intrinsic modes of the tropical atmosphere.