An eigen analysis of the equatorial air-sea coupled model is carried out to understand the mechanism of the slowly varying mode for various zonal phase differences between SST and wind stress. The frequency and growth rate of the slow mode highly depend on the zonal phase difference between SST and wind stress anomalies and the wave scale. For ultra-long waves longer than 20,000 km, the system propagates westward regardless of the position of wind stress. However, for the long waves observed in the Pacific, the slow mode tends to Propagate eastward when the SST and wind stress anomalies are close to each other (within a quadrature phase relationship). On the other hand, when the wind stress is located far away from SST, the slow mode tends to propagate westward. The coupled system produces the unstable modes when the westerly (easterly) wind stress is located in the west of warm (cold) SST. It is noted that for the Pacific basin scale, the eastward propagating unstable waves can be produced when the wind stress is located to the west of SST with a few thousand kilometer distance. Also examined in the present study is the relative role of the thermocline displacement and zonal advection effects in determining the propagation and instability of the coupled system. An eigen analysis of the equatorial air-sea coupled model is carried out to understand the mechanism of the slowly varying mode for various zonal phase differences between SST and wind stress. The frequency and growth rate of the slow mode highly depend on the zonal phase difference Between SST and wind stress anomalies and the wave scale. For ultra-long waves longer than 20,000 km, the system propagates westward regardless of the position of wind stress. However, for the long waves observed in the Pacific, the slow mode tends to Propagate eastward when the SST and wind stress anomalies are close to each other (within a quadrature phase relationship). On the other hand, when the wind stress is located far away from SST, the slow mode tends to propagate westward. The coupled system produces the unstable modes when the westerly (easterly) wind stress is located in the west of warm (cold) SST. It is noted that for the Pacific basin scale, the eastward propagating unstable waves can be p roduced when the wind stress is located to the west of SST with a few thousand kilometer distance. Also examined in the relative role of the thermocline displacement and zonal advection effects in determining the propagation and instability of the coupled system.