A wave front perturbation method is applied to a one-dimensional two-phase flow in a horizontal duct to analyze the interfacial wave stability. The governing equations are thus reduced to a first-order ordinary differential equation (ODE). The linear solution of this ODE agrees with the conclusion from the small amplitude wave perturbation approach. The non-linear theory suggests that the linear stability result is sufficient for predicting instability in wavy how, and may over-predict the critical gas velocity for large disturbances. Accordingly, a weak non-linear stability criterion is obtained. The type of unstable wave, under-cut wave or roll wave, also can be predicted analytically.