In earlier work [Watanabe et al., J. Acoust. Soc. Am. 102, 3484–3496 (1997)] a quasi-one-dimensional nonlinear model for thermoacoustic devices was developed. The model reduces exactly to the well-known Rott theory upon linearization, but numerical work has shown that it is also able to predict nonlinear features in agreement with experiment. In this work, a weakly nonlinear analysis of the model is carried out for conditions close to the linear stability threshold. The level at which the amplitude of the linearly unstable perturbation is predicted to saturate compares favorably with numerical calculations and experiment. On the basis of the result, the effect of several design variables such as nonuniformity of the resonator cross section, stack plate spacing, and others is explored. [Work supported by the Office of Naval Research.]