The linear stability of a thermoacoustic system consisting of a looped tube having a regenerator and a branch resonator is studied numerically and experimentally as a function of the temperature ratio of the hot and cold parts of the regenerator. The characteristic complex angular frequency of the system is obtained as a solution of the basic equations of hydrodynamics. Results show that the system can possess two critical temperature ratios when the regenerator pore radius is smaller than the optimal value. Furthermore, the system can show a transition from a traveling-wave engine to a standing-wave engine as the pore size is increased. The experimental method for deriving the characteristic complex angular frequency of the system and the results are presented to support the validity of the numerical calculations.