Marginal conditions for thermoacoustic oscillations in resonators

This paper examines marginal conditions for the onset of thermoacoustic oscillations in resonators of a Sondhauss tube and a gas-filled, dumbbell-shaped tube. An analysis is performed using a linear acoustic theory based on the first-order boundary-layer approximation. When a parabolic temperature distribution is assumed along the tube's neck, a frequency equation is available analytically, whose complex solutions are examined numerically. In the case of the dumbbell-shaped tube, two modes of oscillations exist, one being an antisymmetric mode and the other a symmetric one for pressure variations in the neck, while in the case of the Sondhauss tube, only a mode corresponding to the antisymmetric one exists due to the boundary condition at the open end. Marginal conditions are sought numerically, not only for the lowest branches of both modes, but also for second higher branches, but they are available only for the lowest branch of the antisymmetric mode. For the Sondhauss tube, marginal conditions are obtained by taking account of radiation into free space. Some discussions are provided in comparison with experiments.