1. Marginal conditions for thermoacoustic oscillations in resonators by Nobumasa Sugimoto, and Ryota Takeuchi Proceedings A Volume 465(2111):3531-3552 November 8, 2009 ©2009 by The Royal Society

2. Illustration of a Sondhauss tube consisting of a straight tube of length L and radius R as a neck, and of a cavity of volume V in the form of a bulb, respectively. Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

3. Illustration of an acoustic field in the neck, which is divided into a boundary layer on the wall and an acoustic main-flow region outside of it, with the tube wall being subjected to a temperature distribution Te(x) axially. Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

4. Illustration of a dumbbell-shaped tube consisting of two identical Sondhauss tubes shown in figure 1 connected at the open ends with their axes common, where the wall temperature Te varies along the neck from T0 in the middle to TL at both junctions symmetr... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

5. Graphs of the angular frequency σ of the neutral oscillations of the lowest antisymmetric and symmetric modes in the dumbbell-shaped tube against the temperature ratio TL/T0 [(1+λ)2] for various values of κ (=0,0.1,0.2,0.5,1,2,4 and 10), where the relation... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

6. Graphs of the angular frequency σ against the temperature ratio TL/T0 [=(1+λ)2] as real solutions to equation (3.50) for the antisymmetric mode in the dumbbell-shaped tube filled with air (C=1.47, CT=1.14) for various values of κ (=0,0.1,0.2,0.5,1,2,4 and 1... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

7. Marginal curves of the temperature ratio TL/T0 [=(1+λ)2] against the tube radius relative to the thickness of the boundary layer at the open end R/(ν0/ω)1/2 for the antisymmetric mode in the dumbbell-shaped tubes filled with air (C=1.47, CT=1.14) for variou... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

8. Graphs of the imaginary part σi of the complex angular frequency σ (= σr+iσi) against the tube radius relative to the thickness of the boundary layer at the open end R/(ν0/ω)1/2 for the antisymmetric mode in the dumbbell-shaped tube filled with air (C=1.47,... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

9. Graphs of the imaginary part σi of the complex angular frequency σ (=σr+iσi) against the tube radius relative to the thickness of the boundary layer at the open end R/(ν0/ω)1/2 for the antisymmetric mode in the dumbbell-shaped tubes filled with air (C=1.47,... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

10. Graphs of the imaginary part σi of the complex angular frequency σ (=σr+iσi) against the real part σr for the second branch of the antisymmetric mode in the dumbbell-shaped tubes filled with air (C=1.47, CT=1.14) for several temperature ratios TL/T0 (=2,3,4... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

11. Graphs of the imaginary part σi of the complex angular frequency σ (=σr+iσi) against the real part σr for the lowest and second branches of the symmetric mode in the dumbbell-shaped tube filled with air (C=1.47, CT=1.14) and κ=1 for several temperature rati... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society

12. Marginal curves of the temperature ratio TL/T0 [(1+λ)2] against the tube radius relative to the thickness of the boundary layer at the open end R/(ν0/ω)1/2 for the lowest branch of the Sondhauss tubes filled with nitrogen gas (C=1.46, CT=1.12) for various v... Nobumasa Sugimoto, and Ryota Takeuchi Proc. R. Soc. A 2009;465:3531-3552 ©2009 by The Royal Society