Previous theories modelling unsteady one-dimensional flames are extended so that the effect of upstream and downstream noise interference is included. Using the fact that the Mach number M is a small number, the principles of matched asymptotic expansions are used to link long wavelength acoustic disturbances at large distances [0(~l) on a diffusion length scale] from an anchored flame to small perturbations in velocity within the combustion zone. This matching process yields a frequency condition governing the flame vibration and a number of cases are illustrated in the paper. Acoustic feedback is modelled by considering particular boundary conditions (on an acoustic scale) at various finite lengths upstream, thus modelling (in a simplified way) practical burner systems. The effect of these conditions on stability predictions is illustrated in particular for the case of "open tube conditions" upstream. Sound emission without feedback is shown to increase the regions of stability, but with feedback present tube modes are introduced and the stability is significantly altered. By including forcing terms, the effect on amplitude and phase on the resulting acoustic signal is investigated. Some comparisons with experimental investigations are made.
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A. C. MclNTOSH (1986) studied this question.
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