Instabilities of confined combustion systems are often discussed in terms of the Rayleigh criterion, which provides a necessary condition for unstable operation and is commonly used to distinguish driving and damping regions. The analysis is also carried out in some cases by making use of an acoustic energy balance in which the Rayleigh term acts as a source. The case of unconfined flames is less well documented but of importance in practical systems used in heating and drying. This study is motivated by problems of self-sustained oscillations of radiant burners for domestic or industrial processes and of various other types of open flames. Application of the Rayleigh criterion and of the balance of acoustic energy to oscillations arising in such unconfined systems is examined. The objective is to see if the Rayleigh condition is fulfilled and to show how the different perturbed variables are linked to each other to develop an unstable oscillation. These issues are investigated by experiments in two geometries. The first case relates to a single "V"- or "M"-shaped flame formed by a burner behaving like a Helmholtz resonator. The second geometry features a collection of conical flames (CCF) established by a multipoint injector. This system is fed by a manifold that features a set of plane modes and resonates like an organ pipe at frequencies corresponding to odd multiples of the quarter wave. The Rayleigh criterion and a related result written in the form of an acoustic energy balance are used to define conditions of instability. A link is established between the pressure signal radiated by the burner and the total heat release rate perturbation yielding the phase lag between these two variables and providing conditions for unstable operation. Systematic experiments carried out in the two burner geometries and model predictions are in good agreement indicating that the Rayleigh source term is positive and that the criterion is well fulfilled by the wavefield component corresponding to the pressure radiated back by the resonator. The gain and loss terms in the acoustic energy budget are estimated in one typical case, indicating that the source term is balanced by the acoustic power radiated away from the flame region, while dissipation of acoustic energy in the burner is shown to be essentially negligible. (c) 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.