## 1 Introduction  
Combustion instability is one of the most significant problems for low-emission combustion chambers and high-energy-density rocket engines currently under development. Combustion instability has a significant meaning in land-based gas turbine installations due to the widespread method of burning lean, preliminarily blended mixtures. This approach allows for significantly reducing the most harmful nitrous oxides (NOx) while sacrificing combustion stability [1]. Under these operating conditions, burners are severely damaged by self-exciting oscillations caused by combustion. This effect has been investigated with different types of combustion chambers in the following works [2; 3; 4]. Particularly, the instability problem emerges in the combustion of such a perspective type of fuel as hydrogen for both aviation engines and land-based gas turbine installations. Combustion instability, a coupling between resonant combustor acoustics and heat release rate fluctuations, is one of the leading challenges in developing and operating both aircraft and power-generation gas turbines [5]. Combustion stability and velocity of heat dissipation, depending also on acoustic oscillations, impact both the engine's characteristics and its reliability. It is also known that the method of vortex burning, which is commonly employed because of its numerous advantages, also has a tendency to instability onset, which is documented in the following sources [6; 7; 8; 9; 10]. Therefore, as for producing a complete mathematical model of self-oscillating processes in the combustion chamber, it is necessary to consider the total energy, including the acoustic component. In some devices, such as thermoacoustic engines and pulsejets, this instability may be used to produce work. In systems where combustion instability is inhibited, such phenomena produce work but are structurally destructive. Pressure oscillations, which occur in the combustion instability process, have high availability to produce work. Moreover, a system that uses pressure oscillations can be added to any internal combustion engine to create additional work. The first study for singing flames was produced by [PERSON] [11]. In his work, the criteria for maintaining the oscillations were elaborated: \"...it follows that at the phase of greatest condensation heat is received by the air, and at the phase of greatest rarefaction heat is given up from it, and thus there is a tendency to maintain the vibrations\". Analyzing combustion stability has been carried out in two ways. The first way was started by work [12] using [PERSON] ideas. The purpose of this work was a physical-based explanation of [PERSON] criteria by dividing the volume of the studied system into small parts and studying the interaction between them. It is assumed that these parts work as the cylinder and piston of a reciprocating engine and move with the main flow. These parts interact with acoustic waves; if parts produce work, then this work is used to increase the energy of acoustic waves; if these parts consume work, then it decreases the energy of acoustic waves. The second way was started from the [PERSON] interpretation [13] of Rayleigh criteria; in this work, mean flow was omitted from the study, and from the linear acoustic analysis for pressure wave equation, it was derived that the mean power added to the acoustic wave for a unit of volume is equal to \(\mathrm{N_{A}}\):  
\[\mathrm{N_{A}}=\lim_{\tau\to\infty}\frac{1}{\tau}\int\limits_{0}^{\tau}\frac{ (\mathrm{k}-1)}{\mathrm{k}\mathrm{P_{0}}}\mathrm{P^{\prime}}\mathrm{Q^{\prime }}\mathrm{dt}. \tag{1}\]  
[PERSON], in his work [14], divided acoustic emission from the combustor into combustion noise and self-excited fluctuation. The combustion noise does not have feedback from the sound emitted behind the combustion region, which does not influence the combustion process or species concentration. The self-excited fluctuations are produced by the right-phase correlation between combustion and reflected sound waves.