Asymptotic theory of nonlinear acoustic waves in a thermoacoustic prime-mover

A nonlinear evolution equation describing stabilization of oscillations in a thermoacoustic prime mover is derived. It predicts that, in the case of q uasiadiabatic interaction of sound and thermal waves inside an inhomogeneou sly heated acoustically thin thermoacoustic stack, sound amplification can be proportional to the square root of frequency and can be frequency-indepe ndent depending on the value of Kramer's constant. It takes into account wa ve velocity dispersion (additional to that caused by thermal and viscous bo undary layers) and also describes the shifts of resonance frequencies induc ed in the acoustic resonator by installation of the stack. The analytical d escription of the spectral characteristics of the thermoacoustic stack (val id for an arbitrary temperature distribution inside the stack) was found by transforming the differential equation for the sound propagation and backs cattering in inhomogeneous media into an equivalent Volterra integral equat ion of the second kind. The latter has a solution in the form of an iterati ve sequence that converges absolutely and uniformly to its exact solution. The derived evolution equation also includes the usual quasi-linear differe ntial term responsible for the nonlinear acoustic process in the gas. Analy tical solution of this equation (in the case of weak dispersion) for the th ermoacoustic prime mover filled with the gas, characterized by a small valu e of Kramer's constant, is found. Characteristic times of shock front forma tion, of wave amplitude stabilization and the characteristic amplitude of t he stationary wave (and also their dependence on the stack heating) are det ermined. Conditions, when a shock front formation caused by nonlinear acous tic processes is the dominant mechanism for the saturation of wave amplitud e are established.