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A mode-based, acoustic low-order network model for rocket chambers with resonator ring is introduced. This model involves effects of dissipation as well as scattering and mode coupling associated with a resonator ring. Discontinuities at the interface between acoustic elements are treated with integral mode-matching conditions. Eigenfrequencies and accordingly the linear system stability can be determined with the generalized Nyquist plot method based on the network model. Because of low computational cost, parameter studies can be performed in a reasonable time. An additional chamber eigenmode is observed for well-tuned resonators. Both the original and the additional mode show favorable stability properties for a generic test chamber. The optimal length and number of cavities are identified for that chamber. The need for analyzing the coupled system of resonators and combustion chamber in the design process is made evident by the discrepancy between detected values and those of an a priori consideration. Derived stability maps demonstrate that the region of stable operation is increased considerably by inserting well-tuned resonators. The destabilizing influence of temperature deviations in the cavities is quantified. Such sensitivities to modifications of design conditions can be extenuated by a ring configuration with several nonidentical resonator types. Moreover, the strong impact of the nonlinear correlation factor in the resonator modeling on the overall system stability is worked out. Nomenclature A = system matrix of network model c = speed of sound d = diameter of quarter-wave tube F mn , G mn = amplitudes of up-and downstream traveling waves f mn , g mn = up-and downstream traveling waves i = imaginary unit equal to −1 p J m = mth Bessel function of first kind k mn = axial wave number L = length of the thrust chamber l, l e , l r = geometric, effective, and equivalent mass length of quarter-wave resonator M = Mach number N = number of modes considered in mode matching n = pressure interaction index, Eq. (16) n R = number of cavities placed in resonator ring p 0 = acoustic fluctuating pressure _ Q = heat release R = radius of thrust chamber; reference length scale r in , r out = radius of curvature at nozzle entrance and throat, respectively T = transfer matrix T = cycle t = time U = mean flow velocity x, r, θ = axial, radial, and tangential coordinates x F , x R = axial position of flame front and resonator ring center, respectively Z = Θ iΨ acoustic impedance α = radial wave numbers Γ = growth rate, Eq. (19) γ = isentropic exponent ΔT = deviation from design temperature ϵ nl = nonlinear resistance factor θ nozzle = nozzle half-angle κ = acoustic balancing parameter defined by Eq. (14) λ mn = nth roots of the derivative of the mth Bessel function of first kind μ = dynamic viscosity Ξ = excess temperature ξ = ratio of specific impedances ρ = density τ = combustion time lag, Eq. (16) Ω = ω iϑ complex valued angular frequency ω ref = angular frequency of the undamped 1T1L mode; used to normalize frequenc...

We develop a linearized imaging theory that combines the spatial, temporal, and spectral components of multiply scattered waves. We consider the case of multiple fixed transmitters and receivers. We incorporate a priori information about the multipathing environment, which in our case consists of a fixed reflector. The scene of interest consists of a distribution of moving objects. We base our imaging technique on a modification of the filteredbackprojection method that produces a phase-space image. We characterize the performance of the imaging system by means of its point-spread function, and we analyze the effect of parameters such as sensor positioning and waveforms. We ultimately find that enhanced phase-space resolution can be achieved using multiple sensors and appropriate waveforms.

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