An acoustically consistent, linear modal analysis-based analytical method is presented to predict the longitudinal and transverse combustion instabilities in a two-dimensional Cartesian dump combustor. At first, rigorous acoustical analysis (without combustion) is performed of two duct configurations with one and two discontinuities in the cross-sectional area. Novel, acoustically consistent jump or matching conditions are developed and applied at the duct cross-sectional interface(s), with distinct forms for the purely axial and nonaxial modes. The effects of uniform and nonuniform mean flows, cross-sectional area ratio, as well as of different types of boundary conditions on the duct acoustic modes are investigated. Acoustic modal frequency predictions are in excellent agreement with the analytical and numerical results of Meissner (“Effect of Cross-Sectional Area Discontinuities in Closed Hard-Walled Ducts on Frequency of Longitudinal Modes,” Archives of Acoustics, Vol. 35, No. 3, 2010, pp. 421–435). In the second part, combustion instabilities of a two-dimensional Cartesian dump combustor are investigated. The instability analysis employs the developed acoustically consistent jump conditions instead of the conventional mass, momentum, and energy balance-based conditions. Effects of the fluctuating heat-release source term in the acoustic wave equation are incorporated directly into the longitudinal wave number, obviating the need for a separate energy matching condition across the flame. A detailed investigation of the parametric space and boundary conditions affecting combustion instabilities is undertaken, and the consistency of the modal analysis with the Rayleigh criterion is explicitly demonstrated. Further, the present approach enables the consideration of arbitrary mean flame shapes in determining the unstable modes. Instabilities are demonstrated for the fundamental longitudinal mode and its harmonics, as well as for the fundamental transverse mode. The effects of the cross-sectional area ratio and flow Mach number on the unstable-mode growth rates are also presented.