Modifications to acoustic eigenmodes in combustion chambers such as those of liquid propellant rocket engines, produced by spatial variations of density and sound speed that arise mainly through progress of combustion processes, are analyzed by using a variational method. The variational principle shows that the eigenvalue is the ratio of a weighted acoustic kinetic energy to a weighted acoustic potential energy, and the eigenfunction is the minimizing function of this ratio. A sample calculation is made for the case in which variations of the properties occur dominantly in the longitudinal direction, with lower temperatures and higher densities prevailing near the injector. The results of the calculation exhibit two major characteristics: the longitudinal density variation aids transfer of acoustic kinetic energy from a lower mode to the adjacent higher mode, so that the pure transverse modes have substantially larger reductions (sometimes exceeding 50%) of their eigenvalues than the combined modes; and variations of the acoustic pressure gradients are found to be larger in high-density regions, so that the acoustic pressure amplitude for purely tangential modes is found to be much higher near the injector than near the nozzle. The higher head acoustic pressure may contribute to the greater sensitivity of acoustic instability to characteristics of the flames near the injectors, as commonly found in engine tests. The improved acoustic eigensolutions can also be helpful in sizing damping devices, such as baffles or acoustic liners.