To describe the acoustic instabilities in the combustion chambers of laterally burning solid propellant rockets the interaction of the mean flow with the acoustic waves is analysed, using multiple scale techniques, for realistic cases in which the combustion chamber is slender and the nozzle area is small compared with the cross-sectional area of the chamber. Associated with the longitudinal acoustic oscillations we find vorticity and entropy waves, with a wavelength typically small compared with the radius of the chamber, penetrating deeply into the chamber. We obtain a set of differential equations to calculate the radial and axial dependence of the amplitude of these waves. The boundary conditions are provided by the acoustic admittance of the propellant surface, given by an existing analysis of the thin gas-phase reaction layer adjacent to the solid-gas interface, and of the nozzle, accounting here for the possible effect of the vorticity and entropy waves. The equations are integrated in closed form and the results provide the growth rate of the disturbances, which we use to determine the conditions for instability of the longitudinal oscillations.