The appearance of nonlinear acoustic oscillations in pulse combustors and other unsteady combustion devices arises from combustion-driven instabilities that excite one or more classical acoustic modes of the system. For sufficiently strong acoustic driving relative to various damping processes, these disturbances grow to finite amplitudes and, owing to the nonlinear coupling between linearly unstable and stable modes, a stable limit-cycle oscillation is typically established. The nonlinear dynamics of these oscillations are formally governed by an infinitely coupled system of evolution equations for the complex mode amplitudes. The linear terms determine the relative growth and decay rates of infinitesimal perturbations, while the nonlinear coupling terms determine the ultimate amplitude of each mode. The present work describes the nonlinear acoustic response of the system by obtaining approximate analytical and numerical solutions of the dynamical system of amplitude equations. In particular, it is shown that, depending on the value of a reduced driving parameter, various finite-mode approximations to the full infinitely coupled system may be employed to describe the acoustic response of the model. The nature of acoustic mode interactions is also investigated, and it is formally demonstrated that a resonance-like coupling exists between any growing mode, whose frequency is ~oj, and its first resonant harmonic, which is the mode whose frequency is 3ยขoj. Thus, the first acoustic bifurcation of the system occurs at a critical value of the driving parameter for which some mode achieves a positive linear growth rate, and is governed by a decoupled subsystem for the two modes corresponding to these frequencies. At larger values of the driving parameter, a second mode may achieve a positive linear growth rate, which in turn may produce a secondary transition in the acoustic response. The results for two cases, corresponding to whether or not one growing mode is the first resonant harmonic of the other, are discussed in terms of the minimal number of modes required to correctly predict the nonlinear acoustic behavior of the system.