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Hydrodynamic modes in the turbulent mixing layer over a cavity can constructively interact with the acoustic modes of that cavity and lead to aeroacoustic instabilities. The resulting limit cycles can cause undesired structural vibrations or noise pollution in many industrial applications. To further the predictive understanding of this phenomenon, we propose two physics-based models which describe the nonlinear aeroacoustic response of a side branch aperture under harmonic forcing with variable acoustic pressure forcing amplitude pa. One model is based on Howe's classic vortex sheet formulation, and the other on an assumed vertical velocity profile in the side branch aperture. These models are validated against experimental data. Particle image velocimetry (PIV) was performed to quantify the turbulent and coherent fluctuations of the shear layer under increasing pa. The specific acoustic impedance Z of the aperture was acquired over a range of frequencies f for different bulk flow velocities U and acoustic pressure forcing amplitudes pa. We show that, once the handful of parameters in the two models for Z have been calibrated using experimental data at a given condition, it is possible to make robust analytical predictions of this impedance over a broad range of f, U and pa. In particular, the models allow prediction of a necessary condition for instability, implied by negative values of the acoustic resistance Re(Z). Furthermore, we demonstrate that the models are able to describe the nonlinear saturation of the aeroacoustic response caused by alteration of the mean flow at large forcing amplitudes, which was recently reported in literature. This effect stabilizes the coupling between the side branch opening and the acoustic field in the cavity, and its quantitative description may be of value for control of aeroacoustic instabilities.