A detailed numerical investigation of the coupling between the vibration of a flexible plate and the ensuing acoustic radiation is described. The non-linear Euler equations are used to describe the acoustic fluid, while the non-linear von Kรกrmรกn plate equations are used to describe the plate response. The plate is excited by a time harmonic, normally incident acoustic source, the amplitude of which is varied. We compute and describe both linear and non-linear time periodic vibrations as well as aperiodic responses. In each case, we also describe the acoustic radiation pattern resulting from the plate vibration. We find that for low excitation levels, so that the plate response is essentially linear, the effect of acoustic coupling is negligible. However, as the level of the loading increases, the radiated pressure from a plate undergoing non-linear or chaotic vibrations is found to propagate non-linearly into the far-field. The non-linearity due to wave propagation is much weaker than that due to the plate vibrations. As the acoustic wave propagates into the far field, the relative difference in level between the fundamental and its harmonics and subharmonics decreases with distance.