On the assumption that the forced modes of vibration of a structure, subjected to pressure fluctuations random in time and space, can be approximated by the composition of the motions of the uncoupled natural modes, a general analysis is made using the ideas of vibration theory and spectrum analysis. The power spectrum, and hence the rms value, of any quantity depending linearly upon structural distortions is derived and it involves a quantity (called the “joint acceptance”) concerning the spacewise structure of the pressure field and of the geometry of the modes of vibration. It is shown how this result may be used (on assuming “normal” randomness) to estimate the fatigue life on the hypothesis of cumulative damage.