Following the methods of Lyon, an analysis of the vibratory response of a plate to a random pressure field is given. The pressure correlation of the random field is assumed to have a scale small compared to the plate size, to decay exponentially, and to convect with constant speed over the plate. Two eases are considered, one in which the convection speed is much less than the speed of free flexural waves in the plate, the other in which the convection speed is the same order as the flexural wave speed. The mean square plate displacement is shown to be relatively independent of convection for speeds much less than the flexural wave speed, and to increase significantly for speeds in the order of the flexural wave speed. It is shown that damping is usually, but not always, an effective means of vibration reduction. In the case of convection speeds much smaller than the flexural speed, the use of hysteretic damping for reduction of the displacement response is shown to be limited by the decay of the assumed random pressure field.