Pressures are determined in a layer of fluid bounded by two parallel elastic plates, one of which is harmonically excited at a point. The response of the coupled plate‐fluid system is obtained for infinitesimal displacements in both the plate and the fluid. The pressures are obtained through a numerical integration of the inverse Hankel transform along an approximation to the steepest descent path. Some numerical results obtained for frequencies both above and below the coincidence frequency are given. These show a complex spatial variation in pressure in addition to that on the scale of the wavelength in the fluid.