The time-dependence of an over-damped local mode of a liquid surface membrane is derived by a method that corrects a previous error. By expressing an arbitrary displacement of the surface in the form of a Fourier-Bessel series, spectra of k values are obtained for Gaussian-dome and cylindrical displacements of the surface, where k is the quantity that corresponds to the wave vector of an under-damped oscillation. At high k values, the results are affected by the discrete small-scale structure of the liquid surface. The root-mean-square surface displacement and the rate of surface-bulk exchange are calculated for the Gaussian modes. Retardation effects, associated with the finite rate of spread of a disturbance from the origin of the displacement, are not significant with this model. An increase in surface roughness at high k values results from the form of the dependence of surface area on the height and variance of the Gaussian. At long times the more rapid decay of high k components causes an arbitrary displacement to tend to the form of a Gaussian dome.