Dead-end filtration tests with a vertically vibrated medium are reported, focussing on the effect of acceleration amplitude. A phenomenological model with constant coefficients and assumed constant solids volume fraction profile is introduced. Its inadequacy in describing the subtleties of the experimental data is discussed. A major shortcoming is the fact that a phenomenological model cannot predict the solids volume fraction profile. A physical theory is then reviewed in which the slurry is described as a dense gas with a position-dependent temperature. The theory is applied to the present problem. A non-dimensional parameter S, which is proportional to the ratio of the mean fluid velocity in the filter to the oscillation velocity amplitude, emerges from the theory (gravitational effects are ignored). Using the physical theory, solutions to the solids volume fraction profile are presented and their sensitivity to the volume of solids in the apparatus and the parameter S is probed.