0 20 40 60 80 100 120 t (s) 10 100 h (mm) Our experimental data is well described by a Boltz- mann sigmoidal curve for γ versus Γ . This fitting rela- tion is given by eq. (1), and it has four fitting param- eters: (a) The higher asymptote γ$_{max}$ (characteristic value for the higher growth rates) (b) The size of the The excitation signal used was a sine wave. The maximum amplitude of oscillation that could be reached was (11 . 1 ± 0 . 1) mm . The excitation was controlled con- sidering the maximum dimensionless acceleration de- fined as Γ = A (2 πf ) $^{2}$/g , where g denotes the accel- eration of gravity, A the amplitude of the oscillations and f the frequency of the oscillations. The value of Γ was varied from 1 to 5 . 75. Lower values did not gen- erate any instability and higher values of this quantity Images of the experiments were recorded using a digital camera (Pixelink PL-B741F). Filming against a back light (see Fig. 1), allowed us to analyze the images using a commercial software to obtain the difference in height Δh between the two free surfaces as a function of time. Fig. 1 Snapshot sequence of the partially filled U-tube in a typ- ical experiment. The grains migrate to the branch of the tube with the initially taller granular column. The images shown cor- respond to an experiment for which the U-tube was shaken at a frequency of 20 Hz and Γ = 4 . 5. The increase of the growth velocity with time can be readily appreciated. The graph of the growth rate γ versus dimension- less Γ is shown in Fig.3. The horizontal axis begins at Γ = 1, because for Γ < 1, γ = 0. For each curve the os- cillation frequency is fixed and the amplitude is varied. For all frequencies investigated there was a seemingly continuous transition from zero growth at the lowest amplitudes to a finite γ at higher amplitudes. It is ap- parent that for the larger frequencies (15 Hz and 20 Hz ) the growth rate reaches a saturation value, for suffi- ciently large amplitudes. For three of the intermediate frequencies (10 Hz , 12 . 5 Hz and 15 Hz ) we see that γ reaches a point where it changes abruptly to zero at a threshold of Γ (see arrows in Fig. 3). This suggests the occurrence of a first order phase transition, but more work needs to be done to characterize this observed abrupt transition. For the lowest frequency measured (7 . 5 Hz ) we could not determine whether there is a sat- uration region or a transition from high nonzero growth rate to zero growth at high amplitudes because of the limitations of our equipment. The height difference Δh between the granular level in both branches of the U-tube grows exponentially with time (see Fig. 2). The growth rate γ is determined from the fit of the exponential Δh = Δh$_{0}$e ( γt $^{)}$, where Δh$_{0}$ is the initial height difference and t is the time. were not possible to achieve due to the limitations of the equipment. For the vibrational system we used a function genera- tor (GW Instek SFG-2110) linked to an audio ampli- fier (Crown XLS 202) feeding two coupled 1000 Watts loudspeakers. For the granular material we used glass spheres with diameter between 250 and 300 µm and bulk static density ρ$_{g}$ = 1440 kg/m $^{3}$. The top of the container was made permeable to air but not to grains (using a 125 µm nylon mesh). The container is a U- shaped tube 200 mm tall, with square internal cross section of 400 mm $^{2}$. The total mass of grains used was 124 g , chosen to completely fill one of the U-tube branches. In this paper we extend the work in reference [5] by exploring the growth rate of the height difference be- tween the two free surfaces, for a wider range of frequen- cies. We obtain a differential equation for the change of the measured growth rates with the dimensionless ac- celeration of the vibrations. 3 Results 2 Experimental setup 2 Fig. 2 Typical time dependence of the height difference Δh , between the free surfaces at the two branches of the tube. This plot is for f = 15 Hz and Γ = 2 . 75.