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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. 

## 2 Experimental setup 

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. 

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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. 

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 

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were not possible to achieve due to the limitations of the equipment. 

## 3 Results 

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. 

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. 

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 

