# Study on Bubble Cavitation in Liquids for Bubbles Arranged in a Column Bubble Group
## 3 Study of Resonance Frequency of Bubbles in a Bubble Group  
From Equation (9) it can be seen that the resonance frequency of the bubbles in a columnar bubble group is related to the distance between the bubbles, the number of bubbles in the bubble group, and the initial radius of the bubbles. Hence, we have been able to numerically calculate the relationship between the distance between the bubbles, the number of bubbles, and the resonance frequency of the bubbles. From Equation (9) we obtain Figures 2-4, where the values of the parameters are, respectively, \(\rho=999\)kg/m\({}^{3}\), \(\gamma=1.33\), \(\sigma=0.072\)kg/s\({}^{2}\), \(p_{0}=1\)bar, and \(p_{v}=2250\)Pa.  
The relationship between the number of bubbles and the resonance frequency of the bubbles is shown in Figure 2. It can be seen from the figure that the more bubbles there are, the smaller the resonance frequency of the bubbles. When the bubble number increases, the resonance frequency of the bubbles changes quickly. When the number of bubbles changes from two to twenty-five the change in bubble resonance frequency is most obvious; the resonance frequency changes from 135.8 kHz to 117.8 kHz, which is a reduction of about 18 kHz. After this, the number of bubbles increases and the resonance frequency tends towards a certain value.  
The relationship between the distance between bubbles and the resonance frequency is given in Figure 3, where the initial radius of each bubble is 20 \(\upmu\)m and the number of bubbles is five. It can be seen from the figure that when the distance between the bubbles increases and the interaction between the bubbles weakens, after that the resonance frequency of the bubbles increases and finally tends towards a fixed value which is the Minnaert frequency of a single bubble. When the distance between  
Figure 2: Relationship between the number of bubbles and the resonance frequency of each bubble in a columnar bubble group.  
bubbles increases to 20 times the initial radius the interaction between the bubbles is almost negligible. According to Figures 2 and 3, it can be concluded that, with regard to the bubbles' initial radius, when the distance between bubbles increases to 20 times the initial radius, or is equal to twenty bubbles in a columnar, the interaction between the bubbles is very weak and can be ignored.  
Figure 4 shows that the resonance frequency of a bubble is affected by the initial radius of the bubbles, the distance between bubbles, and the number of bubbles in the bubble group. It can be seen from Figure 4a,b that when the space between bubbles increases, the inhibitory effect between bubbles becomes weak and the bubble resonance frequency increases until the coupling effect disappears and finally returns to a single bubble resonance frequency. In Figure 4 it can also be observed that for different initial radii and different numbers of bubbles after twenty bubbles, the coupling effect between the first bubble and others becomes very weak. Additionally, when the distance between bubbles increases to 20 times the initial radius of the bubbles, the interaction between bubbles can be neglected. When the initial radius of the bubbles is smaller and the bubbles are closer to each other the coupling force between bubbles will be stronger.  
Figure 4: Relationship between resonance frequency and different number of bubbles as well as different initial radius of bubbles. In (**a**), correspond to different numbers of bubbles (respectively, 10, 20, and 30, where \(R_{0}=10\) μm) and in (**b**) correspond to the different number of bubbles’ initial radii (respectively, 5 μm, 10 μm, and 20 μm, where \(n=10\)).  
Figure 3: Relationship between the distance between bubbles and the bubble resonance frequency in a columnar bubble group, \(n=5\).