## 3 Results and Discussion
The vibrational rarefaction waves under investigation exhibit frequencies around 10 kHz [23; 25], which surpasses the response capabilities of the transducer used for direct oscillation time measurements of the laser bubble via shock wave signals. However, their critical characteristics can be deduced from the dynamics of the laser bubble and subsequent secondary cavitation. The eigenperiod of liquid-mass oscillations (\(T_{\rm lm}\)) is equivalent to the time interval of each vibrational rarefaction wave, as shown in Figure 4. Given that each secondary bubble cloud is driven, respectively, by the initial tensile part of the individual vibrational wave, it can be deduced from the peak-to-peak interval of adjacent secondary bubble clouds. Figure 2c depicts the evolution of the total area of secondary bubble clouds (\(A_{\rm tot}\)) on the upstream side of the laser bubble, which reached a first peak at 103 \(\upmu\)s and a second peak at 227 \(\upmu\)s, ultimately determining \(T_{\rm lm}\) of 124 \(\upmu\)s.  
Considering the almost constant inception time of the first rarefaction wave, the rebound process of the laser bubble will suffer from different parts of the first rarefaction wave with increasing \(T_{\rm osc1}\), leading to the modulation of \(R_{\rm max2}\) and \(T_{\rm osc2}\). Therefore, the eigenperiod of the wall vibrations, \(T_{\rm wall}\), is equivalent to the period of the vibrational rarefaction wave, which can be determined from the modulation of \(R_{\rm max2}\) (\(T_{\rm osc1}\)) and found to have a value of 80 \(\upmu\)s, as shown in Figure 5(b-I). Figure 4: Schematic diagram of the extra pressure wave generation for laser-induced cavitation in the partially confined finite liquid with elastic walls, including bubble expansion-induced compression wave, liquid-mass oscillation-induced pressure change, and wall vibration-induced rarefaction waves. Figure 5: The temporal evol of secondary bubble clouds (**a**) and the evolution of \(R_{\rm max2}\) concerning \(T_{\rm osc1}\) (**b**) for 10 mm inner-sized cuvette with different liquid heights (I: 27 mm; II: 32 mm; III: 37 mm). In the case of a relatively large laser bubble, it is noteworthy that the secondary bubble clouds reach their peak area right around the point in time where the vibrational rarefaction wave transitions from negative to positive (\(t_{1}\) in Figure 4), marking a crucial phase. By considering this, the approximate onset time of the wave (\(t_{0}\)) can be deduced by \(t_{0}=t_{1}-T_{\text{wall}}/2\), resulting in a value of 63 \(\upmu\)s for this specific case where \(d=10\) mm and \(h=27\) mm (here \(t_{1}=103\)\(\upmu\)s and \(T_{\text{wall}}=80\)\(\upmu\)s were calculated from Figures 2c and 5a, respectively). The calculated \(t_{0}\) is closely matched to the \(T_{\text{osc1}}\) of the laser bubble when the ratio \(T_{\text{osc1}}/T_{\text{Ray}}\) reaches a minimum with increasing bubble size, confirming the calculation accuracy. It is worth mentioning that this value closely aligns with the simulated value of 60 \(\upmu\)s from a previous study conducted under similar experimental conditions [23], further underscoring the validity and reliability of this methodology. In general, the parameters of vibrational rarefaction waves, including \(t_{0}\), \(T_{\text{wall}}\), and \(T_{\text{Im}}\), can be determined from secondary bubble dynamics in conjunction with the modulation of \(R_{\text{max2}}\) (\(T_{\text{osc1}}\)). Therefore, in the subsequent analysis, our primary focus revolves around the temporal evolution of the secondary bubble clouds and the modulation of \(R_{\text{max2}}\) (\(T_{\text{osc1}}\)), to delve into the liquid height and cuvette size-related cavitation dynamics. First, we investigated the influence of liquid height on rarefaction waves and cavitation dynamics, and heights of 27 mm, 32 mm, and 37 mm were employed with the same cuvette.