# Frequency Modulation and Erosion Performance of a Self-Resonating Jet
## 3 Results and Discussion  
### Gas-Spring Theory  
The time-resolved chamber pressure signals with various pressure drops are presented in Figure 11, as well as the marginal spectrum obtained from the HHT. Negative pressure oscillations were easily captured by the pressure transducer positioned in the chamber. The absolute values of negative pressure peaks increased with the increasing pressure decline, caused by the intensified turbulent momentum transportation and entrainment between the main jet and the quiescent fluid. Meanwhile, the negative pressure peaks became more intense with higher pressure decline. According to the Gas-Spring Theory, the frequency increase may come from the accumulated time for the cavity cluster in the chamber decreasing as more energy was obtained with a higher pressure decline. To be more specific, the instantaneous energy density (IE) Equation 15 was introduced to check the energy fluctuation (Figure 12). Energy fluctuations were observed in each condition, and the higher the inlet pressure was, the more energy was accumulated for collective bubbles within the same time range. It turned out that the inlet pressure decline affects both the frequency and amplitude of the instantaneous energy density. \[IE(t)=\int_{\omega}H^{2}(\omega,t)d\omega \tag{15}\]  
where \(H(\omega,t)\) is the energy-frequency-time distribution, which is obtained from the HHT, and \(\omega\) is the circular frequency. The numerical simulation was also used to verify the Gas-Spring Theory. The marginal spectrum of the volume fraction of the vapor phase is presented in Figure 13, with a sharp peak at 0.700 Hz being observed. Multiple additional right and left peaks can also be observed, along with the dominant \(f_{a}\), as  
Figure 11: Pressure signal and marginal spectrum of the chamber pressure. Figure 12: Instantaneous energy density of various pressure drops. mentioned in several previous investigations on the cavity flow. They have been denoted \(f_{+}=0.103\) Hz and \(f_{-}=0.400\) Hz when symmetrically distributed around \(f_{a}\). In addition, a peak at low frequency, \(f_{b}=0.267\) Hz, which is close to the experimental results, was observed. The existence of such a low frequency is a common feature in most of the various configurations [36]. The frequency \(f_{b}\) could result from the non-linear interaction between \(f_{a}\) and \(f_{+}\), but \(f_{+}\) could also result from the non-linear combination of \(f_{a}\) and \(f_{b}\), via an amplitude modulation at impingement [37; 38; 39; 40]. Indeed, combinations between the harmonics \(nf_{a}\) of \(f_{a}\) with \(f_{b}\) occur, to produce sum frequencies \(nf_{a}-f_{b}\). For instance, the \(f_{-}=0.400\) Hz is close to the difference frequency \(f_{a}-f_{b}\) = 0.433 Hz. The global structure of the spectral distribution can be viewed as the result of non-linear interactions and modulation processes which may combine, as shown in [PERSON] et al. [41] and [37; 38]. According to the theory of vortex-acoustics, the oscillating frequency could be calculated as follows [42]:  
\[f=\frac{1}{2\pi}\sqrt{\frac{A_{1}\kappa_{p}}{I_{1}V_{c}\kappa_{v}}} \tag{16}\]  
where \(A_{1}\,and\,1_{1}\) are denoted as the cross section area and the length of the upstream nozzle; \(V_{C}\) is denoted as the volume of the oscillating chamber; \(p\), \(k_{a}\) and \(\kappa_{v}\) are denoted as the pressure, the adiabatic exponent of air and the volume fraction of vapor phase, respectively.