# Helmholtz Resonance Frequency of the Synthetic Jet Actuator
## 4 Discussion
### Coupling Ratio in the Analysis of the Model of the Helmholtz Resonance Frequency of an SJ Actuator  
The model presented above was obtained based on the experiments performed using a wide but limited range of SJ actuator geometrical parameters. The graphical presentation of the M3 model compared with the M1 and M2 models (see Table 3) in the considered \(l\)/\(d\) range is presented in Figure 15. The M3 model is shown with a parameter \(\sigma\) equal to the extreme values of the investigated SJ actuator geometry range.  
Figure 14: Dimensionless effective orifice length as a function of the ratio of the orifice area to the cavity cross-section area, and the dimensionless orifice length.  
As can be seen, the presented model differs significantly from the M1 and M2 models, raising the question of whether another parameter affects the frequency. Thus, an additional parameter that influences the Helmholtz resonance frequency of the SJ actuator was taken into account. This parameter is the coupling ratio \(CR\), which represents the ratio of the air stiffness enclosed within the cavity to the diaphragm stiffness [23]. It is given by:  
\[CR=\left(\frac{\omega_{p}}{\omega_{D}}\right)^{2} \tag{19}\]  
where \(\omega_{D}\) represents the structural angular frequency of the diaphragm, and is given by:  
\[\omega_{D}=\sqrt{\frac{k_{D}}{m_{D}}} \tag{20}\]  
where \(k_{D}\) is the equivalent spring stiffness of the diaphragm and \(m_{D}\) is the diaphragm mass.  
In addition, \(\omega_{p}\) is the natural frequency of the pneumatic spring made of the air in the cavity and of the diaphragm mass \(m_{D}\)[18]. It is defined as:  
\[\omega_{p}=\sqrt{\frac{\kappa\cdot A_{D}{}^{2}\cdot p_{a}}{V\cdot m_{D}}} \tag{21}\]  
where \(A_{D}\) is a diaphragm wall area.  
If \(CR\) takes small values, the system of the diaphragm dynamics is decoupled from the acoustic oscillator [23]. Then, the Helmholtz and the diaphragm resonance frequency positions are far apart [24] and the uncoupled case occurs. For larger values of \(CR\), the dependence of the centerline characteristic velocity of the SJ actuator on the excitation frequency has a form similar to that of Figure 5, and the coupled case occurs.  
Based on the characteristics of the STX W.18.200.8.FGX-type loudspeaker used in the present investigations, the \(CR\) parameter was obtained for the investigated geometries of the SJ actuator. The coupling ratio was changed from about \(CR\) = 20 for the largest cavity length, to about \(CR\) = 100 for the smallest cavity length. These findings indicate that coupling occurred in the considered system, and a single peak of characteristic velocity emerged, which was mainly dominated by the diaphragm resonance, as shown in Figure 5.  
Figure 15 compares the obtained M3 model with the literature data, and shows a large discrepancy. This may be due to the large value of \(CR\) in the case considered. Model M1,  
Figure 15: The dependence of the dimensionless effective length as a function of the dimensionless orifice length.  
with \(\Delta l=0.62d\), was used by researchers [23] for investigations of the frequency response of SJ actuators. The authors obtained a good agreement between the numerical simulations and the experimental and analytical findings. In their tested cases, the CR ratio ranged from 0.04 to 1.88. In addition, the specifications of the investigated SJ actuators quoted in [15] enabled estimation of the \(CR\) parameter, which had a value of about 3. In this case, the experimental results were consistent with M2 model.  
Based on the above considerations, it can be assumed that the type of SJ actuator system considered (uncoupled or coupled) has a significant impact on the modeling of the Helmholtz resonance frequency.