## 2 Mechanism of Low-Frequency Pressure Fluctuation
### Resonance Mechanism of Low-Frequency Pressure Fluctuation
#### 2.2.2 Planar Standing Wave in Plenum  
* Plenum standing wave as an excitation source in LAWT  
Recently, [PERSON] et al. [22] discovered that the significant low-frequency pressure fluctuation in the plenum of the LAWT manifests itself as a planar pressure standing wave in the transverse direction, the amplitude of which can be expressed as:  
\[P(y)=A\sin(2\pi y/\lambda)\cos(2\pi ft). \tag{4}\]  
where \(A\) denotes the pressure maximum, \(y\) is the width-direction coordinate (the plenum is symmetric about the jet center in the transverse direction), \(\lambda=2L_{y}\) is the wavelength, \(L_{y}\) is the width of the plenum, \(f\) is the frequency of the standing wave, and \(t\) is the time. A diagram illustrating the pressure standing wave is presented in Figure 2. It can be seen that the waveform has a sinusoidal pattern with an anti-symmetric distribution along the \(y\)-axis. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Resonance Mechanism** & **Wind Tunnel** & **Frequency (Mode) (Intensity)** & **Suppression Methods** \\ \hline \multirow{6}{*}{Planar standing waves of plenum} & LAWT [22; 23; 24; 25] & 5.4 Hz (\(n_{y}=1\)) (120 dB) &
\begin{tabular}{c} CVG; \\ Changing plenum dimension \\ (Simulation only) \\ \end{tabular} \\ \cline{2-4}  & & 127 Hz (\(u_{x}=1\)) (\(\times\)); & Changing plenum size; \\  & SKLA-01 [25] & 109 Hz (\(u_{y}=1\)) (\(\times\)); & Increasing collector inlet size \\  & & 169 Hz (\(u_{z}=1\)) (\(\times\)) & \\ \cline{2-4}  & & 18 Hz (\(u_{x}=1\)) (130 dB); & \\  & IVK-MWT [19] \({}^{\ddagger}\) & 22 Hz (\(u_{y}=1\)) (134 dB); & \(\times\) \\  & & 239 Hz (\(u_{z}=1\)) (118 dB) & \\ \cline{2-4}  & DNW-LLF [4] \({}^{\ddagger}\) & 5.5 Hz (\(u_{y}=1\)) (\(\times\)); & \(\times\) \\ \hline Helmholtz modes of plenum & S2A [8] & 1.6 Hz (\(\times\)) (100 dB) & Passive flap \\ \cline{2-4}  & 1/20-scale AAWT [26] & 45 Hz (\(\times\)) (99 dB) & ARC \\ \hline \multirow{3}{*}{Reverberation modes of plenum} & \multirow{3}{*}{IVK-MWT [19]} & 22 Hz (0,1,0) (134 dB) \({}^{*}\) & \\  & IVK-MWT [19] & 39 Hz (0,0,1 or 2,0,0) (118 dB) \({}^{**}\) & FKFS-laps (movable) \\ \cline{1-1} \cline{2-4}  & & 42 Hz (1,0,1 or 0,1,1) (111 dB) \({}^{**}\) & \\ \hline \hline \end{tabular}  
* \({}^{\dagger}\times\) denotes the unavailability or nonexistence of the data. \({}^{\ddagger}\) The type of resonance mechanism is speculated by the authors of this review.
* (0,1,0) means \(n_{x}=1\), \(n_{y}=1\), \(n_{z}=0\). \({}^{**}\) The authors believe these two modes were excited by edgetone feedback (please refer to Section 2.2.2).  
\end{table} 
Table 1: _Cont._* Mechanism and prediction of plenum standing wave  
It was found that the planar pressure standing wave resonated with the edgetone feedback, resulting in significant low-frequency pressure fluctuations in the LAWT [22]. On the basis of this, the condition for the maximization of the low-frequency pressure fluctuation caused by the standing wave can be derived as follows [22]:  
\[\frac{L_{y}}{L_{j}}=\frac{1}{2n}\bigg{(}\frac{1}{V_{a}/a}+\frac{1}{0.65 Ma} \bigg{)}. \tag{5}\]  
where \(L_{y}\) denotes the width of the plenum, \(Ma\) is the jet Mach number, and the definitions of the remaining symbols correspond to those in Equation (1). Equation (5) was derived in the literature [22] by making the edgetone frequency given by Equation (2) equal to the standing wave frequency in the transverse direction of the plenum:  
\[f_{sw}=\frac{a}{2L_{y}}. \tag{6}\]  
Note that the contribution of \(\Delta L\) is not considered in Equation (5). In the literature [22], only the external feedback is considered, resulting in \(V_{a}/a=1\), and the number of modes, \(n\), is taken as the number of large vortex pairs in the jet (\(n=2\)), under the operating conditions of the LAWT wind tunnel. When the discrepancy between \(L_{y}/L_{j}\) and the predicted value of Equation (5) falls below 20%, the resonance of the standing wave with edgetone feedback is likely to occur.