## 3 FEM Modeling of Standing Wave  
The three dimension wave equation describes the sound pressure and velocity relationship. The wall of enclosure is treated as hard sound boundary condition, which means velocity is zero. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Driver** & **Passive Radiator** \\ \hline \(R_{E}\) (Ohm) & 6.58 & 0 \\ \(L_{E}\) (mH) & 0.038 & 0 \\ \(M_{ms}\) (g) & 0.9 & 0.638 \\ \(R_{ms}\) (kg/s) & 0.1 & 0.12 \\ \(C_{ms}\) (mm/N) & 1.11 & 0.278 \\ \(Bl\) (N/A) & 0.161 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter identification. Figure 6: Equivalent circuit of speaker enclosure with passive radiator. Figure 7: Experiment and simulation comparison. The three dimension wave equation is solved based on the boundary conditions. According to the sound pressure solution, the sound pressure resonance is calculated as [12]  
\[f_{n_{x}n_{y}n_{z}}=\frac{c}{2}\sqrt{\left(\frac{n_{x}}{l_{x}}\right)^{2}+\left( \frac{n_{y}}{l_{y}}\right)^{2}+\left(\frac{n_{z}}{l_{z}}\right)^{2}} \tag{10}\]  
where \(f_{n_{x}n_{y}n_{z}}\) is the acoustic-enclosure resonance frequency; \(n_{x}\), \(n_{y}\), and \(n_{z}\) are the integers for the corresponding resonance modes; \(l_{x}\), \(l_{y}\), and \(l_{z}\) are the lengths of the enclosure in the \(x\), \(y\), and \(z\) directions, respectively; and \(c\) is the air velocity. Thus, the resonance frequency of the standing wave is related to the dimensions of the speaker box. The resonance decreases with increase in speaker-box dimensions. When the enclosure dimension is substituted into Equation (10), the resonance of the speaker box is calculated as 3253 Hz, which is caused by the longest dimension of the speaker box. This frequency is close to the experimental resonance peak frequency of 3150 Hz. The difference in values is because the speaker box is not an exact rectangular box. The standing wave can be explained as an acoustic wave that stands at a certain position and does not move in the speaker box. Before the frequency reaches 3150 Hz, the sound pressure under the diaphragm of the driver reaches a minimum, which means that the force acting on the driver diaphragm is a minimum. Thus, the acceleration is at a minimum, causing a dip in the SPL curve. Similarly, the peak at 3150 Hz occurs because the standing wave under the diaphragm of the driver is at a maximum. Thus, the force and acceleration are at a maximum, causing a peak in the SPL curve. Figure 8 shows the sound-pressure distribution. The sound pressure under the driver diaphragm is at the minimum at 3000 Hz and at the maximum at 3150 Hz; the latter contributes to the peak in the SPL curve. Figure 9 shows the force curve in terms of the frequency domain. The current force is defined as the force caused by the input current in the coil. It is at a minimum around the resonance frequency because the current is at a minimum. The dip and peak of the pressure force at approximately 3000 Hz is because of the sound-pressure distribution in the enclosure. The detail explanation is listed in the following sentence. Figure 8: Sound pressure distribution (without screen). The total force acting on the driver is the summation of the current force and pressure force on the driver. The acceleration of the driver and passive radiator is demonstrated in Figure 10, based on the forces acting on them. The dip and peak remain at approximately 3000 Hz because of the acting force. The dip and peak in the SPL curve can be explained with the acceleration which is caused by the sound-pressure distribution in the enclosure as a result of the standing wave. The dips and peaks caused by the standing wave need to be reduced to improve the response curve of the speaker box with the passive radiator. The screen is used to cancel the standing-wave effect. The screen, demonstrated in Figure 11, is made using ventilation material that has small holes to allow sound to penetrate.