# Improved Sound Radiation of Flat Panel Loudspeakers Using the Local Air Spring Effect
## 4 Prototyping and Simulation of the Modal Related Air Compliance
### Modal and Sound Pressure Related Decomposition of Optimized Construction  
Figure 12 illustrates the modal-related decomposition and the sound pressure-related decomposition of the model with a separation plate. This shows the individual modal stiffening through the air compliance. \(SPL_{\textit{Model3}}\) is clearly suppressed compared to \(SPL_{\textit{Model}}\), which is visible in fewer levels of \(SPL_{\textit{Model3}}\) and fewer anti-phase components. The number of in-phase components is not larger than that in the standard version. However, the reduced amount of anti-phase components increases the resulting sound pressure. In this case, the system works very well because dips and low-level areas are amplified and peaks are minimized. A detailed analysis is performed below, where the response is divided into five frequency areas. At very low and high frequencies, in Areas I and IV/V, respectively, the acoustic performances of both configurations are very similar. At low frequencies, the sound radiation is dominated by \(SPL_{\textit{Model}}\), which is not influenced by the additional air stiffness. In the upper frequency range, higher order modes with lower amplitude occur and in the influence of the separation plate is negligible. The standard construction has a significant dip between 100 and 200 Hz due to the reversed phase behavior of \(SPL_{\textit{Model}}\) and \(SPL_{\textit{Model3}}\). Because of the separation  
Figure 11: Visualization of the displacement and pressure in the acoustic chamber of the standard against the separation version. The displacement scale is adjusted for every frequency, but both versions are scaled to the same range. The pressure scale is in dB and the frequencies are similar for all cases. plate \(SPL_{Mode3}\) is shifted in phase at a lower frequency, the proportion of the anti-phase component in the overall sound pressure is reduced. Consequently, the frequency response of the standard configuration is smoothed by using an additional modal air stiffness. The same working principle is responsible for smoothing the transfer function up to approximately 500 Hz. Due to the modal stiffness of the additional air volume, certain modes are suppressed, which either leads to dips or peaks in the frequency response. A detailed analysis is performed below. * **Area I (20-70 Hz):** In the lower frequency range, \(SPL_{Mode1}\) and \(SPL_{Mode3}\) predominantly contribute to the overall performance. Furthermore, the \(SPL_{Total}\) is almost identical to the \(AAL_{Total}\); consequently, the radiation is nearly ideal. The sound pressure response of all modes is lower than the response of \(SPL_{Mode1}\), as there are anti-phase components evoked by \(SPL_{Mode3}\). * **Area II (70-90 Hz):** This band shows a significant dip at 80 Hz. \(SPL_{Mode1}\) and \(SPL_{Mode3}\) contribute nearly equally, which increases the anti-phase components \(AAL_{Anti}\). * **Area III (90-170 Hz):** This band shows the large difference between separation and standard construction. The sound pressure response of all modes is below the pressure response of the single \(SPL_{Mode1}\). \(SPL_{Mode3}\) relates to anti-phase components. However, the significant change is the pressure difference of the in-phase components \(AAL_{In}\) and the anti-phase components \(AAL_{Anti}\), which is approximately 5 dB or more. Even with a lower total displacement of the panel, the sound pressure can be higher due to the reduced amount of anti-phase components. This results in a flatter response compared to the standard construction with higher sensitivity. * **Area IV (170-220 Hz):** This band shows substantially high movement of the panel with less output. This is a superposition of \(SPL_{Mode1}\), \(SPL_{Mode3}\), \(SPL_{Mode8}\) and \(SPL_{Mode11}\). in high in-phase and high anti-phase components. The resulting pressure response is lower than that in Area III. In comparison to the standard construction, the deviation is reduced.