## 3 Numerical Analysis of Damped HPL Panels
### Numerical Results of Damped HPL Panels  
The previous section explained that damping increases the proportion of traveling waves relative to standing waves. Furthermore, an amplitude redistribution of the motion quantities on the panel occurs. Both effects produce a change in the radiation pattern in the critical frequency range \"below coincidence\" or \"precoincidence\". This section shows simulated results in the frequency range between 1 kHz and 5.0 kHz. The panel is driven centrally by an exciter and is fixed at the edges. The damping of the entire panel is varied to implement a low (\(\eta=0.02\)), moderate (\(\eta=0.08\)) or high (\(\eta=0.2\)) damping loss factor. This theoretical simulation experiment aims to visualize the surface motion and radiated sound field as a function of panel damping.  
Figure 9 visualizes the panel motion for four frequencies below coincidence for the three different damping levels. This figure clearly reveals an amplitude redistribution towards the center of the panel with increased damping. The proportion of standing waves to traveling waves is reduced. The bending wavenumber (represented by the number of nodal lines) remains the same regardless of damping, but as damping increases, the deflection becomes concentrated at the point of force application. With rising frequency, this effect is visibly enhanced.  
These observations are also supported by the analysis of the radiated sound field, as shown in Figure 10. Each frequency clearly shows that an increased damping value reduces the sources at the edge of the panel compared to the center of the panel. Accordingly, with increasing damping, the radiation from the center of the panel or the excitation point of the structure dominates. At 3.6 kHz and 5 kHz, the off-axis sound radiation is reduced. Additionally, at frequencies of 1.4 kHz and 2.2 kHz, the tendency of lower off-axis radiation becomes visible. Simultaneously, the sound radiation on-axis or in the LW is almost unaffected, even at high panel damping. The interference effects are reduced with increased damping. Both constructive and destructive interference are reduced and the  
\begin{table}
\begin{tabular}{c c c} \hline Outer dimensions & \multicolumn{2}{c}{855 \(\times\) 467 \(\times\) 120 (mm)} \\ Moving surface dimensions & \multicolumn{2}{c}{775 \(\times\) 387 (mm)} \\ Exciter & Dayton Audio DAEX25 FHE-4 \\ Panel material & & HPL \\ Panel thickness & \(h\) & 3.05 (mm) \\ Young’s modulus & \(E\) & 16.4 (GPa) \\ Density & \(\rho\) & 1595 (kg/m\({}^{3}\)) \\ Poisson’s ratio & \(\
u\) & 0.3 \\ Damping loss factor & \(\eta\) & 0.02 \\ \hline \end{tabular}
\end{table}
Table 1: Overview of the prototype dimensions and panel properties.  
Figure 9: Simulated frequency-dependent surface displacement scaled according to the individual maximum value of a panel with different damping loss factors applied. With increased damping, the traveling wave pattern becomes dominant.  
-directivity response becomes more homogeneous. Accordingly, damping reduces off-axis radiation, only slightly affecting sound radiation in the LW. As a result, the DI below coincidence can be improved by additional damping.