## 4 Simulation Study on Foam Acoustic Parameters
### Effect of Foam Viscous Characteristic Length on Acoustic Radiation Performance  
The viscous characteristic length \(\Lambda\) is a parameter that characterizes the energy loss due to the viscosity of the fluid inside porous materials. It can be defined as follows:  
\[\Lambda=2\frac{\int_{V}\upsilon_{i}^{2}(r)dV}{\int_{A}\upsilon_{i}^{2}(r_{w}) dA}, \tag{3}\]  
here \(\upsilon_{i}(r)\) represents the velocity of the non-viscous fluid within the pore, \(\upsilon_{i}(r_{w})\) stands for the velocity of the fluid at the interface between the pore and the material skeleton, \(A\) represents the pore surface area, and \(V\) denotes the volume of the pore.  
Keeping all other parameters unchanged, the foam's viscous characteristic length was, respectively, assigned to its original value, twice the original value, and half the original value, that is, 0.037 mm, 0.074 mm, and 0.0185 mm. The sound radiation power was then calculated at different viscous characteristic lengths, as depicted in Figure 12.  
According to Table 5, at approximately 854 Hz, the sound powers increase from small to large viscous characteristic lengths as follows: 41.26 dB, 40.44 dB, and 39.83 dB. The corresponding sound powers around 1467 Hz are 16.15 dB, 17.78 dB, and 18.01 dB, while near 1588 Hz the corresponding sound powers are 40.00 dB, 42.42 dB, and 42.89 dB. Calculation demonstrates that, at approximately 854 Hz, there are consecutive reductions of 0.82 dB and 0.61 dB, suggesting a gradual decline in sound radiation suppression per unit viscous characteristic length. Technical abbreviations are defined on their first use. In contrast, at around 1467 Hz, the sound radiation power increases consecutively by 1.63 dB and 0.23 dB; and at about 1588 Hz, the sound radiation power rises by 2.42 dB and 0.47 dB, respectively. This suggests that the rise in the length of the viscous feature has minimal effect on the improvement of sound radiation at higher frequencies.  
In conclusion, when the thermal characteristic length is kept constant, the impact of the viscous characteristic length on the radiated sound power varies at different frequencies. Additionally, the viscous characteristic length solely has a notable impact at peak  
Figure 12: Comparative graph of radiated sound power level curves of foam-filled panels at various viscous characteristic lengths and thermal characteristic lengths.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Peak Frequency** & **Sound Power Level** \\ \hline \hline \multirow{3}{*}{0.037 mm} & 854 Hz & 41.26 dB \\  & 1467 Hz & 16.15 dB \\  & 1588 Hz & 40.00 dB \\ \hline \multirow{3}{*}{0.074 mm} & 854 Hz & 40.44 dB \\  & 1467 Hz & 17.78 dB \\  & 1588 Hz & 42.42 dB \\ \hline \multirow{3}{*}{0.0185 mm} & 854 Hz & 39.83 dB \\  & 1467 Hz & 18.01 dB \\  & 1588 Hz & 42.89 dB \\ \hline \multirow{3}{*}{0.119 mm} & 854 Hz & 40.33 dB \\  & 1467 Hz & 17.45 dB \\  & 1588 Hz & 41.96 dB \\ \hline \multirow{3}{*}{0.238 mm} & 854 Hz & 40.45 dB \\  & 1467 Hz & 17.75 dB \\  & 1588 Hz & 42.43 dB \\ \hline \multirow{3}{*}{0.0595 mm} & 854 Hz & 40.69 dB \\  & 1467 Hz & 17.95 dB \\ \cline{1-1}  & 1588 Hz & 42.67 dB \\ \hline \hline \end{tabular}
\end{table}
Table 5: Table of radiated sound power level results for foam-filled panels at varying viscous characteristic lengths and thermal characteristic lengths.  
frequencies. Around 854 Hz, the radiated sound power decreases continuously as the viscous characteristic length increases, while around 1467 Hz and 1588 Hz, the radiated sound power increases with an increase in the viscous characteristic length.