## 4 Experimental Validations  
In order to verify the accuracy of the FEM analysis, an experimental platform for sound transmission loss measurement of ABH circular plates was constructed, which is shown in Figure 5. The four-microphone method of sound transmission in the standing wave tube [26; 27] was adopted to measure and calculate the TLs of ABH circular plates. The signal generator generated a sinusoidal excitation signal with a frequency of 20 Hz\(\sim\)10 kHz, which was amplified by a power amplifier to drive the speaker to generate sound waves. Then, the sound wave entered the sound source tube to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Young’s Modulus (Pa)** & **Density (kg/m\({}^{3}\))** & **Poisson’s Ratio** & **Loss Factor** \\ \hline Aluminum 6061 & \(7\times 10^{10}\) & 2700 & 0.35 & \(5\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ABH FEA material properties. Figure 4: TL of a circular plate embedded with a diameter of 60 mm acoustic black hole and the uniform circular plate: **(a)** TL values from 0 to 10 kHz; **(b)** TL differences of the two plates. generate a plane incident wave. When the sound wave propagated to the ABH circular plate, part of the sound wave will be reflected to form a plane reflection sound wave, and the other part of the sound wave would be transmitted through the ABH circular plate into the receiving tube to form a plane transmitted sound wave. Two acoustic sensors were placed in front of and behind the ABH plate to measure the sound pressure at the location. The edge of the ABH circular plate was bonded to the standing wave tube with the transparent glass glue, which not only could fix the position of the ABH plate, but also ensured there was no gap between the edge of the ABH plate and the inner wall of the standing wave tube. The signals of the four acoustic sensors were collected and processed using the \(m\) + \(p\) acquisition terminal and the SO Analyzer data processing software. The sound pressure transmission coefficient can be calculated by:  
\[t_{p}=\frac{\sin[k(X_{1}-X_{2})]}{\sin[k(X_{4}-X_{3})]}\times\frac{p_{3}e^{jk( X_{4}-X_{3})}-p_{4}}{p_{1}-p_{2}e^{-jk(X_{1}-X_{2})}}\times e^{jk(X_{2}+X_{3})} \tag{20}\]  
Figure 5: Experimental setup for TL measuring: (**a**) Schematic of the experiment system; (**b**) photograph of the experiment system. Mic. :Microphone, Vib. :Vibration, SO:SmartOffice  
where \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\) are the sound pressures measured by four sensors and they can be expressed as \(p=Ae^{j_{out}}\) and \(A\) and \(\omega t\) are the amplitude and the phase of the sound pressures. Taking the sound pressure at Microphone 1 as a reference, the phases of the other three sound pressures are represented by the phase difference from that of the Mic. 1, and the transmission coefficient can be expressed as:  
\[t_{p}=\frac{\sin[k(X_{1}-X_{2})]}{\sin[k(X_{4}-X_{3})]}\times\frac{A_{3}e^{j( \omega\Delta t_{3})}e^{jk(X_{4}-X_{3})}-A_{4}e^{j(\omega\Delta t_{4})}}{A_{1}- A_{2}e^{j(\omega\Delta t_{2})}e^{-jk(X_{1}-X_{2})}}\times e^{jk(X_{2}+X_{3})} \tag{21}\]  
where \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\) are the amplitudes of the sound pressures measured by the four sensors; \(\omega\Delta t_{2}\), \(\omega\Delta t_{3}\), \(\omega\Delta t_{4}\) are the phase differences between the sound pressures at Mic. 2, 3, and 4 and the sound pressure at Mic. 1. Then, the transmission loss can be obtained by:  
\[TL=-20\lg\big{|}t_{p}\big{|}(dB) \tag{22}\]  
A fast Fourier transform was performed on the sound pressures measured at sensors 1, 2, 3, and 4. Taking the sound pressure at the sensor 1 as a reference signal, the amplitudes and relative phases at the corresponding frequencies in the spectrum can be obtained by FFT transform and crosspower-spectral calculation, and then substituted into Equations (21) and (22) for TL calculation.