## 4 Acoustic Black Hole Theory and Verification of the ABH Effect  
A two-dimensional (2-D) ABH was embedded on the rectangular elastic plate; the schematic diagram of the structure is shown in Figure 8. The thickness of the ABH plate can be expressed by the following equation:  
\[h(x)=\left\{ \begin{array}{ll}\varepsilon x^{m}-b,&\text{ABH region}\\ h_{p},&\text{Uniform region}\end{array} \right. \tag{4}\]  
where \(\varepsilon\) is a constant, and \(m\) is the power exponent of the thickness change in the ABH region, \(m\geq 2\), \(b\) is the distance between the center of the ABH region and the coordinate origin in the \(z\)-axis direction, and the thickness of the uniform region of the ABH plate is \(h_{p}\). However, due to the limitation of the processing technology, the ideal ABH structure cannot  
Figure 8: Schematic diagram of the 2-D ABH structure. Figure 7: Coherence function curve between the radiated sound of the back wall of the refrigerator and the vibration acceleration on the back wall of the refrigerator. be produced in the actual manufacturing process, so there must be a truncation existing in the actual ABH structure. For the model in Figure 8, the thickness of the truncated region can be expressed as:  
\[h_{t}=\frac{h_{p}}{2}-b \tag{5}\]  
The existence of the truncation will significantly increase the reflection coefficient of the flexural wave. For an isotropic thin plate, the propagation speed of the flexural wave in the plate is:  
\[c_{B}=\left(B\omega^{2}/m\right)^{1/4} \tag{6}\]  
where \(B\) is the bending stiffness of the ABH plate material, \(B=Eh(x)^{3}/\left[12(1-v^{2})\right]\), \(E\) is the Young's modulus of the ABH plate material, \(v\) is the Poisson's ratio; \(\omega\) is the angular frequency, \(m\) is the areal density of the ABH plate, \(m=\rho_{m}h(x)\), and \(\rho_{m}\) is the density of the ABH plate material. When the flexural wave propagates from the uniform region of the ABH plate to the ABH region, as the thickness of the plate gradually decreases, the velocity of the flexural wave also gradually decreases [32]. At this time, the wavelength is compressed, and the amplitude of the flexural wave gradually increases so as to achieve the concentration effect of the flexural wave. In order to verify the concentration effect of the ABH plate on flexural waves, a square plate embedded with a 2-D ABH was designed. The geometric size of the ABH plate is 200 \(\times\) 200 mm, the thickness of the uniform region is 4 mm, and the coordinate origin is established at the geometric center of the middle plane in the thickness direction of the ABH plate. The thickness equation of the ABH region is \(h(x)=0.006x^{2}-1.5\). According to Equation (5), the thickness of the truncation of the ABH region is 0.5 mm, and the outer diameter of the ABH region is about 48.3 mm. The multi-physics simulation software COMSOL is used to carry out FEM modeling and numerical calculation of flexural wave propagation in the ABH plate. To ensure the calculation accuracy, the extremely fine level is used to mesh the 2-D ABH plate, and the ABH regions are encrypted to ensure that the number of grids in each wavelength is more than six. The FEM model of the ABH plate is shown in Figure 9. The material of the 2-D ABH plate is 1060 aluminum, the Young's modulus is 69 GPa, the Poisson's ratio is 0.33, and the density is 2710 kg/m\({}^{3}\). The left side face of the ABH plate is clamped, the remaining faces and edges are set as free boundaries, and the excitation point is located at the midpoint of the right side of the upper surface of the ABH plate. A simple harmonic excitation force perpendicular to the ABH plate is applied to the excitation point, the amplitude of the excitation force is 100 N, the frequency is 80 MHz, and the action time is 0\(\sim\)2 \(\times\) 10\({}^{-5}\) s. The transient calculation of the flexural wave propagation in the ABH plate was carried out.