## 2 Theoretical Background
### Theory of Transmission Loss  
When sound waves are incident on a wall or partition, some of them will be reflected and some will be transmitted through the wall or partition. Figure 2 shows the propagation procedure of the  
Figure 1: A circular plate embedded with a two-dimensional acoustic black hole. sound wave when transmitting through an ABH circular plate. The transmission coefficient is defined as the ratio of transmitted energy to incident energy:  
\[\tau=\frac{E_{transmitted}}{E_{incident}} \tag{4}\]  
With the condition that the fluids on both sides of the plate are the same, the sound transmission loss of the circular plate can be expressed as:  
\[TL=-10\text{lg}\tau(dB) \tag{5}\]  
For an infinite plate, the transmission coefficient at the (\(\theta\), \(\varphi\)) angle to the normal to the surface of the plate is:  
\[\tau(\theta,\varphi)=\left|1+\frac{Z\cos\theta}{2\rho_{0}c_{0}}\right|^{-2} \tag{6}\]  
where \(\theta\) is the angle between the incident sound wave and the normal of the plate surface; \(\varphi\) is the angle of incidence with respect to the axis of the minimum stiffness of the plate; \(Z\) is the impedance of the flexural wave,; \(\rho_{0}\) is the density of the fluid; \(c_{0}\) is the velocity of the sound wave in the fluid. The diffuse field transmission coefficient can be obtained by determining the weighted average of the transmission coefficients of all incident angles:  
\[\tau_{d}=\frac{1}{\pi}\int_{0}^{2\pi}d\varphi\int_{0}^{\pi/2}\tau(\theta, \varphi)\cos\theta\sin\theta d\theta \tag{7}\]  
For an isotropic plate, Equation (7) can be simplified to:  
\[\tau_{d}=\int_{0}^{1}\tau(\theta)d(\sin^{2}\theta) \tag{8}\]  
Figure 2: Conceptual diagram of the sound wave propagating through an acoustic black hole (ABH) circular plate. When the sound wave is incident normally and the thickness of the plate is much smaller than the wavelength of the sound wave, the transmission loss can be expressed as:  
\[TL_{N}=10\lg[1+(\frac{\pi fm}{\rho_{0}c_{0}})^{2}] \tag{9}\]  
[PERSON]'s research showed that for a single-layer plate, the transmission loss in the range of mass law can be obtained by using a constant value of \(\theta=78^{\circ}\) and the relationship between the sound transmission loss of field incidence and that of normal incidence in one-third octave band is [25]:  
\[TL=TL_{N}-5.5\ (\mathrm{dB}) \tag{10}\]  
The sound transmission loss of a plate can be divided into several characteristic intervals by its first-order resonance frequency and critical frequency. According to the theory of elastic vibration, the natural frequency of a thin circular plate freely vibrating under the boundary condition that the edge is clamped can be expressed as:  
\[f_{nm}=\alpha_{nm}\frac{h}{r^{2}}\sqrt{\frac{E}{\rho(1-\
u^{2})}} \tag{11}\]  
where \(h\) is the thickness of the circular plate; \(r\) is the diameter of the circular plate; \(E\) is the Young's modulus of the material; \(\rho\) is the density; \(\
u\) is Poisson's ratio; \(\alpha_{nm}\) is the coefficient. The first-order resonance frequency of the uniform circular plate can be obtained according to Equation (11). The circular plate has a critical frequency (also known as the coincidence frequency), at which the velocity of the flexural wave in the plate is equal to the velocity of the sound wave propagating in the fluid. The critical frequency can be obtained by:  
\[\tau=\frac{E_{transmitted}}{E_{incident}} \tag{4}\]0  
where \(c\) is the velocity of sound in the fluid, \(\sqrt{E/\rho_{m}}\) is the velocity of the longitudinal wave in the thin rod. In practice, the TL of a plate is often lower than the calculated value according to the mass law, because the sound insulation performance of the plate is also related to its stiffness and damping. The sound insulation characteristics of a typical plate can be divided into four parts according to the frequency, which are called the stiffness-controlled region, the mass-controlled region, the coincidence-controlled region, and the damping-controlled region.