# A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption
## 2 Models and Methods  
### Theoretical Calculation  
Figure 1a shows the structure of a PPSA absorber, which consists of a PPSA and an air cavity supported by a rigid wall. The PPSA is perforated with a series of stepped holes, which are composed of two unequal circular apertures in the coaxial line. For each circular aperture, the ratio of diameter to depth is greater than 1 and the diameter is not less than 1.5 mm. The diameter, thickness and perforation ratio of the small aperture are \(d_{1}\), \(t_{1}\), \(\sigma_{1}\), those of the large aperture are \(d_{2}\), \(t_{2}\), \(\sigma_{2}\), respectively, and the air cavity depth is \(D\)  
Structurally, PPSA can be viewed as a serial combination of two perforated panels PP1 and PP2 without spacing. Based on the transfer matrix method [27], for the one-dimensional acoustical system element of PPSA absorber, as shown in Figure 1b, the sound pressure \(p_{1}\) and particle velocity \(u_{1}\) on the left side and the sound pressure \(p_{3}\) and particle velocity \(u_{3}\) on the right side can be expressed as:  
\[\left[ \begin{array}{c}p_{1}\\ u_{1}\end{array} \right]=\left[ \begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array} \right]\left[ \begin{array}{c}p_{3}\\ u_{3}\end{array} \right], \tag{1}\]  
where \(T_{11}\), \(T_{12}\), \(T_{21}\) and \(T_{22}\) are the four pole parameters of the total transfer matrix \([T]\). The total transfer matrix \([T]\) is obtained by multiplying the unit transfer matrix of PP1, PP2 and air cavity, written as:  
\[[T]=[T_{\text{PP1}}][T_{\text{PP2}}][T_{Air}], \tag{2}\]  
For PP1 and PP2, the transfer matrix \([T_{pp}]\) can be given as:  
\[[T_{\text{PP1,2}}]=\left[ \begin{array}{cc}1&Z_{\text{PP1,2}}\\ 0&1\end{array} \right], \tag{3}\]  
The acoustic energy loss of the perforated panel mainly includes the air viscous dissipation inside the hole and the end correction caused by viscous friction and sound radiation. Therefore, the acoustic impedance of perforated panel is the sum of the acoustic impedance inside the hole and the end correction outside the hole. According to the viscous motion theory in the tube derived by [PERSON] and simplified by [PERSON] [28], the acoustic impedance in the circular hole is expressed as:  
\[z_{\text{hole}}=j\omega\rho_{0}t\left[1-\frac{2}{k\sqrt{-j}}\frac{J_{1}(k\sqrt {-j})}{J_{0}(k\sqrt{-j})}\right]^{-1}, \tag{4}\]  
Figure 1: A perforated plate with stepwise apertures (PPSA) absorber. (**a**) Structure diagram; (**b**) One-dimensional acoustical system. According to [PERSON] and [PERSON] theory [29], the corrections of acoustic resistance and reactance at both ends of the hole are \(\sqrt{2\omega\rho_{0}\eta}/2\) and \(0.85d\), respectively. Due to the continuity of the surfaces of PP1 and PP2, the corrections of acoustic impedance at one end is considered. Thus, the acoustic impedance of PP1 and PP2 is written as:  
For the air cavity, the transfer matrix [\(T_{Air}\)] can be written as:  
\[Z_{P1,2}=\frac{1}{\sigma_{1,2}}\left(j\omega\rho_{0}t_{1,2}\Bigg{(}1-\frac{2} {k_{1,2}\sqrt{-j}}\frac{I_{1}(k_{1,2}\sqrt{-j})}{I_{0}(k_{1,2}\sqrt{-j})}\Bigg{)} ^{-1}+j\omega 0.425d_{1,2}+\frac{\sqrt{2\omega\rho_{0}\eta}}{4}\right), \tag{5}\]  
\[[T_{Air}]=\left[ \begin{array}{cc}\cos(k_{0}D)&j\rho_{0}c_{0}\sin(k_{0}D)\\ j\sin(k_{0}D)/\rho_{0}c_{0}&\cos(k_{0}D)\end{array} \right], \tag{6}\]  
where \(\omega=2\pi f\) is the angular frequency, \(\eta\) is the dynamic viscosity coefficient of the air, \(k=d\sqrt{\omega\rho_{0}}\eta/2\) is the ratio of the inner radius to the viscous boundary layer thickness inside the tube, for square arrangement, perforation ratio \(\sigma=0.785d^{2}/b^{2}\), \(d\) and \(b\) are the diameter of holes and the spacing between holes, \(J_{0}\) and \(J_{1}\) are the Bessel functions of the zero order and the first order, respectively. \(\rho_{0}\) is the air density, \(c_{0}\) is the sound speed in the air, \(k_{0}=\omega/c_{0}\) is the air wave number. Since the air cavity is supported by a rigid wall, the particle velocity on the rigid wall is \(u_{3}=0\).