## Materials and Methods  
### Numerical Simulation  
In order to investigate acoustic propagation through small holes, a time-accurate, three dimensional, numerical solution was obtained for the propagation of a white noise signal in an impedance tube, in the absence of any mean flow or bias flow, under the assumption that the acoustic perturbations may be considered to remain in the laminar flow regime. The propagation of an acoustic plane wave in an impedance tube, an incident normal to a perforated plate, as show in Figure 3, may be represented as  
Figure 1: Fok’s function. Figure 2: Illustration of the Hole–Hole interaction effect, redrawn, based upon Melling [10]. an unsteady compressible laminar flow. Simulation of such flow regimes does not require significant modeling approximations, and therefore, in the context of this application, may be considered to be a direct numerical solution to the governing Navier-Stokes equations, provided satisfactory temporal and spatial resolution have been demonstrated. Computational Fluid Dynamics (CFD) is based upon the numerical solution of the underlying governing equations for fluid flow under specified boundary conditions. The method has been successfully applied by a number of authors to study acoustic radiation effect for multi-perforated plate absorbers [16; 17; 18; 19]. The simulated acoustic source applied in this work was a white noise signal ranging between 100 Hz and 1000 Hz with a 100 dB overall sound pressure level. The governing equations may be expressed, for brevity alone, in axisymmetric cylindrical polar coordinates as:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{x})}{\partial x}+\frac{ 1}{r}\frac{\partial(\rho ru_{r})}{\partial r}=0 \tag{5}\]  
\[\frac{\partial(\rho u_{x})}{\partial t}+\frac{1}{r}\frac{\partial(\rho ru_{x} u_{x})}{\partial x}+\frac{1}{r}\frac{\partial(\rho ru_{r}u_{x})}{\partial r}= \frac{\partial\tau_{rx}}{\partial r}-\frac{\partial p}{\partial x}+S_{x} \tag{6}\]  
\[\frac{\partial(\rho u_{r})}{\partial t}+\frac{1}{r}\frac{\partial(\rho ru_{x} u_{r})}{\partial x}+\frac{1}{r}\frac{\partial(\rho ru_{r}u_{r})}{\partial r}= \frac{\partial\tau_{xx}}{\partial x}-\frac{\partial p}{\partial r}+S_{r} \tag{7}\]  
\[\frac{\partial(\rho h)}{\partial t}+\frac{1}{r}\frac{\partial(\rho ru_{x}h)}{ \partial x}+\frac{1}{r}\frac{\partial(\rho ru_{r}h)}{\partial r}=\frac{1}{r} \frac{\partial}{\partial x}\left(r\Gamma\frac{\partial h}{\partial x}\right)+ \frac{1}{r}\frac{\partial}{\partial r}\left(r\Gamma\frac{\partial h}{\partial r }\right)+S_{h} \tag{8}\]  
where \(x,r\) are the axial and radial directions of the cylindrical coordinate system and \(u_{x},u_{r}\) are the axial and radial velocities, \(t\) represents time, \(\tau_{rx}\) is the axial direction viscous stress due to flow velocity gradient in radial directions. \(S_{x},S_{r}\) are momentum source terms; \(h\) is the enthalpy and \(S_{h}\) the source term of the energy equation. The ideal gas equation of state, \(p=\rho RT\), is employed to determine the density of the compressible fluid, where \(p,\rho,T,R\) represent the absolute pressure, density, temperature of the medium, and the gas constant, respectively. The discretized spatial terms were approximated by a second order biased upwind scheme, which provides both the numerical accuracy of the second order upwind scheme and the convergence robustness of the first order scheme [20]. The discretized temporal terms were approximated by a bounded second order accurate, implicit scheme, to robustly resolve the rapidly fluctuating pressure signal [20]. All simulations were undertaken with the double precision version of ANSYS FLUENT 17.2. A general non-reflecting boundary condition proposed by [PERSON] et. al. [21; 22] is applied to represent anechoic boundaries in CFD simulations. A two-microphone transfer function was employed for acoustic analysis [23].