## 3 Labyrinthine Slit Perforations in a Rigid-Porous Matrix  
With sufficient scale separation between the characteristic matrix pore size and the labyrinthine slit width, a labyrinthine perforation in a porous matrix may be regarded as a dual-porosity system. If there is a high permeability contrast between the matrix and the labyrinth, the resulting bulk complex density, \(\rho_{apL}\), and complex compressibility, \(C_{dpL}\), are [6],  
\[\rho_{dpL}(\omega)=\biggl{[}\frac{1}{\rho_{L}(\omega)}+\frac{(1-\Omega_{B})}{ \rho_{m}(\omega)}\biggr{]}^{-1},\;C_{dpL}(\omega)=C_{L}(\omega)+(1-\Omega_{B}) F_{d}(\omega)C_{m}(\omega), \tag{11}\]  
where \(\rho_{m}\) and \(C_{m}\) represent the complex density and compressibility of the rigid-porous matrix and \(F_{d}(\omega)\) is the pressure diffusion function. For a high permeability contrast, \(F_{d}(\omega)\), is given by [6,7]  
\[F_{d}(\omega)=\Bigl{[}1+i\frac{\omega}{\omega_{d}}\frac{D(\omega)}{D(0)} \Bigr{]},\;D(\omega)=\frac{D(0)}{-i\frac{\omega}{\omega_{d}}+\sqrt{1-i\frac{M( \omega)}{2}}\frac{\omega}{\omega_{d}}},\;D(0)=(1-\Omega_{B})\biggl{(}\frac{(L/ 2)^{2}}{3}\biggr{)} \tag{12}\]  
\[M(\omega)=\frac{8D(0)}{{\Lambda_{d}}^{2}(1-\Omega_{m})},\;\Lambda_{d}=\frac{2 (l+w)((N+1)w+Nb)}{(N+2)w+Nb+l},\;\omega_{d}=\frac{(1-\Omega_{B})P_{0}}{\Omega _{m}\sigma_{m}D(0)} \tag{13}\]  
where \(D(\omega)\) is a dynamic pressure permeability function, \(D(0)\) is the static pressure permeability involving half the width, \(L\), between perforation openings at the surface and \(\Lambda_{d}\) is a pressure diffusion length. \(\Omega_{m}\) and \(\sigma_{m}\) denote the porosity and flow resistivity of the porous matrix, respectively. In a horizontal-wall labyrinth, the distance between adjacent slit openings, \(L\), is \((l+w)\). In a vertical-wall labyrinth, the distance between adjacent slit openings is \([(N+1)w+Nb]\). In Figure 2a, \(|F_{d}(\omega)|\) is plotted as a function of frequency for a 10-fold vertical-wall labyrinthine slit perforation, in which a 3 mm wide labyrinthine slit is formed by 1 mm wide, 45 mm long walls, in a porous matrix in which the pores have the form of micro-slits (0.04 mm wide slits inclined at 70\({}^{\circ}\) to the normal, porosity 0.9). The acoustical properties of a matrix of parallel identical slits of width \(b_{s}\) inclined at \(\theta\) to the normal are given by Equations (9) and (10) with \(\Omega_{B}\) replaced by \(\Omega_{m}\), \(b\) replaced by \(b_{s}\) and \(T=T_{m}=1/(cos(\theta))^{2}\)[5]. The flow resistivity, \(\sigma\), due to slit-like pores is calculated from [5]  
\[\sigma=\frac{12\mu T}{\Omega b^{2}} \tag{14}\]  
where \(T,~{}\Omega,~{}b\) are appropriate for the labyrinthine slit or inclined slit matrix. The flow resistivity of the labyrinthine slit in a non-porous matrix (3.5 kPa s m\({}^{-2}\)) is much smaller than that (1297 kPa s m\({}^{-2}\)) of the porous matrix. Figure 2b shows the corresponding prediction for the imaginary part of the bulk modulus, \(K_{dpl}(\omega)=1/C_{dpl}(\omega)\), which shows a peak near the pressure diffusion frequency (97.5 Hz). Figure 3 shows predictions of normal incidence absorption spectra assuming 5.8 cm thick hard-backed layers of rockwood or melamine foam with parameters according to the Johnson-Champoux-Allard-Lafarge (JCAL) model (see Table 1) with and without horizontal-wall labyrinthine slit perforations. The frequency range plotted in the figures is restricted to below 1 kHz since the low-frequency performance is of most interest. As with other forms of perforation [1; 2], the higher permeability contrast between the perforation and the porous matrix yields the more significant change in low-frequency absorption. Figure 3: Normal incidence absorption spectra predicted for 5.8 cm thick hard-backed porous layers of rockwood and melamine foam described by the JCAL parameters listed in Table 1 with horizontal-wall labyrinthine slit perforations (\(w\) = 5 mm, \(b\) = 5.5 mm, \(l\) = 40 mm, \(N\) = 5 as shown on the right).