## 4 Alternating-Width Vertical-Wall Labyrinthine Slit Perforation in a Porous Matrix  
If a slit has abrupt changes in cross section along its length, the fluid streamlines are perturbed and, as a result, the tortuosity is increased. Consider the vertical-wall labyrinthine slit shown in Figure 6 in which the slit width alternates between \(b\) and \(b(1+\delta)\). The broken line in Figure 6 indicates the assumed zigzag approximation to the streamline path. The tortuosity of this labyrinthine architecture, \(T_{AWZ}\), is the product of that resulting from the zig-zag approximation of the streamline path and that due to the changes in cross  
Figure 5: Normal incidence absorption coefficient spectra predicted for 0.05 m thick hard-backed layer composed from a inclined slit matrix (0.04 mm wide slits at 70\({}^{\circ}\) to the normal, porosity 0.9) with (**a**) varying wall length in a horizontal-wall labyrinthine slit perforation (\(w\) = 1 mm, \(b\) = 5.1 mm, \(N\) = 8); \(l\) = 0.04 m (solid line), \(l\) = 0.08 m (dotted line), \(l\) = 0.12 m (dash-dot line) or (**b**) varying slit folding number in a vertical-wall labyrinthine slit perforation (\(w\) = 1 mm, \(b\) = 3 mm, \(l\) = 45 mm, \(N\) = 5 (continuous line), \(N\) = 10 (blue dotted line), \(N\) = 15 (black broken line). The low absorption dash-dot lines represent predictions for the respective labyrinthine slit perforations in a non-porous solid matrix. Figure 6: A vertical-wall labyrinthine slit with alternating widths. section [8] which is the second factor in Equation (15). Similarly, the tortuosity, \(T_{AWC}\), with the slit centre line path approximation is given by Equation (16). \[ \begin{split}& T_{AWZ}\\ &=\left\{\frac{\frac{(N-1)\left[\sqrt{\ell^{2}+(b(1+\delta)+w)^{2}} +\sqrt{\ell^{2}+(b+w)^{2}}\right]+\sqrt{\ell^{2}+(b+w)^{2}}/4}+\sqrt{\left(l+ b/2\right)^{2}+(b(1+\delta)+w)^{2}/4}+w/2-(l+b+2w)}{l+b+2w}\right\}^{2}\\ &\times\left[\frac{(2+\delta)^{2}}{4(1+\delta)}\right]\\ & T_{AWC}=\left\{\frac{\frac{Nb}{2}(2+\delta)+(N-1)(w+l)}{l+b+2 w}\right\}^{2}\left[\frac{(2+\delta)^{2}}{4(1+\delta)}\right]\\ &\end{split}  \tag{16}\]  
The corresponding complex density and compressibility are given by [9],  
\[ \begin{split}&\rho(\omega)=T_{AW}\frac{(1+\delta)\rho_{1}(\omega)+ \epsilon\rho_{2}(\omega)}{1+\delta+\epsilon},\;C(\omega)=\frac{C_{1}(\omega)+ (1+\delta)C_{2}(\omega)}{2+\delta}\;\;\rho_{1,2}(\omega)=\rho_{0}/\left[1- \frac{\tanh\left(\lambda_{1,2}\sqrt{(-i)}\right)}{\lambda_{1,2}\sqrt{(-i)}} \right],\\ &\lambda_{1}=(b/2)\sqrt{\omega\rho_{0}/\mu},\;\lambda_{2}=(b/2)( 1+\delta)\sqrt{\omega|\rho_{0}/\mu},\;C_{1,2}(\omega)=\left(\rho_{0}{\varsigma _{0}}^{2}\right)^{-1}\left[\gamma\;-\;(\gamma\;-\;1)H\left(\lambda_{1,2}\, \sqrt{(N_{PR})}\right)\right]\end{split}  \tag{17}\]  
For the alternating-width vertical-wall labyrinth shown in Figure 6, the distance between adjacent slit openings is \([(N+1)w+Nb]\). The pressure diffusion function for the alternating-width labyrinthine slit perforation, \(F_{dAW}(\omega)\), is obtained from Equations (12), (13) and (18). \[D(0)=(1-\Omega_{B})\left\{\frac{[(N+1)w+Nb+(N+1)(b/2)]^{2}}{12}\right\},\; \Omega_{B}=\frac{b(N+1)(w+b+1)+\frac{b}{(l+b)(\delta-1)}}{(l+b+2w)\left[(N+2) w+(N+1)b\left(1+\frac{\delta}{2}\right)\right]},\;\Lambda_{d}=\frac{2[(N+1)w+ Nb]((N+2)w+(N+1)b]}{(N+3)w+(N+1)b(1+\delta/2)+l} \tag{18}\]  
A consequence of the increased tortuosity due to alternating slit width is that, for a given layer thickness, wall length and porous matrix, a low-frequency absorption peak at a target frequency can be achieved with a lower slit folding number than required with a uniform labyrinthine slit width.