## 2 Labyrinthine Slits Formed by Walls Parallel and Normal to the Layer Surface  
Consider a plane wave at normal incidence on a layer of contiguous rectangular unit cells, each of which contains a single labyrinthine slit of width \(b\) with \(N\) folds in the slit formed by walls of length \(l\) and width \(w\). The walls are either parallel to (Figure 1a) or normal to (Figure 1b) the surface on which the wave is incident. In this paper, the former arrangement is called a horizontal-wall labyrinth and the latter arrangement is called a vertical-wall labyrinth. Associated parameters are given H or V subscripts.  
The shortest paths through the slit labyrinths are approximated by the zigzag paths represented by broken lines in Figure 1a,b. The corresponding tortuosity values (\(T_{HZ}\), \(T_{VZ}\)) are underestimates since the fluid streamlines must pass around the internal walls. On the other hand, calculations of tortuosity (\(T_{HC},\ T_{VC}\)) using piecewise linear paths along the centres of the labyrinthine slits are overestimates since the streamlines will curve around corners. Expressions for labyrinth tortuosity corresponding to Figure 1a,b and according to these streamline path length estimates are given in Equations (1) to (4), respectively. Predictions in this paper assume the zigzag path approximations.  
\[T_{HZ}=\bigg{\{}\frac{(N-1)\sqrt{(l+b)^{2}+(b+w)^{2}}+\sqrt{(l+b/2)^{2}+(b+w)^ {2}/4}+w/2-[(N+1)w+N]}{(N+1)w+Nb}\bigg{\}}^{2}, \tag{1}\]  
\[T_{HC}=\bigg{[}\frac{Nl-w-b/2}{(N+1)w+Nb}\bigg{]}^{2} \tag{2}\]  
\[T_{VZ}=\bigg{\{}\frac{(N-1)\sqrt{\beta^{2}+(b+w)^{2}}+\sqrt{\beta^{2}+(b+w)^{ 2}/4}+\sqrt{(l+b/2)^{2}+(b+w)^{2}/4}-w-b/2-l}{l+b+2w}\bigg{\}}^{2}, \tag{3}\]  
\[T_{VC}=\bigg{[}\frac{N(l+b)+(N-1)w}{l+b+2w}\bigg{]}^{2} \tag{4}\]  
The expressions for bulk, \(\Omega_{B}\), and surface, \(\Omega_{S}\), porosities of the horizontal- and vertical-wall labyrinth structures in Figure 1 are given in Equations (5) to (8),  
\[\Omega_{BH}=\{[Nb(l+b+w)]/[((N+1)w+Nb)(l+b+2w)]\}, \tag{5}\]  
Figure 1: (**a**) Unit cells containing a single labyrinthine slit, formed by walls either (**a**) parallel to or (**b**) normal to the insonfied surface, detailing the associated dimensions, zigzag approximations to streamline paths (broken lines) and path lengths.  
\[\Omega_{SH}=b/(l+b+2w) \tag{6}\] \[\Omega_{BV}=\{[(N+1)(l+b+w)]/[((N+2)w+(N+1)b)(l+b+2w)]\},\] (7) \[\Omega_{SV}=b/((N+2)w+(N+1)b) \tag{8}\]  
The complex density, (\(\rho_{L}(\omega)\)), complex compressibility, (\(C_{L}(\omega)\)), propagation constant, (\(k_{L}(\omega)\)), and normalised impedance (\(Z_{L}(\omega)\)) of an effective fluid medium equivalent to a labyrinthine slit structure are, respectively [5],  
\[\rho_{L}(\omega)=(T/\Omega_{B})\rho(\omega),\;\rho(\omega)=\rho_{0}/H(\lambda ),\;H(\lambda)=1-\tanh\Bigl{(}\lambda\sqrt{-i}\Bigr{)}/\Bigl{(}\lambda\sqrt{- i}\Bigr{)},\;C_{L}(\omega)=\Omega_{B}C(\omega) \tag{9}\]  
\[\begin{array}{l}C(\omega)=\bigl{(}\rho_{0}c_{0}^{2}\bigr{)}^{-1}\Bigl{[} \gamma\;-\;(\gamma\;-\;1)H\Bigl{(}\lambda\sqrt{(N_{\rm PR})}\Bigr{)}\Bigr{]}, \;\lambda=(b/2)\sqrt{\omega\rho_{0}/\mu},\\ k_{L}(\omega)=\omega\sqrt{\rho_{L}(\omega)C_{L}(\omega)},\;Z_{L}(\omega)=( \Omega_{S}\rho_{0}c_{0})^{-1}\sqrt{\rho_{L}(\omega)/C_{L}(\omega)}\end{array} \tag{10}\]  
\(\rho_{0},\;c_{0},\;N_{\rm PR}\) and \(\mu\) are density, sound speed, Prandtl number and dynamic viscosity in air, respectively, \(\Omega_{B}\) is either \(\Omega_{BH}\) or \(\Omega_{BV}\), \(\Omega_{S}\) is either \(\Omega_{SH}\) or \(\Omega_{SV}\) and \(T\) is either \(T_{HZ}\) or \(T_{VZ}\). Time dependence \(e^{-i\omega t}\) is understood where \(i=\sqrt{(-1)}\).