## 3 Results
### Two Continuous Layers in Series  
First, we consider a two-layer continuous system with a unit cell matrix given by Equation (8), with \(\mathbf{M}_{\kappa}=\mathbf{M}_{1}\) and \(\mathbf{M}_{\beta}=\mathbf{M}_{2}\), as in Equation (5); the dispersion relation takes the form:  
\[\cos\phi=\cos\!\left(\frac{d_{1}}{c_{1}}\omega\right)\cos\!\left(\frac{d_{2}}{c _{2}}\omega\right)-\frac{1}{2}\!\left(\frac{Q_{1}}{Q_{2}}+\frac{Q_{2}}{Q_{1}} \right)\sin\!\left(\frac{d_{1}}{c_{1}}\omega\right)\sin\!\left(\frac{d_{2}}{c _{2}}\omega\right), \tag{11}\]  
where the subscripts 1 and 2 indicate the two continuous phases. Relation (11) corresponds to the equation obtained by [PERSON] and [PERSON] [40] (their Equation (38), in which their parameter \(dh\) is now \(\phi\)) and by [PERSON] et al. [41] (their Equation (14)). The dispersion relation (11) is shown in Figure 3 (diagrams have been created in Wolfram Mathematica (version 12.0)). It presents propagating and no-propagating bands in which the group velocity \(\partial\omega/\partial k\) can be either positive or negative. In Equation (11), \(\phi\) plays the role of the Floquet-Bloch parameter, so that the nature of the solution changes from pass band when \(\phi\) is real, that is for \(|\cos\phi|\leq 1\) and \(|\text{tr}\mathbf{M}|\leq 2\), to stop band when \(\phi\) is complex, that is for \(|\cos\phi|>1\) and \(|\text{tr}\mathbf{M}|>2\). When \(\phi\) is complex, we take \(\phi=\pi-i\zeta\) or \(\phi=i\zeta\) (\(\zeta>0\)), where \(\zeta=\log(|\cos\phi|+\sqrt{\cos^{2}\phi-1})\) is the decay exponent of the exponentially decaying solution. In the quasi static limit, for \(\omega\to 0\) and \(k\to 0\), the structured medium appears to be non dispersive with homogenised phase and group velocity both equal to \(\sqrt{\bar{\mu}/\bar{\rho}}\), where:  
\[\bar{\mu}=\frac{\mu_{1}\mu_{2}}{d_{1}\mu_{2}+d_{2}\mu_{1}}d\quad\text{ and }\quad\bar{\rho}=\frac{d_{1}\rho_{1}+d_{2}\rho_{2}}{d}, \tag{12}\]  
are the homogenised shear modulus and mass density, respectively, with \(d=d_{1}+d_{2}\). Note that the homogenised shear modulus and mass density are equal to the harmonic and arithmetic means, respectively, of the phases values. Figure 3: Continuous laminate. Dispersion diagram and decay exponent \(\zeta\) for a two-layer continuous system. The Bloch parameter \(Re[\phi]\) (continuous black lines) and the decay exponent \(\zeta\) (dashed grey lines) are given as a function of the angular frequency \(\omega\). The material and geometric parameters are: shear moduli, \(\mu_{1}=33\times 10^{3}\) MPa, \(\mu_{2}=201\times 10^{3}\) MPa; mass densities \(\rho_{1}=2300\) kg/m\({}^{3}\), \(\rho_{2}=7800\) kg/m\({}^{3}\); phase lengths \(d_{1}=0.27\) m, \(d_{2}=0.73\) m.  
We introduce now the homogenised characteristic frequency:  
\[\bar{\omega}=\sqrt{\frac{\bar{\mu}}{\bar{\rho}}}\ \frac{1}{d}, \tag{13}\]  
which contains the length-scale factor \(d\); correspondingly, we define the characteristic frequencies:  
\[\omega_{1}=\sqrt{\frac{\mu_{1}}{\rho_{1}}}\frac{1}{d_{1}},\qquad\qquad\omega_{2 }=\sqrt{\frac{\mu_{2}}{\rho_{2}}}\frac{1}{d_{2}}, \tag{14}\]  
having ratio \(\eta_{c}=\omega_{1}/\omega_{2}\) and the following elastic impedances:  
\[q_{1}=d_{1}\sqrt{\mu_{1}\rho_{1}},\qquad\qquad q_{2}=d_{2}\sqrt{\mu_{2}\rho_{ 2}}, \tag{15}\]  
having ratio \(\xi_{c}=q_{1}/q_{2}\), so that the quantities \(|\ln\eta_{c}|\) and \(|\ln\xi_{c}|\) measure the contrast between the two phases.