## 3 Results
### Single Oscillator in Series  
We consider a discrete oscillator arranged in series between the two continuous phases as in the inset of Figure 11. The transfer matrix of the unit cell can be obtained by direct multiplication of the transfer matrix of the single phases as in Equation (8), namely \(\mathbf{M}_{S}=\mathbf{M}_{\beta}\,\mathbf{M}_{\chi}\,\mathbf{M}_{\kappa}\), where \(\mathbf{M}_{\kappa}\) and \(\mathbf{M}_{\beta}\) are in the the form of (5) and \(\mathbf{M}_{\chi}\) in the form of (6). The dispersion relation is:  
\[\cos\phi = \frac{1}{4}\bigg{[}\frac{(1+\xi_{c})^{2}}{\xi_{c}}\cos\phi_{+}\,- \frac{(1-\xi_{c})^{2}}{\xi_{c}}\cos\phi_{-}\bigg{]}\bigg{[}1-2\big{(}\frac{ \eta_{m}}{\delta}\big{)}^{2}\Omega^{2}\bigg{]}+\] \[-\frac{1}{4\xi_{m}}\bigg{[}(1+\xi_{c})\sin\phi_{+}\,+(1-\xi_{c}) \sin\phi_{-}\bigg{]}\frac{\eta_{m}}{\delta}\Omega+\] \[-\frac{\xi_{m}}{\xi_{c}}\bigg{[}(1+\xi_{c})\sin\phi_{+}\,-(1-\xi_ {c})\sin\phi_{-}\bigg{]}\frac{\eta_{m}}{\delta}\Omega\bigg{[}1-\big{(}\frac{ \eta_{m}}{\delta}\big{)}^{2}\Omega^{2}\bigg{]}.\]  
In this case, the characteristic frequency of the discrete oscillator is \(\omega_{m}=\sqrt{2\gamma/m}\) and its elastic impedance is \(q_{m}=\sqrt{2\gamma\,m}\); they define the frequency \(\eta_{m}=\omega_{1}/\omega_{m}\) and impedance \(\xi_{m}=q_{1}/q_{m}\) ratios as previously. In Figure 11, the dispersion diagram and the decay exponent are shown for the bi-layered laminate with the discrete oscillator arranged in series. The presence of the discrete oscillator drastically alters the dispersion properties in the whole frequency range: as a consequence of its introduction, narrower and narrower pass bands are obtained at increasing frequency; on the other hand, the central frequency of each band is determined by the continuous phases. By inspection of Equation (26), it is clear that the polynomial terms in \((\eta_{m}/\delta)\Omega\), associated with the discrete oscillator, change the dispersive behaviour of the micro-structured system. Polynomial terms in \((\eta_{m}/\delta)\Omega\) increase in magnitude with the frequency and generate wider and wider stop bands, where \(\cos\phi>1\). Such a polynomial dependence induces larger attenuation, as evidenced by the decay exponent \(\xi\) in Figure 11, which has increasing maximum values passing from one stop band to the next one. In turn, the frequency position of the narrower and narrower pass bands are determined by the continuous phases, associated with the trigonometric terms in \(\phi_{+}\) and \(\phi_{-}\), which make the absolute value of the right-hand side of Equation (26) less than \(1\). The continuous nature of the micro-structured system is also linked to the fact that the system is not a low-pass filter and, in principle, it can support waves without a frequency upper limit. The distributions of the bands in term of the eigenfrequency ratio \(\eta_{m}\) and the impedance ratio \(\xi_{m}\) are detailed in Figure 12a,b, respectively. The oscillator eigenfrequency \(\Omega=\delta/\eta_{m}\), indicated in dashed grey line, is a threshold between two different behaviours: below this critical eigenfrequency, the system has a \"continuous\" dispersive character, where the  
Figure 11: Oscillator in series. Dispersion curves (continuous black lines) and decay exponent (dashed grey lines). Normalised frequency \(\Omega\) is given for \(\xi_{c}=0.22\), \(\eta_{c}=0.5\), \(\xi_{m}=8.71\) and \(\eta_{m}=1.265\). sizes of pass and stop bands are in general of the same order. In particular, in the limit \(\eta_{m}\to 0\), the dispersion Equation (26) reduces to (16) associated with the laminate described in Section 3. Above this threshold value, the system has the \"discrete-continuous\" character described in Figure 11, with highly localised pass bands over the all frequency axis that are positioned accordingly to the continuous phases.