## 3 Results
### Phases in Parallel
#### 3.2.1 Single Oscillator in Parallel  
We start with a discrete oscillator as in the insets of Figure 6b or Figure 6c (from now on, for visualisation purposes, we indicate the spring horizontally). The transfer matrix  
Figure 5: Continuous laminate. Band distribution as a function of the impedance ratio \(\xi_{c}\). Normalised frequency \(\Omega\) is given for the characteristic frequency ratio \(\eta_{c}=0.5\). In the upper right, a detail of the band distribution is given. On the bottom right, the same results are presented in term of \(\omega/\omega_{1}\). Filled area indicates propagating bands. of the unit cell is the matrix \(\mathbf{M}_{T}\) described at the end of Section 2.2, where \(\mathbf{M}_{\alpha}\) and \(\mathbf{M}_{\beta}\) are as in Equation (5) and \(\mathbf{M}_{\chi}\) as in Equation (6). We introduce the characteristic frequency \(\omega_{m}=\sqrt{\gamma/m}\), the impedance \(q_{m}=\sqrt{\gamma\,m}/d_{m}\) (\(d_{m}\) is a length parameter that can be taken equal to unity) and the frequency and impedance ratios \(\eta_{m}=\omega_{1}/\omega_{m}\) and \(\xi_{m}=q_{1}/q_{m}\), respectively. Then, the dispersion relation has the normalised form:  
\[\cos\phi = \frac{1}{4}\bigg{[}\frac{(1+\xi_{c})^{2}}{\xi_{c}}\cos\phi_{+}- \frac{(1-\xi_{c})^{2}}{\xi_{c}}\cos\phi_{-}\bigg{]}+\] \[-\frac{1}{4\xi_{m}}[(1+\xi_{c})\sin\phi_{+}\,+(1-\xi_{c})\sin\phi _{-}\,]\frac{\frac{\eta_{m}}{\delta}\Omega}{1-\big{(}\frac{\eta_{m}}{\delta} \big{)}^{2}\Omega^{2}}.\]  
Note that Equation (20) is the same as Equation (16) plus an additional term on the right-hand side with a singularity at \((\eta_{m}/\delta)\Omega=1\) (i.e., \(\omega=\omega_{m}\)). In Figure 6, dispersion diagrams are shown for the simple bi-layered laminate (part (a)) and for the cases where two different oscillators are included in the unit cell (parts (b) and (c)). For the laminate, \(\xi_{c}=0.22\) and \(\eta_{c}=0.5\), whereas the two oscillators are characterised by the impedance and frequency ratios \(\xi_{m}=8.71\), \(\eta_{m}=1.260\) (case (b)) and \(\xi_{m}=8.71\), \(\eta_{m}=0.525\) (case (c)). It is evident that the introduction of the oscillator induces a localised effect in the neighbourhood of its characteristic frequency \(\omega_{m}\) without altering the dispersion properties of the system in the rest of the frequency spectrum. In particular, it creates a thin stop band if \(\omega_{m}\) is in a pass band of the laminate (see case (b) in Figure 6) and a pass band if \(\omega_{m}\) is in a stop band of the laminate (see case (c) in Figure 6). Note that, for \(\Omega=\delta/\eta_{m}\), the decay exponent \(\zeta\to\infty\), as shown in the insets of Figure 6b,c, annihilating a propagating wave at this particular frequency. Note also the slope inversion effect of the dispersion curves. In fact, for the simple laminate, different branches alternate the slope (namely, the group velocity) from positive to negative and vice versa. The introduction of the discrete resonator generates two consecutive curves with group velocity of the same sign. In Figure 7, the distribution of the bands are given as a function of the ratio \(\eta_{m}\) for \(\xi_{c}=0.22\), \(\eta_{c}=0.5\) and \(\xi_{m}=8.71\). It is shown that in the whole range of \(\eta_{m}\) considered, the perturbation induced by the oscillator is always localised closed to the characteristic frequency of the oscillator and slightly larger when \(\omega_{m}\) approaches the upper or lower limit frequencies of one band. Note also that in the limit \(\xi_{m}\to\infty\), Equation (20) tends  
Figure 6: Oscillator in parallel. Dispersion curves (continuous black lines) and decay exponent (dashed grey lines). Normalised frequency \(\Omega\) is given for \(\xi_{c}=0.22\) and \(\eta_{c}=0.5\). (**a**) Bi-layered laminate. (**b**) Bi-layered laminate with oscillator, with \(\xi_{m}=8.71\), \(\eta_{m}=1.260\). (**c**) Bi-layered laminate with oscillator, with \(\xi_{m}=8.71\), \(\eta_{m}=0.525\).