## 3 Results
### Phases in Parallel
#### 3.2.2 Continuous Phase in Parallel  
We investigate now the effect of a continuous phase arranged in parallel at the interface of the bi-layer laminate system, as indicated in the inset of Figure 9a. The transfer matrix \(\mathbf{M}_{T}\) of the unit cell can be built as for the case of a single oscillator in parallel with the difference that \(\mathbf{M}_{T}\) is now in the form of (5). We introduce the additional characteristic frequency and impedance ratios as \(\eta_{p}=\omega_{1}/\omega_{3}\) and \(\xi_{p}=q_{1}/q_{3}\), respectively, where \(\omega_{3}=\sqrt{\mu_{3}/\rho_{3}}/d_{3}\) and \(q_{3}=d_{3}\sqrt{\mu_{3}\rho_{3}}\) refer to the phase in parallel, as shown in the inset of Figure 9a. The dispersion relation takes the form:  
\[\cos\phi = \frac{1}{4}\Big{[}\frac{(1{+}\xi_{c})^{2}}{\xi_{c}}\cos\phi_{+} \,-\frac{(1{-}\xi_{c})^{2}}{\xi_{c}}\cos\phi_{-}\,\Big{]}+ \tag{22}\] \[-\frac{1}{4\xi_{p}^{2}}\Big{[}(1{+}\xi_{c})\sin\phi_{+}\,+(1{-} \xi_{c})\sin\phi_{-}\,\Big{]}\tan\frac{\eta_{3}}{\delta}\Omega.\]  
The terms within the first square bracket on the right-hand side of Equation (22) correspond to the right-hand side of the dispersion Equation (11), describing the effect of the two continuous phases in series. The comparative analysis between the dispersion Equations (20) and (22) shows that they only differ in the terms including the singularity, namely:  
\[\frac{\eta_{m}\Omega}{1-\big{(}\frac{\eta_{p}}{\delta}\big{)}^{2}\Omega^{2}} \qquad\text{and}\qquad\tan\frac{\eta_{p}}{\delta}\Omega. \tag{23}\]  
The two above expressions have simple poles at the corresponding characteristic frequencies, a single frequency \(\Omega=\delta/\eta_{m}\) for the discrete case (we do not consider the negative frequency \(\Omega=-\delta/\eta_{m}\)) and the multiple frequencies:  
\[\tilde{\Omega}\simeq\frac{1+2\,n}{2}\,\pi\,\frac{\delta}{\eta_{p}},\qquad n \in\mathcal{N}_{0}, \tag{24}\]  
Figure 9: Continuous phase in parallel. Results are given for \(\xi_{c}=0.22\), \(\eta_{c}=0.5\). (**a**) Dispersion curves (continuous black lines) and decay exponent (dashed grey lines) for \(\xi_{p}=1.20\), \(\eta_{p}=0.60\). (**b**) Band distribution, where \(\Omega\) is given as a function of the impedance ratio \(\xi_{p}\) (\(\eta_{p}=0.60\)). for the continuous one. Similar to the dispersion equation for the discrete resonator, the second term within square brackets on the right-hand side of Equation (22) reflects the presence of the continuous phase in parallel. The comparative analysis between the dispersion diagrams of Figures 6 and 9a reveals that the presence of the continuous phase in parallel introduces multiple flat bands in the vicinity of the normalised eigenfrequencies:  
\[\hat{\Omega}\simeq\frac{1+2\,n}{2}\,\pi\,\frac{\delta}{\eta_{p}},\qquad n\in \mathcal{N}_{0}, \tag{25}\]  
when \(\xi_{p}\) is sufficiently large. These are the eigenfrequencies of the finite system indicated in the top right inset in Figure 9b, where the single continuous phase in parallel has clamped and free boundary conditions. This effect is similar to the case of Section 3.2.1: for the discrete oscillator, a single additional band is created, whereas for the continuous phase, multiple bands are generated in the neighbourhood of the eigenfrequencies reported in (25), the constant frequency interval between eigenfrequencies being \(\Delta\hat{\Omega}=r\delta/\eta_{p}\). Indeed, this is particularly evident by looking at the band distributions as a function of the characteristic frequency ratios \(\eta_{m}\) in Figure 7 and \(\eta_{p}\) in Figure 10, which reveal that the bands created by the bi-layered laminate (which can be recognised looking at the limit \(\ln(\eta_{m}),\ln(\eta_{p})\to-\infty\), respectively) are perturbed in correspondence of the eigenfrequencies of the system arranged in parallel: a single eigenfrequency for the discrete oscillator, multiple ones for the continuous phase.