## 4 Energy Distribution
Dashed lines correspond to the bi-layered system, while the continuous line corresponds to the bi-layered system with the oscillator disposed in parallel (indicated in black). Material and geometrical parameters are the same as Figure 6a,b. Pass band frequency intervals are indicated at the top and at the bottom of the plot. The energy \(E_{\chi_{c}}\) of the continuous phase arranged in parallel can be computed by implementing the relevant transfer matrix as given in Equation (10). Finally, the energy \(E_{S_{d}}\) of the single oscillator arranged in series is given by:  
\[E_{S_{d}} = \frac{1}{2}\frac{\gamma}{(2\gamma-m\omega^{2})^{2}}\bigg{[}\frac{2 \gamma^{2}-\gamma\,m\omega^{2}+m^{2}\omega^{4}}{2}(A_{d}\bar{A}_{d}+B_{d}\bar{B }_{d})- \tag{34}\] \[-(2\gamma^{2}-3\gamma m\omega^{2})\mathcal{R}e(A_{d}\bar{B}_{d}) \bigg{]}.\]  
where \(A_{d}\) and \(B_{d}\) are the complex displacement amplitudes. In Figures 13-15, we present the distribution of the energies within the phases as a function of the frequency \(\omega\) for the mechanical systems described in the previous section, where the additional phase is arranged in parallel and in series. The energies are normalised by the total energy stored in the unit cells and are presented with dashed lines for the bi-phase system and with continuous lines for the different tri-phase systems. We note that the energies corresponding to waves propagating in opposite direction are coincident in the propagating bands and distinct in the stop bands. In the last case, the upper curves correspond to the propagation direction. In Figure 13, we present the effect of the discrete oscillator arranged in parallel. The comparative analysis with the two-phase system shows that, in correspondence of the characteristic frequency \(\omega_{m}\), the energy is almost entirely localised in the discrete oscillator, whereas, departing from this frequency, the behaviour tends to be practically coincident to the bi-laminate. This is fully consistent with the outcomes of Figure 7.  
The effects of the continuous phase arranged in parallel are shown in Figure 14. It can be seen that the continuous phase arranged in parallel localises most of the energy in correspondence of its eigenfrequencies (indicated with three arrows), even if the amount of stored energy does not reach the same peak as for the discrete oscillator. Finally, in Figure 15, the effect of the discrete oscillator arranged in series is shown. As for the dispersion diagram, the introduction of the discrete phase radically changes the dynamic behaviour of the continuous phases. It is noted that the energy is concentrated in the oscillator in correspondence of its characteristic frequency, but not in the thin pass band  
Figure 14: Energy distribution between different phases as a function of the radian frequency \(\omega\). Dashed lines correspond to the bi-layered system, while the continuous line corresponds to the bi-layered system with the continuous phase disposed in parallel (indicated in black). Results are given for \(\xi_{c}=0.22\), \(\eta_{c}=0.5\), \(\xi_{p}=0.75\) and \(\eta_{p}=1.47\). Pass band frequency intervals are indicated at the top and at the bottom of the plot. Arrows indicate resonance frequencies of the phase arranged in parallel to the bi-layered system. opened at higher frequencies. Furthermore, the high energy concentration in the phases in the higher frequency stop bands is an indication of the high level of attenuation, as shown in Figure 11.