# Theoretical Optimization of Trapped-Bubble-Based Acoustic Metamaterial Performance
## 3 Results and Discussion
### Numerical Validation  
Now, let us examine dispersion relations for all of the frequencies obtained above. Introduce non-dimensional quantities \(p=d/a\) and \(q=h/a\) for convenience and consider the behavior of modes \((m,1)\) normalized by mode \((0,1)\). The case of 1 interface without reflector is shown in Figure 2. Figure 2 represents the case without interface and without reflector (given by Equation (18)) and it is provided for reference. Figure 2,d correspond to \(\omega_{01}\) from Equation (24) and \(\omega_{02}\) from Equation (22). The plots look pretty similar due to relatively low value of dissipation factor (\(\delta=0.05\)) and demonstrate the presence of bandgaps while the values of the resonant peaks increase with increasing \(p\) until hardly recognizable maximum (compare \(p=10\) and \(p=50\)) and then reaches saturation. Large values of \(p\) correspond to the case of independent scatterers. Finally, Figure 2 corresponds to \(\omega_{02}\) from Equation (22). The behavior of dispersion curve changes dramatically as \(p\) increases from \(p=3\) to \(p=4\). More specifically switching between the modes happens. Hereafter, under \"switching\" between modes we understand changing dispersion curve position along horizontal axis that corresponds to a different value of \(\kappa a\). Moreover, the value of the resonant peak increases before mode switching and starts to decrease after it. Figure 3 shows the case of 1 interface with reflector featuring amplitude reflection coefficient \(r_{0}~{}=~{}0.5\). Figure 3,c correspond to \(\omega_{02}\) from Equations (32) and (35), respectively. The difference between them is that Figure 3 neglects the dissipation, while Figure 3 accounts for it. The switching between modes is observed in both cases, while it happens for higher values of \(p\) if dissipation is considered. For \(\omega_{01}\) from Equations (32) and (35) (Figure 3,d, respectively) no switching is observed and the solutions look pretty much similar. If compared with Figure 2,d, no maximum value of the peak is visible, while only increasing with \(p\) until saturation in case of independent scatterers. It can also be seen that in presence of reflector peak values reduce significantly. Figures 4 and 5 provide the similar plots for two interfaces. For \"negative\" branch, there is a usual switching between modes that, however, also differs in case of absence (Figure 4,c) and presence of reflector (Figure 5,c) with the later exhibiting lower values of the resonant peak. As for \"positive\" branch of the solution, there is, again, no switching between the mode is observed while the peak values visibly increase, reach maximum, and decrease down to saturation with increasing \(p\). Next, we consider the effect of reflector explicitly, by fixing one of the geometry parameters \(p,q\) and varying another one along with the amplitude reflection coefficient, \(r_{0}\). Let us first fix \(q=1\) (the smallest possible value that represents the case of a  
Figure 3: Dispersion relation for metascreen (**a**) w/1 interface and w/reflector (\(r_{0}=0.5\)), \(k\,h\ll\,1,\delta\ll\,1\), “negative” branch; (**b**) w/1 interface and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), “negative” branch; (**c**) w/1 interface and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), “negative” branch; (**d**) w/1 interface and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), “positive” branch. For all plots \(n=1,q=2\). Dashed lines in (**a**,**c**) are used to demonstrate switching between modes; dotted lines in (**b**,**d**) demonstrate the largest resonant peak for various configurations of metamaterial and compare it with the case of independent scatterers (black curve). Figure 2: Dispersion relation for metascreen (**a**) w/o interface and w/o reflector; (**b**) w/1 interface and w/o reflector, \(kh\ll 1\), \(\delta\ll 1\); (**c**) w/1 interface and w/o reflector, \(kh\ll 1\), “negative” branch; (**d**) w/1 interface and w/o reflector, \(kh\ll 1\), “positive” branch. For all plots \(n=1,q=2\).