# Theoretical Optimization of Trapped-Bubble-Based Acoustic Metamaterial Performance
## 3 Results and Discussion
### Numerical Validation
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 4: Dispersion relation for metascreen (**a**) w/o interface and w/o reflector; (**b**) w/2 interfaces and w/o reflector, \(kh\ll 1\), \(\delta\ll 1\); (**c**) w/2 interfaces and w/o reflector, \(kh\ll 1\), ”negative” branch; (**d**) w/2 interfaces and w/o reflector, \(kh\ll 1\), ”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 5: Dispersion relation for metascreen (**a**) w/2 interfaces and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), \(\delta\ll 1\), ”negative” branch; (**b**) w/2 interfaces and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), \(\delta\ll 1\), ”positive” branch; (**c**) w/2 interfaces and w/reflector (\(r_{0}=0.5\)), \(kh\ll 1\), ”negative” branch; (**d**) w/2 interfaces 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). Now, let us examine the behavior of the system resonant frequency normalized by that of a single bubble with respect to geometry parameters \(p\) and \(q\). In Figure 8, the dependence of normalized system frequency from \(p\) for case of 2 interfaces with and without reflector is shown. For all values of \(q\) and all solution branches normalized frequency tends to 1 for large values of \(p\) that corresponds to the case of independent scatterers. For \"positive\" branch of the solution (Figure 8a,c) the presence of the reflector flattens the resonant peak observed for small values of \(p\) (strongly coupled bubbles). Figure 6: Dispersion relation for metascreen w/1 interface and w/reflector (**a**) \(q=1,p=2\); (**b**) \(q=1,p=4\); (**c**) \(q=1,p=6\); (**d**) \(q=1,p=10\). For all plots \(n=1,r_{0}=0.5,kh\ll 1,\delta\ll 1\). “Positive” branch only is shown. Figure 7: Dispersion relation for metascreen w/1 interface and w/reflector (**a**) \(q=1,p=3\); (**b**) \(q=2,p=3\); (**c**) \(q=5,p=3\); (**d**) \(q=10,p=3\). For all plots \(n=1,r_{0}=0.5,kh\ll 1,\delta\ll 1\). “Positive” branch only is shown. The behavior of normalized frequency also changes from resonant to monotonically increasing until saturation with increasing \(q\). A completely different behavior is observed for \"negative\" branch of the solution (Figure 8b,d). In particular, the system experiences abrupt monotonic decay of normalized frequency starting from a certain value of \(p\) that is almost independent from \(q\) if the reflector is absent. The situation changes when the reflector is added. In this case, the starting point of abrupt decay shifts to higher values of \(p\) with increasing \(q\). Figure 9 shows the dependence of normalized system frequency as a function of \(q\). Similar to above, we consider the case with two interfaces with and without reflector. For \"positive\" branch of the solution (Figure 9a) the presence of reflector (Figure 9c) does not effect the behavior of the system. The normalized frequency decreases monotonically with increasing \(q\) and tends to \(1\) with increasing \(p\). However, the situation changes for \"negative\" branch of the solution. If the reflector is absent (Figure 9b) the normalized frequency does not change with \(q\) and decreases with increasing \(p\). If the reflector is present (Figure 9d), the normalized frequency increases abruptly with increasing \(q\).