#### 3.1. Metascreen Modes

To determine the resonant frequencies of the array, we should first know the resonant frequencies of the individual scatterer (trapped bubble). Instead of commonly used Minnaert frequency, we employ an approach proposed in [

19] and developed in our previous work [

18]. The modes of the interface vibrations read as [

18]:

where

${J}_{n}({\kappa}_{mn}a)$ is the Bessel function of the first type of the order

n, with

${\kappa}_{mn}$ being mode-dependent radial wave number. Here, we only account for the viscous damping. It is well known that for the clamped membrane multiple modes can be excited. However, we only pick the modes

$(m,1)$ for our calculations. Once the resonant frequency of the individual scatterer was identified, we apply all of the analysis [

13,

14,

20] to make the corrections to the resonant frequency of the system due to the strong coupling between the bubbles that allows shifting the resonant frequencies of the system.

The general approach to find resonant frequencies of the metamaterial includes two steps: (1) find

$\Omega $ from extremal problem for energy transmission coefficient; and, (2) find

$\omega $ from Equation (

7). Let us first consider an array of bubbles positioned in the liquid bulk. Solving Equation (

9), we arrive at:

or in terms of resonant frequency:

where

$A=\frac{2\sqrt{\pi}a}{d}$,

a, and

d stand for the bubble radius and distance between the centers of adjacent bubbles. Hereafter, we will use the notions of “negative” and “positive” branches of the solutions. These branches correspond to the values of

${\Omega}_{i}$ that has “−” and “+” in front of the “square root” terms. For instance, in Equation (

19)

${\Delta}_{0}-\sqrt{{\delta}^{2}+{\Delta}_{0}^{2}}$ is a “negative” branch of the solution, whereas

${\Delta}_{0}+\sqrt{{\delta}^{2}+{\Delta}_{0}^{2}}$ is a “positive” one. The transmission dip given by Equation (

18) is always present in the system, regardless of its complexity.

Next, let us assume the presence of the interface where the acoustic impedance changes abruptly from one side of the bubble array and bulk fluid from the other side. The interface can be formed by two media with different acoustic properties (e.g., water-air interface). Subsequently, solving Equation (

18), we get:

where

${\Delta}_{0}=(B(1+{h}^{2}{k}^{2})+2hk\delta )/2$. Here,

$B=\frac{4\pi ah}{{d}^{2}}$ and

h is the distance between the bubble’s center and adjacent interface. Subsequently, the resonant frequencies of the system read as:

Now, let us assume that

$hk\ll 1$. Subsequently, Equation (

19) becomes:

and the resonant frequencies of the system read as:

Finally, if both

$hk\ll 1$ and

$\delta \ll 1$, the solution reduces to:

that results in two resonant frequencies of the system:

Similarly, if there are interfaces with abrupt acoustic impedance change from both sides of the bubble array, we obtain from Equation (

13):

where

${\Delta}_{1}=(B+4hk\delta )/4$. Now, let us assume that

$hk\ll 1$, Subsequently, Equation (

25) becomes:

and the resonant frequencies of the system read as:

Finally, if both

$hk\ll 1$ and

$\delta \ll 1$, the solution reduces to:

that results in two resonant frequencies of the system:

Solutions given by Equations (

24) and (

29) are identical to Equation (

2) from [

14], where the dissipation factor was neglected.

#### 3.3. 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 2a represents the case without interface and without reflector (given by Equation (

18)) and it is provided for reference.

Figure 2b,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 2c 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

${r}_{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$.

Figure 3a,c correspond to

${\omega}_{02}$ from Equations (

32) and (

35), respectively. The difference between them is that

Figure 3a neglects the dissipation, while

Figure 3c 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 3b,d, respectively) no switching is observed and the solutions look pretty much similar. If compared with

Figure 2b,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.

Figure 4 and

Figure 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 4a,c) and presence of reflector (

Figure 5a,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 reflector positioned just next to the bubble array) and vary $p=2\dots 10$ and ${r}_{0}=0\dots 1$.

Figure 6 shows the corresponding dispersion relation. As can be seen from it, no switching between modes is observed while the value of the resonant peak decreases (from red to black plot) monotonically with increasing

${r}_{0}$. This trend remains the same with

p increasing. At the same time, the effect of variation in amplitude reflection coefficient becomes smaller as

p increases.

Alternatively, we fix

$p=3$ and vary

$q=1\dots 10$ along with

${r}_{0}=0\dots 1$, as shown in

Figure 7. In this case, the maximum values of the peak decrease monotonically with both

${r}_{0}$ and

q increasing and several bandgaps are present, as previously.

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). 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. However, the trend becomes more moderate with increasing

p until it becomes independent of

q for large values of

p.

Finally, consider the case of varying reflection coefficient,

${r}_{0}$. It is shown in

Figure 10. Here, we fix

$q=1$ (lowest possible value) and examine the behavior of normalized frequency of the system with

p. In

Figure 10a,c the “positive” branch of the solution is presented without (

Figure 10a) and with dissipation (

Figure 10c). The trend is overall repeatable with normalized frequency reaching saturation for large

p. The dramatic difference is that for small dissipation the rate of reaching saturation decreases with increasing

${r}_{0}$ whereas it increases with increasing

${r}_{0}$ when dissipation is taken into account. For “negative” branch of the solution (

Figure 10b,d) the trend is almost the same: abrupt decrease of normalized frequency with increasing

p. However, the starting point of abrupt decay moves to lower values of

p with increasing

${r}_{0}$, opposite to the trend observed in

Figure 8d.