## 2 Formulation of the Problem  
For the purposes of this paper, it is believed that phononic crystals are one-dimensional, heterogeneous, two-component media periodic in space, and the components can be both elastic and liquid or gaseous media. An inhomogeneous one-dimensional periodic medium consists of two components, medium 1 and medium 2, and the characteristics of these media have an index of 1 or 2, respectively (Figure 1). \(L\) is the smallest spatial period, \(X\) is a dimensional variable, \(x=X/L\) is a dimensionless spatial variable, here the period is 1. The unit cell is a set of such \(x\) that \(x^{*}<x<x^{*}+1\) where \(x^{*}\) is any number. Media 1 consists of a chain of drops, and media 2 consists of a chain of bubbles. The unit cell is \(0<x<1\). The coordinates of the borders of drops and bubbles should be as follows:  
\(\Omega_{1}=\ \{x\): \(0\leq x\leq k_{1}\}\ \ \ \Omega_{2}=\ \{x\): \(k_{1}\leq x\leq k_{1}+k_{2}\}\),  
\(\Omega_{3}=\ \{x\): \(k_{1}+k_{2}\leq x\leq k_{1}+k_{2}+k_{3}\}\), \(\Omega_{4}=\ \{x\): \(k_{1}+k_{2}+k_{3}\leq x\leq 1\}\)  
\(k_{i}\)--the size of the drop or bubble under the index <<>. Modeling of inhomogeneous media using chains of inclusions of one medium into another is one of the possible approaches for studying wave propagation. The wave propagation equations are described by an elementary wave packet, in which there is a circular frequency, a wave number and a wave amplitude:  
\(\mathrm{A}(x)e^{i(\xi,\ x-\omega t)}=u(x,t)\)  
\(\mathrm{A}(x+1)=\mathrm{A}(x)\) is amplitude factor. Fluctuations in one spatial period are described by the equations:  
\(\Omega_{1}:\ p_{1}^{\prime}+\lambda^{2}p_{1}=0,\ \lambda=\frac{\omega L}{c_{1}} \ \ \ \kappa=\frac{c_{1}}{c_{2}}\)  
\(\Omega_{2}\): \(\ p_{2}^{\prime}+\lambda^{2}k^{2}p_{2}=0\),  
\(\Omega_{3}:\ p_{3}^{\prime}+\lambda^{2}p_{3}=0\),  
\(\Omega_{4}:\ p_{4}^{\prime}+\lambda^{2}k^{2}p_{4}=0\),  
where \(\kappa=\frac{c_{1}}{c_{2}}\) is the ratio of sound velocities, \(\omega\) is the circular oscillation frequency, \(\lambda=\frac{\omega L}{c_{1}}\) is the spatial frequency, indices 1, 3 and 2, 4--correspond to media 1 and 2, with boundary conditions:  
\(p_{1}(0)e^{i\xi}\ =p_{4}(1)\), (1)  
\(p_{1}(k_{1})\ =p_{2}(k_{1})\), (2)  
\(p_{2}(k_{2})\ =p_{3}(k_{2})\), (3)  
\(p_{3}(k_{3})\ =p_{4}(k_{3})\), (4)  
Figure 1: The unit cell. \[\tau_{12}\frac{\partial p_{1}}{\partial x}(k_{1}) = \frac{\partial p_{2}}{\partial x}(k_{1}),\ \tau_{12}=\
icefrac{{p_{2}}}{{\rho_{1}}}, \tag{5}\] \[\tau_{23}\frac{\partial p_{2}}{\partial x}(k_{2}) = \frac{\partial p_{3}}{\partial x}(k_{2}),\ \tau_{23}=\
icefrac{{p_{3}}}{{\rho_{2}}},\] (6) \[\tau_{34}\frac{\partial p_{3}}{\partial x}(k_{3}) = \frac{\partial p_{4}}{\partial x}(k_{3}),\ \tau_{34}=\
icefrac{{p_{4}}}{{\rho_{3}}},\] (7) \[\tau_{12}\frac{\partial p_{1}}{\partial x}(0)e^{i\xi}=\frac{ \partial p_{4}}{\partial x}(1) \tag{8}\]  
where \(\tau\) is the ratio of the density of air (gas) to the density of water (liquid). \(\xi\) is the phase shift of the oscillations in the adjacent cells of the chain. Sound waves are described by an acoustic pressure disturbance, respectively, in these media. Helmholtz equations are used to solve the equations of elastic and acoustic waves. The general form of the solution of the Helmholtz equation for calculating the pressure in the cells \(\Omega_{1}\), \(\Omega_{2}\), \(\Omega_{3}\), \(\Omega_{4}\) can be written in the form:  
\[p_{1}=a_{1}\exp(i\lambda x) + b_{1}\exp(-i\lambda x), \tag{9}\] \[p_{2}=a_{2}\exp(i\lambda kx) + b_{2}\exp(-i\lambda kx),\] (10) \[p_{3}=a_{3}\exp(i\lambda kx) + b_{3}\exp(-i\lambda kx),\] (11) \[p_{4}=a_{4}\exp(i\lambda kx) + b_{4}\exp(-i\lambda kx) \tag{12}\]  
Dynamic and kinematic conditions at the boundary (\(x_{1}=c_{1},x_{2}=c_{1}+c_{2},x_{3}=c_{1}+c_{2}+c_{3}\)) of media.