## 3 Dispersion Relations  
The wave motion in the unit cell of a polydisperse phononic crystal (Figure 1) is described using four equations in the corresponding domains and conditions at the contact boundaries of two (Media1 and Media2) media of continuity of acoustic pressure and acoustic velocity. In addition, the phase shift condition must be met at the boundaries of the unit cell.  
These relations can be considered as a system of algebraic equations for unknown coefficients \(a_{1}\), \(b_{1}\), \(a_{2}\), \(b_{2}\), \(a_{3}\), \(b_{3}\), \(a_{4}\), \(b_{4}\). Since this system of equations is homogeneous, the solution of the system of equations with the matrix \(M(\lambda,\xi)\) exists if the determinant of the matrix is zero _Det \((M\ (\lambda,\xi))=0\)_. For all specific phononic crystals with known parameters media of Media1 and Media2, relation  
\[DetM(\lambda,\xi))=0 \tag{21}\]is the dispersion relation in a phononic crystal for elementary wave packets of the form  
\[A(x)e^{i(\xi\ x-\omega t)},A(x+1)\ =A(x) \tag{22}\]  
For example, acoustic waves in a polydisperse phononic crystal made of water and air or polyurethane foam  
\[\rho_{air}=1.204,\ c_{air}=331,\ \rho_{water}=999,\ c_{water}=1500\]  
\[\tau_{12}=\frac{\rho_{water}}{\rho_{air}},\ \tau_{23}=\frac{\rho_{air}}{\rho_{ water}},\ \tau_{34}=\tau_{12},\ \kappa_{12}=\frac{c_{air}}{c_{water}}\ =\varkappa,\ \lambda=\frac{\omega L}{c_{1}}\]  
With the help of dispersion relations (21), taking into account the accepted reference thermodynamic characteristics of water and air, using numerical methods for investigating functions given implicitly, it is possible to graphically describe the dependence of the upper frequency of the first bandwidth on the concentration of water in the air for polydisperse crystals. Since for the first bandwidth, the lowest frequency is 0, the upper value of the bandwidth is the width of this bandwidth.  
The dependence of the frequency bandwidth on the linear concentration \(k\) for fixed values of the parameters of two media (water-air) is shown in Figure 2. Here, the polydispersity of the medium is taken into account using the expressions \(k_{1}+k_{3}=k\), \(k_{3}=0.2*k_{1}\), \(k_{4}=0.2*k_{2}\). The graph coincides with the case of a monodisperse distribution of media, \(k_{3}=k_{4}=0\).  
The most significant thing is that when \(k=0.5\) there is a global minimum of the width of the first frequency band as a function of linear concentration.  
For an elementary wave packet (22), the phase velocity of wave propagation in a phononic crystal is determined using the ratio  
\[c_{ph}(\xi,\varkappa,\tau)\ =\lambda(\xi,\varkappa,\tau)/\xi, \tag{23}\]  
which is true for mono and polydisperse phononic crystals. Here, \(\xi\) is a dimensionless wave number, \(\lambda\) is a dimensionless cyclic oscillation frequency, and k is the concentration of the medium. It should be noted that the waves in any phononic crystal are always dispersing, since the phase velocity depends on the frequency of the wave (or on the wave number).  
Figure 3 shows the dependence of the phase velocity on the wave number for equally large volume (linear, \(k=0.5\)) concentrations of water and air. Graphs are for monodisperse \(k_{1}=k_{2}=0.5\); \(k_{3}=k_{4}=0\) and polydisperse \(k_{1}=k_{2}=0.4\); \(k_{3}=k_{4}=0.1\) phononic crystals.  
Figure 2: The upper frequency of the bandwidth for the 1 st band or the width of this band, depending on the concentration of water in the air. Polydispersity \(k_{1}+k_{3}=k\), \(k_{3}=0.2*k_{1}\), \(k_{4}=0.2*k_{2}\) is taken into account here. The graph coincides with the case of a monodisperse distribution of media, \(k_{3}=k_{4}=0\).