## 4 The Effect of Polydispersity on the Dispersion Ratio and Bandwidth for Long Waves  
The dispersion relation for low-frequency waves (or for small frequency values) is the form  
\[\lambda_{poly}(\xi,\ k,\varkappa,\ \tau)\ =\sqrt{\frac{2\tau[1-\cos(\xi)]}{((k \tau-(k-1)\varkappa^{2})((k-1)\tau-k))}} \tag{24}\]  
This relation defines the dimensionless frequency \(\lambda\) as a function of the system parameters and associates it with the parameter \(\xi\) wavenumber, which determines the wavelength of an elementary wave packet traveling through a phononic crystal. Here, \(k=k_{1}+k_{3}\), and the parameter \(\varkappa\) is the ratio of the speed of sound in media 2 (gas or air) to the speed of sound in media 1 (water or polyurethane). This explicit expression of the dependence of the dimensionless frequency on the wavenumber \(\xi\) is obtained by direct calculation from the dispersion relation (22) for low frequencies, \((\lambda_{poly})^{4}\approx 0\). This expression does not depend on the polydispersity of phononic crystals. This means that polydispersity does not affect the propagation of low-frequency waves in phononic crystals. It should be noted that it is assumed here that polydispersity has little or no effect at all on the change in the size (length) of the unit cell of the phononic crystal. Comparative Analysis of Dispersion Ratios of Two-Component Mono and Polydisperse Phononic Crystals at a Fixed Concentration of Media  
This section presents the results of numerical and analytical studies of the effect of crystal polydispersity on dispersion relations for acoustic waves in crystals. Throughout, the volume (linear) concentration of the components is constant and equal to 0.5 in fractions or 50% in percent, unless otherwise specified. Tables 1 and 2 describe the quantitative characteristics after crushing the components in the unit cell. The first column contains the phononic crystal number, the second the characteristics (lengths) of alternating connected components of PCs, and the third column, the color of the graph of the corresponding dispersion ratio (as indicated on Figures 4-6). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cell Number** & **Component Length** & **The Color of the Graph** \\ \hline N 1 & 0.4 + 0.4 + 0.1 + 0.1 & Black \\ N 2 & 0.4 + 0.25 + 0.1 + 0.25 & Green \\ N 3 & 0.5 + 0.5 + 0 + 0 & Blue \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Small perturbations of a monodisperse crystal. Description of the unit cell [0,1]. Figure 3: Phase velocity as a function of the wavenumber 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. The graphs match. When describing the properties of the dispersion relations shown in Figure 4, it is necessary to note that perturbation of a monodisperse crystal by crushing its components in the unit cell expands the first bandwidth. The dispersion relations on the interval \((0,2\pi)\) are shown. Waves in the interval \((0,\pi)\) propagate in the positive direction along the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cell Number** & **Component Length** & **The Color of the Graph** \\ \hline N 4 & 0.25 + 0.24 + 0.25 + 0.26 & Brown \\ N 5 & 0.3 + 0.2 + 0.2 + 0.3 & Red \\ N 6 & 0.25 + 0.25 + 0.25 + 0.25 & Yellow \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Small perturbations of a monodisperse crystal with a period 2 times smaller. Description of the unit cell [0,1]. Figure 4: Small perturbations of a monodisperse crystal with a constant concentration of components of the form \(0.5+0.5+0+0\): N1 crystal of the kind \(0.4+0.4+0.1+0.1\), color = black; N2 crystal of the form \(0.4+0.25+0.1+0.25\), color = green; N3 component of the form \(0.5+0.5+0+0\), color = blue. Figure 5: Small perturbations of a monodisperse crystal with a constant concentration of components of the form \(0.25+0.25+0.25+0.25\). N4: \(0.25+0.24+0.25+0.26\), color = brown; N5: \(0.3+0.2+0.2+0.3\), color = red; N6: \(0.25+0.25+0.25+0.25\), color = yellow.