## 4 The Effect of Polydispersity on the Dispersion Ratio and Bandwidth for Long Waves
Figure 6: General view for comparison. coordinate axis (x). At the interval (\(\pi\), \(2\pi\)), the waves propagate in the negative direction of the coordinate axis (x). For long waves (\(|\vec{\xi}|\to 0\)), the polydispersity or fragmentation of the components does not affect the dispersion relations for acoustic waves in phononic crystals. This fully confirms and coincides with the analytical, explicit form of dispersion relations for long waves (24). It should be noted that the graphs of dispersion relations for phononic crystals N4 and N6 coincide well. We performed comparative analysis of dispersion relations for acoustic waves in polydisperse phononic crystals obtained by crushing components in the unit cell of a monodisperse phononic crystal. Figure 7 shows graphs of dispersion relations for four phononic crystals, N3-N6, in the wavenumber interval (0, 4\(\pi\)). The appearance of phononic crystals with a smaller unit cell size as a result of crushing significantly changes the dispersion relations for acoustic waves. It should be noted that splitting the phononic crystal components into 2, 3, \(\dots\) N equal parts in the unit cell always leads to a catastrophic rearrangement of the dispersion relations for acoustic waves in these crystals. The Effect of the Fragmentation of Components in the Unit Cell of a Phononic Crystal on the Phase Velocities of Acoustic Waves  
Hereafter, the following terminology is used unless otherwise stated:  
1. N5 is a basic monodisperse phononic crystal with an equal volume (linear) concentration of components, and all the others are obtained from it by crushing the components. 2. N6 is a phononic crystal with a smaller spatial period obtained by crushing the components of crystal N5. This crystal catastrophically changes the dispersion relations for acoustic waves of phononic crystal N5. Figure 8 shows graphs of the phase velocities of acoustic waves in phononic crystal as functions of the wavenumber for six phononic crystals in the wavenumber interval (0, \(\pi\)). The appearance of phononic crystals with a smaller unit cell size as a result of crushing causes an increase in the velocity of acoustic waves in the phononic crystal. Properties of acoustic velocities in phononic crystals N4 (0.25 + 0.3 + 0.25 + 0.2, coral) and N6 (0.25 + 0.25 + 0.25 + 0.25, violet) (Table 3) are signs of a catastrophic restructuring of the dispersion properties of these crystals. Figure 7: The appearance of a smaller spatial period during the crushing of the components of a monodisperse crystal with a constant concentration of the form 0.25 + 0.25 + 0.25 + 0.25. Here: N3: 0.5 + 0.5 + 0 + 0, color = blue; N4: 0.25 + 0.24 + 0.25 + 0.26, color = red; N5: 0.3 + 0.2 + 0.2 + 0.3, color = green; N6: 0.25 + 0.25 + 0.25 + 0.25, color = violet. Splitting the components in half, each component is divided into 2 equal parts (two halves). The graphs show the dependence of the upper boundary of the first (lower) band of dimensionless frequencies on the size of one part of the crushing of the second component of a two-component phononic crystal, which is shown in Figure 1.  
The graph of the dependence of the width of the first bandwidth for phononic crystals of type N2 is in good agreement with the appearance of a unit cell with a smaller size as a result of crushing phononic crystals, which significantly changes the dispersion relations for acoustic waves. This means that splitting the corresponding components in half primarily causes (or causes) the greatest expansion of the first bandwidth in comparison with a monodisperse phononic crystal with the same concentration of components. The Effect of the Crushing of Components in the Unit Cell of a Phononic Crystal on the Width of the First Forbidden Band  
Graphs of the dependence of the width of the first forbidden frequency band for phononic crystals (Figure 9) agree well with the graph of the dependence of the width of the first bandwidth for phononic crystals (Figure 10).