## 1 Introduction  
The relevance of the research is determined by the increasing need to model and describe the averaged characteristics of waves in heterogeneous media. One of the methods of averaging the properties of a heterogeneous medium is the description of long waves using phononic crystals (PCs). These methods are based on the fact that most heterogeneous media have a periodic or almost periodic structure with a characteristic spatial period. This approach was used by [PERSON] when modeling acoustic waves in the air using a weight-spring chain. It should be noted that all composite materials with a woven reinforcing base have a spatially periodic structure. It is usually assumed that the unit cell of a phononic crystal modeling a two-component inhomogeneous heterogeneous medium contains one component of each of the media; such crystals are, for convenience, considered monodisperse. In this paper, studies of the influence of polydispersity (at the same concentration) on the main characteristics of the propagation of long acoustic waves in phononic crystals are carried out. A phononic crystal will be called polydisperse if the number of matrix elements and reinforcing material elements in the unit cell is greater than one and there are at least two matrix elements or two reinforcing material elements of different sizes. An element of the matrix material or reinforcing material is the connected part of the corresponding material. Sometimes, the intuitive terminology of \"drop\" is used for the reinforcing material and \"bubble\" for the mixture matrix. Using this terminology, polydispersity means that there are either two drops or two bubbles of different sizes in the unit cell. Phononic crystals take into account the properties of waves, such as scattering, interference and resonant phenomena in one-dimensional periodic chains of inhomogeneities, to form \"band-gaps\", that is, ranges of wavelengths or frequencies within which waves cannot propagate through the structure [1]. Due to its spatial periodicity, the propagation of acoustic waves in phononic crystals has special properties that can later be used in various devices and techniques. There are several classes of phononic crystals that differ mainly in the physical nature of the inclusions and the matrix. Among them, attention has been paid to solid/solid, liquid/liquid and mixed solid/liquid composite systems. Such media have transmission frequency bands and forbidden frequency bands of transmission spectra for which the propagation of sound or vibration can be strictly prohibited in all directions. Using the band gap principle, phononic crystals make it possible to regulate the propagation of elastic or acoustic waves [2]. In other words, they play the role of ideal scatters for elastic or acoustic waves in the band gap frequency range. Thus, phononic band gap can be used to filter, limit or direct acoustic energy and, therefore, can be used in various applications. Acoustic waves themselves are the object of research in seismology and seismic exploration. There are features and differences of wave propagation in a solid medium and in a porous one (when the medium is inhomogeneous and heterogeneous). The propagation of acoustic waves in porous media is described in [3]. In [4; 5; 6; 7; 8; 9], studies of the soundproofing and wind-proofing properties of phononic crystals were carried out. These studies have shown that devices such as filters, resonators, waveguides, switches and other phononic metamaterials can be created on the basis of phononic crystals. Acoustic wave propagation and resonance phenomena in monodisperse phononic crystals are described in [10]. Modeling of a one-dimensional periodic medium with the help of mechanical analogs of a weight-spring is described in the fundamental work [11]. This approach can be used for acoustically contrasting heterogeneous one-dimensional periodic media. In fact, the mechanical analogy [11] is an approximate model of a phononic crystal with contrasting media.