The main purpose of the paper is to present an extended analysis of the coupled system of linear equations describing wave propagation in fluid-saturated porous materials used for the determination of their fundamental characteristics. The present paper is the first part, mainly devoted to the analysis of the system of equations describing transverse wave propagation. The analysis of equations for compressional waves will be presented in a separate paper. The starting point of considerations are nonlinear equations describing the dynamic behavior of the medium formulated in the spirit of the theory of interacting continua in which parameters of isotropic pore space structure are explicitly present in the model, and constitutive equations are formulated for each physical component separately. They contain all fundamental mechanical couplings between components of this medium in the general nonlinear form and of clear physical meaning. Linear equations of this theory for barotropic fluid and hyperelastic skeleton are the subject of the analysis performed in the paper. They are equivalent to Biot's model, widely used in the description and analysis of wave propagation in saturated porous materials and wave interaction with the boundary of such media. A new method of analysis is proposed in the paper, based on spectral decomposition of the system of equations in the two-dimensional vector space of rotations of solid and fluid particles. It is shown that each choice of the base in this vector space defines the division of the medium into two new components and transforms the system of equation to the form describing the dynamic behavior of these components, and the requirement of the basic dynamic properties of the medium to be invariant in each chosen base uniquely determines the mass densities and bulk modules of the new components. Such an approach enabled decomposition of the system of equations for particle rotations to two uncoupled scalar equations, the only one of which describes propagation of the transverse wave. This also allowed obtaining expressions characterizing transverse wave propagation in fluid-saturated porous materials: velocity, effective mass densities and wave impedance of the medium associated with this wave and its frequency characteristics: phase velocity and attenuation coefficient. It is shown that both frequency characteristics are fully defined by the effective mass densities of both components associated with the transverse wave, the tortuosity of the pore space, and the coefficient characterizing the viscous interaction of the fluid with the skeleton.

本文的主要目的是提供线性方程组耦合系统的扩展分析，该线性方程组描述用于确定其基本特性的流体饱和多孔材料中的波传播。本文是第一部分，主要致力于分析描述横波传播的方程组。压缩波方程的分析将在另一篇论文中介绍。考虑的出发点是描述相互作用连续性的精神所描述的介质动力学行为的非线性方程式，其中模型中明确存在各向同性的孔隙结构参数，并分别为每个物理成分制定了本构方程式。它们包含一般非线性形式且具有明确物理含义的该介质的各个组件之间的所有基本机械耦合。正压流体和超弹性骨架的该理论线性方程是本文进行分析的主题。它们等效于Biot模型，广泛用于描述和分析饱和多孔材料中的波传播以及与此类介质边界的波相互作用。本文提出了一种新的分析方法，该方法基于固体和流体颗粒旋转的二维矢量空间中方程组的谱分解。结果表明，在此向量空间中对碱基的每次选择都将介质划分为两个新的成分，并将方程组转换为描述这些成分的动态行为的形式，以及对基本动态特性的要求。每个选定基中不变的介质唯一地确定了新组件的质量密度和体积模块。这种方法可以将粒子旋转方程组分解为两个非耦合的标量方程，其中唯一一个描述了横波的传播。这也允许获得表征横向波在流体饱和的多孔材料中传播的表达式：与该波有关的介质的速度，有效质量密度和波阻抗及其频率特性：相速度和衰减系数。结果表明，两个频率特性完全由与横波相关的两个分量的有效质量密度，孔隙空间的曲折度以及表征流体与骨架的粘性相互作用的系数来完全定义。