## 2 Basic Equations and Problem Formulation
### Dispersion Equations
\(I_{0}\) is the zero-order Bessel function of the first kind of virtual quantities, which describes the reflection field from the interface. \(K_{0}\) is the Bessel function of the second kind of the zero-order virtual component, which represents the wave propagating from the acoustic excitation source to infinity. Three kinds of wave modes with different velocities are obtained in a homogeneous porous medium: a shear wave and two longitudinal waves, often referred to as the fast and slow waves, respectively. In the case of the fast wave, the skeletal frame and the pore fluid move in-phase. This wave is similar to the longitudinal wave in an elastic solid medium. In the case of the slow wave, the frame and the fluid move relative to each other in antiphase. This wave is unique to [PERSON]'s model and all similar porous models. Their wave numbers are \(k_{PSIf}=\omega_{PSIf}\), \(k_{PSIs}=\omega_{PSIs}\), and \(k_{PSt}=\omega_{PSt}\), respectively. Here, \(\omega\) is the angular frequency, and \(s_{PSIf}\), \(s_{PSIs}\), \(s_{PSt}\) can be written as  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]1  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]2  
where \(\Delta=P(\rho_{f}-\rho_{12})+R\big{(}\rho_{s}-\rho_{12}\big{)}-2\rho_{12}Q\), \(P=A+2N\), and \(\rho_{12}\) is the coupling density of the solid and liquid. \(A\), \(N\), \(R\), and \(Q\) are all poroelastic constants [30; 31]. The poroelastic constants \(A\) and \(N\) correspond to the familiar Lame constants of a purely elastic solid, and the constant \(N\) also represents the shear modulus of the solid. The constant \(R\) is a measure of the pressure required on the liquid to force a certain amount into the aggregate, while the total volume remains constant. The constant \(Q\) represents the coupling between the volume change of the solid and that of the liquid. The calculated parameters in the porous medium are  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]3\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]4 \[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]5  
where \(K_{PS}\), \(K_{PL}\), and \(K_{Pb}\), are the bulk moduli of the solid matrix, fluid, and solid skeleton, respectively, while \(N\) and \(\beta\) denote the shear modulus and porosity in porous medium. With regard to the liquid phase in a porous medium, the equation consists of two parts, namely the longitudinal and the shear wave potential function,  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]6  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]7  
where \(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\) are the participation coefficients of the fast longitudinal wave, slow longitudinal wave and shear wave, respectively [32; 33]. According to [PERSON], a porous medium may be considered as a porous skeletal frame that is permeated by a fluid. [PERSON]'s equations quantify the momentum balance within the skeletal frame, within the pore fluid, and between them. It can be shown that as porosity tends to zero, [PERSON]'s equations reduce to the wave equation of a solid, and conversely, as the porosity tends to 1, they reduce to the wave equation of a fluid. The displacement of the porous medium is composed of two basic components: solid displacement and pore fluid displacement.