## 1 Introduction
When detecting pipelines with coatings, this method only requires the removal of a small part of the coating to test a significant length of the pipeline; it allows coverage of the inspected structure in both cross-sectional and longitudinal directions; due to the lack of relative motion between the probe and the pipe wall, the wear on the probe is minimal, and there is no need for motion or rotation mechanisms or other control facilities; and it can inspect pipelines in use (in water, with coatings or insulation layers, etc. ), making it particularly suitable for detecting dangerous defects such as internal and external corrosion in pipelines in use, as well as dangerous defects in joints [14]. In real-world pipeline engineering projects, it is common to apply a coating layer on the outer surface of the pipeline or to use double-layer pipes to achieve insulation, corrosion protection, and extended service life [15]. [PERSON] et al. conducted theoretical analysis on the longitudinal modal propagation characteristics in viscoelastic-coated pipes and found that the attenuation dispersion curve can serve as a theoretical basis for mode selection in pipeline defect detection. [PERSON] et al. proposed a new procedure to improve the efficiency and accuracy of exploring longitudinal guided wave dispersion characteristics in multi-layered pipes [16]. [PERSON] et al. analyzed the dispersion curves of the fundamental antisymmetric Lamb wave modes considering different thicknesses and material attenuation of the substrate and coating [17]. The propagation of ultrasonic guided waves has been extensively studied worldwide, and various models for different geometric shapes and materials have been proposed [18]. Maintenance issues caused by failure of composite structures due to low-velocity impact damage are of significant concern as they may result in nearly invisible and undetectable damage [19]. [PERSON] achieved damage localization imaging of composite materials through simulation and experiments, providing an effective solution for damage localization in composite materials [20]. [PERSON] proposed a simulation-based Lamb wave propagation method for non-destructive monitoring of layered composites by accurately estimating the damping coefficient [21]. [PERSON] accurately described the elastic properties of wood using Lamb wave propagation [22]. [PERSON] and [PERSON] compared six selected methods for calculating dispersion curves in viscoelastic materials [23]. [PERSON] proposed an algorithm and code for solving the dispersion equation of cylindrical multilayered elastic structures [24]. [PERSON] developed a spectral matching method for calculating dispersion curves in generally anisotropic viscoelastic media in both planar and cylindrical geometries [25]. [PERSON] et al. analyzed the effects of average uniform fluid flow with cylindrical geometry on the vibration of the pipe wall driven by acoustic waves through analytical, numerical, and experimental analysis [26]. [PERSON] provided reliable mathematical solutions for wave propagation in anisotropic hollow cylindrical bodies [27]. [PERSON] and [PERSON] analyzed general prism waveguides [28]. [PERSON] used mathematical physics and classical instruments to deal with the two-dimensional problem of elastic wave propagation in rectangular solids, resulting in a highly attractive analytical model [29]. [PERSON] and [PERSON] improved the traditional polynomial method for solving the propagation problem of guided waves in multilayered hollow cylinders [30]. [PERSON] derived the dispersion characteristic equation in cylindrical coordinates by introducing the state vector form of displacement and stress components, solving the wave propagation problem of partially complex anisotropic cylindrical bodies [31]. In recent years, there has been increasing research on mode conversion. [PERSON] and [PERSON] proposed a computationally efficient time-domain spectral finite element method and crack model to consider the propagation, scattering, and mode conversion of guided waves in pipes [32].