# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 1 Introduction
In addition, the cylinder may be laminated, which substantially increases numerical and computational challenges. The dynamics of thick, laminated, cylindrical structures with FSI can only be modelled accurately if the through-thickness deformation is included. This typically results in very complex analytical/numerical models, which are, however, often limited to low-frequency applications and beam-type modes. In any case, even for simple cases, the analysis is very complex due to the interdependence of the parameters and the transcendental dispersion equations to which explicit solutions do not exist. Relatively recently, coupled analysis has been conducted using the finite element (FE) method [14]. This has made it possible to simulate the dynamic motion of an acoustic fluid coupled with a flexible structure with arbitrary geometry. However, standard FE analysis cannot model large structures at high frequency or with short wavelengths: there must be at least six elements per wavelength to accurately model the behaviour, and the model size becomes very large. In the low/low-mid frequency range, computational fluid-structure interaction issues can be handled by applying special reduction techniques [15] or using boundary element (BEM) and finite element (FEM) discretisation [16]. An important issue regarding applications involving wave propagation is the ability to develop accurate and computationally efficient models. The computational constraints of a full FE analysis of the structure and the complexity of analytical formulations have inspired the development of semi-analytical and numerical methodologies to analyse wave propagation in both structure and acoustic fluid, using new finite element formulations for high-frequencies. In the context of cylindrical waveguides with FSI, we can cite, for example, the Spectral-Finite-Element method, e.g., [17; 18], which allows the evaluation of wave propagation using a semi-analytical FE formulation. This method has the advantage of being able to predict wave behaviour up to a very high frequency with high accuracy; however, it requires the formulation of new finite elements that must be realised for each model, resulting in a substantial effort, especially for complicated constructions. AnotherFE-based approach is the Wave Finite Element (WFE) method, which has seen application to a variety of problems due to its ability to model, in a straightforward way, periodic and homogeneous waveguides with complex characteristics. The unit cell of the structure is taken and discretised using a conventional FE formulation, typically using commercial FE software. Periodicity conditions are then applied to the FE equations of motion as developed by [PERSON] in his PhD thesis [19], and a quadratic or linear eigenvalue problem is obtained. Solutions to this eigenvalue problem give the dispersion curves (the relation between the wavenumbers and frequency) and the wavemodes (the cross-section deformation caused by the passage of a wave through the structure). These provide information for further analysis, such as forced response, the estimation of the parameters in a Statistical Energy Analysis model, the vibroacoustic response and implementation of SHM and NDT techniques. When both the structure and the fluid are taken into account, the WFE method enables a very significant decrease in the size of the problem and in computation time. The approach has been described in previous studies, which have shown its practical applicability to periodic and continuous structures. Examples include laminated plane panels [20], cylindrical structures [21], forced vibration [22], vibroacoustic response [23] and metamaterial [24]. The aim of this paper is to investigate the application of the WFE method to model wave propagation in fluid-filled, laminated cylinders of complex construction, with and without pre-stress conditions. The method can be implemented at a very low computational cost and exploits the versatility of commercial FE software.