# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 2 2D WFE Formulation for Fluid-Filled Cylinders
Assuming time-harmonic behaviour and omitting the time-dependence \(e^{i\omega t}\), Equation (1) can be rewritten in the form  
\[\left(-\omega^{2}\mathbf{M}+i\omega\mathbf{C}+\mathbf{K}\right)\mathbf{q}= \mathbf{f}\quad\text{or}\quad\mathbf{D}\mathbf{q}=\mathbf{f}, \tag{4}\]  
where \(\mathbf{D}\) is the dynamic stiffness matrix, \(\mathbf{q}^{T}=[\mathbf{q}_{s}^{T}\ \ \mathbf{q}_{f}^{T}]\), \(\mathbf{f}^{T}=[\mathbf{f}_{s}^{T}\ \ \mathbf{f}_{f}^{T}]\) where \(T\) denotes the matrix transpose, and the matrices are those of the coupled system in Equation (1), typically obtained using FE commercial software. Following the WFE formulation in [25; 27], a 4-noded super-element is formed by concatenating all the acoustical and structural degrees of freedom (DOFs) at the four corners of the prismatic segment in Figure 1c. The total vectors of the DOFs \(\mathbf{q}\) and nodal forces \(\mathbf{f}\) are ordered as  
\[\mathbf{q}=[\mathbf{q}_{1}^{T}\ \ \mathbf{q}_{2}^{T}\ \ \mathbf{q}_{3}^{T}\ \ \mathbf{q}_{4}^{T}]^{T},\ \ \mathbf{f}=[\mathbf{f}_{1}^{T}\ \ \mathbf{f}_{2}^{T}\ \ \mathbf{f}_{3}^{T}\ \ \mathbf{f}_{4}^{T}]^{T}. \tag{5}\]  
The local coordinates must be rotated as described in [25], Figure 1d. This is required to model the curvature of the cylinder and can be easily accomplished by calculating the rotation matrix \(\mathbf{R}\); the new FE matrices in the global system of reference become \(\mathbf{R}^{T}\mathbf{MR}\), \(\mathbf{R}^{T}\mathbf{CR}\) and \(\mathbf{R}^{T}\mathbf{KR}\). Using periodicity conditions, the relations between the nodal DOFs at the four corners are  
\[\mathbf{q}_{2}=\lambda_{a}\mathbf{q}_{1};\quad\mathbf{q}_{3}=\lambda_{y} \mathbf{q}_{1};\quad\mathbf{q}_{4}=\lambda_{a}\lambda_{y}\mathbf{q}_{1}, \tag{6}\]  
where  
\[\lambda_{a}=e^{-k_{B}\alpha};\ \ \lambda_{y}=e^{-ik_{y}L_{y}}. \tag{7}\]  
Equation (6) then gives  
\[\mathbf{q}=\Lambda_{R}\mathbf{q}_{1};\ \ \ \Lambda_{R}=[\mathbf{I}\ \ \ \lambda_{a} \mathbf{I}\ \ \lambda_{y}\mathbf{I}\ \ \ \lambda_{a}\lambda_{y}\mathbf{I}]^{T}, \tag{8}\]  
where \(\mathbf{I}\) denotes the identity matrix. Considering internal equilibrium and periodicity conditions, the relationship between the nodal forces can be written as  
\[\Lambda_{\mathbf{I}}\mathbf{f}=\mathbf{0};\ \ \ \Lambda_{L}=[\mathbf{I}\ \ \ \lambda_{a}^{-1}\ \ \ \lambda_{y}^{-1}\mathbf{I}\ \ \ (\lambda_{a}\lambda_{y})^{-1}\mathbf{I}]. \tag{9}\]  
Using Equations (8) and (9), the WFE reduced equation of motion becomes  
\[[-\omega^{2}(\Lambda_{L}\mathbf{M}\Lambda_{R})+i\omega(\Lambda_{L}\mathbf{C} \Lambda_{R})+(\Lambda_{L}\mathbf{K}\Lambda_{R})]\mathbf{q}_{1}=\mathbf{0} \tag{10}\]and hence  
\[\sigma_{ax}=pR_{int}^{2}/(R_{ext}^{2}-R_{int}^{2}), \tag{2}\]0  
where \(\overline{\mathbf{K}}\), \(\overline{\mathbf{M}}\) and \(\overline{\mathbf{C}}\) are the WFE reduced stiffness, mass and damping matrices whose size are \(m\times m\), \(m\) being the total number of nodal DOFs \(\mathbf{q}_{1}\) at corner 1, Figure 1c. For closed cylindrical structures, the number of wavelengths around the circumference must be integral and therefore the circumferential wavenumber \(k_{\theta}\) must be an integer \(n=0,1,2,\ldots\). It follows that  
\[\sigma_{ax}=pR_{int}^{2}/(R_{ext}^{2}-R_{int}^{2}), \tag{2}\]1  
for a given circumferential mode order \(n\). Equation (11) thus becomes an eigenvalue problem in \(\lambda_{y}\) and \(\omega\), whose solutions give the WFE predictions of the wavenumbers in the \(y\) direction (through \(\lambda_{y}\)) and the wavemodes (through \(\mathbf{q}_{1}\)). The form of the eigenproblem depends on the physical problem and on whether the frequency or the wavenumber is assumed to be real [28]. In this paper, the frequency is assumed to be real, and a quadratic eigenvalue problem in \(\lambda_{y}\) is obtained as shown in Appendix A. In particular, the eigenvalue formulation solved in this paper (to obtain the complex dispersion curves and wavemodes) is a linear companion form of the quadratic eigenvalue problem, see Equations (A3)-(A5).