# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 3 Numerical Examples
### Wire-Reinforced Hard Rubber Pipe under Internal Pressure
Analysis of the deformation of the cross-section and the energy associated with the individual DOFs has shown that different motions are involved under the passage of a wave. Consider the branches corresponding to the helically wire-reinforced pipe for \(n=0\), thick lines in Figure 7a. The first branch in Figure 7a starts propagating as a quasi-extensional wave. The cross-sectional displacement and the relative contributions of the kinetic energy in the axial (\(y\)) direction are significantly larger than those in the circumferential and radial directions. At \(\omega\approx 0.1\Omega_{ring}\) the branch veers, involving motion primarily in the \(x\) direction, but there are also cross-sectional displacements in the \(y\) and \(z\) directions. As the frequency increases, the behaviour of branch 1 changes again, involving motion primarily in the radial (\(z\)) direction. The second branch in Figure 7a starts propagating at \(\omega=0.1\Omega_{ring}\) as a predominantly shear wave, then it veers, involving primarily extensional motion in the axial direction, and then its behaviour changes again, involving motion in both the \(y\) and \(x\) directions. Higher-order branches represent waves which are predominantly shear and extensional in nature. Generally, the behaviour is very complicated because also radial motion is involved. Figure 7 shows that, as expected, the helical reinforced pipe exhibits a stiffer behaviour than the non-reinforced pipe, in particular in terms of extensional propagation of disturbances. Figure 7 also shows that the stiffness of the pipe depends, in different ways, on the lay angle of the reinforcing wires. At lower frequencies, the behaviour of the helically reinforced pipe (\(\phi=20^{\circ}\)) is similar to that of the circumferentially reinforced pipe (\(\phi=0^{\circ}\)). As the frequency increases, the first branch shows that the helically reinforced pipe is less stiff with respect to flexural wave propagation. The third branch of the helically reinforced pipe has a behaviour that is intermediate between those of the circumferentially reinforced pipe and the non-reinforced pipe. Branch 4 is similar to that of the circumferentially reinforced pipe below \(kR=2\). However, as the frequency and wavenumber increase, branch 4 and branch 5 corresponding to the \(\phi=0^{\circ}\) case veer, due to coupling between radial and axial motion, while for \(\phi=20^{\circ}\) branch 4 and branch 5 have almost constant phase velocities. In the frequency and wavenumber range considered, as \(n\) increases the behaviour of  
Figure 7: Dispersion curves for the helically wire–reinforced pipe. Circumferential modes (**a**) \(n=0\), (**b**) \(n=1\):helically reinforced pipe \(\phi=20^{\circ}\);...... reinforced pipe \(\phi=0^{\circ}\);...... non reinforced pipe. the helically reinforced pipe starts becoming more similar to that of the circumferentially reinforced pipe. This is particularly true for small values of the wavenumber, that is for long wavelengths, although some differences in the dispersion curves are still evident. The presence of a high internal pressure considerably stiffens the pipe, and helical reinforcement can be used to bear the radial stress induced by internal pressure. Figure 8 shows the dispersion curves for circumferential mode orders \(n=0\) and \(n=1\) when the residual stress due to high internal pressure is included in the model. Dispersion curves for the same helically reinforced pipe without internal pressure are shown for comparison. It can be noticed that the internal pressure considerably stiffens the dynamic behaviour of the pipe, hence decreasing the wavenumbers. For \(n=0\), only the quasi-extensional wave propagates below \(\omega/\Omega_{ring}\approx 0.4\). As the frequency increases, the pipe starts to be less stiff in the radial direction and the quasi-flexural branch cuts-off.