Flexible pipes are used in the oil and gas industry to transport various kinds of fossil fluid. One application is to use a pipe as a riser, transporting crude oil or natural gas from the seabed to an oil rig or a tank ship hovering in surface of the sea. A common issue with these kinds of pipes is generation of vibrations due to the fluid flowing inside. These vibrations can be transmitted through the pipe as elastic waves potentially resulting in fatigue failure of the components attached to the ends of the pipe. With the aim of reducing these vibrations at the ends, a study of the waveguide properties of such pipes is needed. The first step in this study is, in this thesis, taken by investigating the wave guide properties of a infinite toroidal shell. The waveguide properties of an infinite toroidal shell can later be used to determine the wave guide properties of a small section of the torus by means of boundary integral equations method. The small section of the torus can be thought of as representing a bend thin walled pipe section. It has been chosen to model the pipe as a shell in order take the flexibility of the cross section into account. The toroidal shell model has been benchmarked against, respectively, classical Bernoulli-Euler beam theory and curved beam theory. The first comparison is relevant when the radius of the torus is very large. Due to same reason a shell model of a thin walled cylinder has also been established and benchmarked against the toroidal shell. The benchmarking is made between dispersion curves for some of the simplest vibration modes covered by both theories. The dispersion curves are obtained by enforcing a trial solution on the differential equations governing the system and a system of equally many equations and unknowns is obtained by imposing Galerkin's method. Different trial solutions representing, respectively, in-plane bending and out-off-plane bending, have been enforced on the toroidal shell model in fashion of truncated complex Fourier series and the converge of the dispersion curves related to some of the simplest vibrations modes has been studied. From these studies it can be concluded that the governing differential equations of the toroidal shell, which have been derived, are valid.