The goal of this paper is to theoretically investigate the propagation of elastic waves in helical waveguides. In the context of non-destructive evaluation for structural health monitoring, this study is motivated by the need for inspecting helical structures such as cables or springs. A numerical method is chosen based on a semi-analytical finite element technique. The proposed method relies on a non-orthogonal curvilinear coordinate system that is translationally invariant along the helix centreline, so that a Fourier transform is explicitly performed and the problem is reduced to two dimensions. Some useful expressions are also derived for the averaged energy and flux in order to directly compute the energy velocity. The convergence and accuracy of the proposed method are then assessed by comparing finite element results with reference solutions. A dispersion analysis inside a 7.5° helical wire, typically encountered in civil engineering cables, is realised including attenuation due to material damping. Some dispersion curves are finally presented for a wide range of lay angles and for several centreline radii. Significant differences with the infinite cylinder are observed.