Keywords: waveguide, wave propagation, radiation, infinite domain.

Classically, waveguides are structures with uniform sections or are periodic along an axis. Vibrations of uniform waveguides can be described by wave modes and propagation constants relating the displacements and forces in two sections of the waveguide. For general waveguides with a complex cross-section, the displacements in the cross-section can be described usingthe finite element method while the variation along the axis of symmetry is expressed as an analytical wave function. Following these ideas, the authors developed, for instance, the spectral finite element approach. However, more general waveguides can be studied by considering periodic structures. It was showed that the solution can be decomposed into an equal number of positive and negative-going waves. The approach is mainly based on Floquet's principle or the transfer matrix and the objective is also to compute propagation constants relating the forces and displacements on the two sides of a single period and the waves associated to these constants. For complex structures, finite element models are used for the computation of the propagation constants and waves leading to the wave finite element method (WFE).

In this paper, waveguides of non uniform sections are considered, more precisely, with sections of a size increasing proportionally to the distance from an origin. This is for instance the case of a domain exterior to a convex body. Using a finite element model of a small section of the guide, the WFE can be used to find propagation constants and wave modes relating the displacements and forces at the two extremities of the finite element section. Then, it is shown that the wave modes and propagation constants in a section are simply related to the same quantities in other sections but for different frequencies. From this set of waves, general solutions in the waveguide can be computed. This approach allows an efficient solution by limiting the discretization to a very small part of the waveguide using only classical dynamic stiffness matrices, obtained using any finite element software.

This approach can then be used for computing wave radiation in domains exterior to a convex body. The solution is projected onto these waves, allowing a clear separation between incoming and outgoing waves. Conserving only the outgoing part of the solution, wave radiation can be computed. It is shown that the approach is more efficient for high frequencies and can be complementary to the usual methods involving traditional finite or boundary elements.

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