A Study of Waveguides with Sections of Proportional Sizes: Application to Wave Radiation

Classically, waveguides are structures with uniform sections or are periodic along an axis. Vibrations of these waveguides can be described by wave modes and propagation constants relating the displacements and forces in two sections of the waveguide. Here, waveguides of non uniform sections are considered. More precisely, we consider waveguides which have sections with size increasing proportionally to the distance from an origin. This is for instance the case for a domain exterior to a convex body. Using a finite element model of a small section of the guide, the wave finite element method (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 by 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. Keeping 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 of the usual methods involving traditional finite or boundary elements.