A modal formulation for the propagation of guided waves in straight and curved pipes and the scattering at their junction

The well-established semi-analytic finite element (SAFE) method for predicting guided wave propagation in straight guides is extended to deal with curved pipes using curvilinear coordinates system, following developments by Treyssède and Laguerre [J. Sound Vib. 329 1702-16 (2010)] for helical wires. The modal solution in the elbow can be obtained and the bi-orthogonality relation known for straight guides still holds. The influence of elbow curvature is exemplified by a short parametrical study. The scattering at the junction between a straight guide and an elbow sharing a common section is then considered. Assuming the modal solutions in the straight part and in the curved one are known, the mode-matching method is specifically derived to compute the scattering matrix which deals with waves arising from one part or the other, that is to say, to predict reflection and transmission coefficients at such a junction. Interestingly, the overall formulation requires a single mesh of the section to compute both the modal solutions and the scattering. The scattering matrix computation makes it necessary to take into account evanescent modes locally at the junction of both parts. Again, few examples of numerical results are shown to illustrate specific phenomena arising in this complex scattering case