Normal Modes in External Acoustics. Part I: Investigation of the One-Dimensional Duct Problem

One of the simplest problems of external acoustics is investigated. It is the one dimensional duct problem with fully absorbing ends. This is equivalent to an internal problem with mixed or Robin type boundary conditions. The one-dimensional time-harmonic boundary value problem can be solved analytically. It is possible to determine eigenvalues. These eigenvalues go to infinity if non-reflecting boundary conditions are considered. After this analytic solution, the problem is reformulated and discretized by means of finite elements. The algebraic eigenvalue problem is derived and the solution is represented in terms of normal modes. The applications section contains discussion on eigenvalue distribution in complex plane and on convergence of eigenvalues in terms of mesh size. Then, appearance of complex modes will be checked. Finally, we investigate solution convergence for mode superposition techniques. For that, three sorting criteria will be compared for the fully absorbing case and for the lightly damped case as well.