Carrilho J.

Coupled sets of eigenvalue equations can occur in a variety of problems. A simple example is a duct of rectangular cross-section, with uniform mean flow and absorbing walls. Analytic modal solutions may be found for this problem but the equations for the transverse eigenvalues become coupled in the presence of flow, through the dispersion relation, and must be solved simultaneously. This requirement does not pose special difficulties since both shooting methods and discretization methods are readily applicable to systems of coupled equations. Nevertheless, there still remains the need to ensure that all eigenvalues of interest have been found. This is far more difficult to achieve with a coupled set of equations. In much the same way as for equations in a single complex variable, the exact number of roots of a system of coupled equations, within a bounded region of the complex plane, can be determined by using the extension of the Argument Principle to several complex variables. Once the number of roots within the given region is known, these can be refined using a simple Newton-Raphson iteration. We present and discuss in detail the algorithm for a reliable and efficient solver for coupled eigenvalue problems. The application of the coupled eigenvalue solver is then demonstrated through examples. The sets of eigenvalues for a three-dimensional rectangular duct with uniform flow are found for the case where all four walls are lined with different specific acoustic admittance. This requires the solution of a coupled set of two equations in the transverse eigenvalues. The results will be verified by comparison with finite element simulations. Particular choices of parameters are used to demonstrate the reliability of the solver, and also its efficiency. In addition, other coupled eigenvalue problems arising in duct acoustics applications will be discussed.