Propagation multimodale dans un pavillon à section variable

Some authors have tried to calculate propagation in a varying cross section duct by slicing it into cylindrical elements with length small compared to wavelength. This enables to solve the wave equation with the multimodal decomposition formulation. The results compared favorably with experimental data. However, this approach have certain limitations and we, therefore, wanted to study this problem by writing continuous equations. First of all, we develop a matricial horns equation extending the classical Horn equation. Then, we find a matricial Riccati equation for the impedance matrix which is symmetric and only depends on the axial coordinate. This equation for the impedance avoid spurious divergence due to vanishing modes. Finally were are able to get the pressure or the velocity field by a calculation of a kind of argument matrix (similar to the usual jkx for plane wave) with another continuous equation slave to the matrix impedance variation. The computations are performed with a Runge-Kutta method of order 3 and applied to a particular geometry in the case of monopole source the solution of which is known for a straight duct