Some authors have tried to calculate propagation in a varying cross se
ction duct by slicing it into cylindrical elements with length small c
ompared to wavelength. This enables to solve the wave equation with th
e multimodal decomposition formulation. The results compared favorably
with experimental data. However, this approach have certain limitatio
ns and we, therefore, wanted to study this problem by writing continuo
us equations. First of all, we develop a matricial horns equation exte
nding the classical Horn equation. Then, we find a matricial Riccati e
quation for the impedance matrix which is symmetric and only depends o
n the axial coordinate. This equation for the impedance avoid spurious
divergence due to vanishing modes. Finally we are able to get the pre
ssure or the velocity field by a calculation of a kind of argument mat
rix (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 are apllied to a p
articular geometry in the case of monopole source the solution of whic
h is known for a straight duct.