Propagation of disturbances through a uniform region of a relaxing gas in a duct with spatially varying cross section is analysed using the methods of relatively undistorted waves and weakly nonlinear geometrical optics. Particular attention is focused on situations when the disturbance amplitude is finite, arbitrarily small and not so small. In certain situations a complete history of the evolutionary behaviour of waves including weak shocks can be traced out. The asymptotic decay laws for weak shocks in a non-relaxing gas are exactly recovered. The damping effects of relaxation and non-planar wavefront configurations on the distortion, attenuation and shock formation of pulses, as they propagate, are described in detail. In the small-amplitude high-frequency limit, a solution up to the second order is obtained and numerical computations are carried out for typical values of the physical parameters involved in the solution. Transport equations are derived for signals having all possible wave modes which are mutually coupled and interact resonantly among themselves. The progressive wave approach describes the far field behaviour which is governed by the generalized Burger’s equation.