The perturbation analysis method of multiple scales is applied to the problem of linear sound propagation in a rectangular waveguide with viscous and thermal dissipation effects at the boundary. In the past, the method of multiple scales has been applied to the corresponding problem of finite level fluctuations in a waveguide. The second‐order solution (representing the first correction due to nonlinearity) cited in these investigations would however preclude the solution of the linear problem, as it grows without bound with the axial coordinate of the waveguide. In this investigation an alternate solution at second order is proposed, which is bounded throughout the spatial domain of the duct. This property of the alternate solution allows the linear problem to be solved, including the region arbitrarily close to the cutoff frequency for higher‐order modes. In addition, the uniform solution may be directly applied to the analysis of finite level fluctuations, with consequences as yet unknown.