In one-dimensional granular-liquid mixture flow, we investigate travelling waves that emerge for Froude numbers below a critical threshold ( \(Fr<Fr_{cr}\) ), where the flow smooths out minor perturbations and maintains stable uniform flows. Our analysis utilizes the model introduced by Fei et al. (Appl Math Model 119:763–781, 2023), which incorporates the classical Saint-Venant approach in conjunction with the \(\mu (J)\) rheology, to accurately capture both friction and viscous-diffusion terms. Through a detailed dynamical systems analysis, we reveal the theoretical existence of stable undular bores propagating down inclined channels, in a wide, analytically determined, parameter range. We further explore the influence of the flow parameter values, utilizing the tools of dynamical systems theory. Numerical experiments, based on the Saint-Venant partial differential equations, confirm the stability of these waves and their ability to spontaneously emerge from a smooth profile. We conclude this paper by providing a brief analysis of the additional travelling waveforms emerging in the \(Fr<Fr_{cr}\) regime.
