Waves are omnipresent in avalanches on Earth and other planets. The dynamic nature of waves makes them dangerous in geological hazards such as debris flows, turbidity currents, lava flows, and snow avalanches. Extensive research on granular waves has been carried out by using theoretical and numerical approaches with idealized assumptions. However, the mechanism of waves in realistic complex situations remains intangible, as it is notoriously difficult to capture complex granular waves on real terrain. Here, we leverage a recently developed hybrid Eulerian-Lagrangian numerical scheme and an elastoplastic constitutive model to investigate the processes involved in waves of snow avalanches, including erosion, deposition, and flow instability induced by terrain irregularity. This enables us to naturally simulate roll-waves, erosion-deposition waves, and their transitions in a single large-scale snow avalanche on real terrain. Simulated wave features show satisfactory consistency with field data obtained with different radar technologies. Based on a dimensionless analysis, the wave mechanics is not only controlled by the Froude number and local topography but also by the mass of the wave which governs the entrainment propensity. This study offers new insights into wave mechanisms of snow avalanches and provides a novel and promising pathway for exploring transient waves in granular mass movements.