Evolution of the bending mode and the breathing mode in the single-impact model. First panel: bending and breathing amplitude ratio as a faction of time and the galactocentric radius. The horizontal axis indicates the time since the pericentre passage of the dwarf. A vertical dotted line indicates the pericentre time t–t₀ = 0. Dashed curve represents R/σ_R(R). Second panel: bending amplitude as a function of time and the galactocentric radius. The amplitude is normalised by the maximum amplitude at each radius. Solid curves represent (1, 3, 5, 7) × π/v(R)−50 Myr. Third panel: same as the second panel but for the breathing mode. Solid lines represent (1, 2, 3, 4) × π/Ω(R). The bending mode decays rapidly due to horizontal mixing, while the breathing mode persists longer, driven by spiral arms. This difference of the two modes causes a transition to breathing dominance over time on the timescale of horizontal mixing (R/σ_R).