Radial profiles of velocity perturbations ∆V_R (top row) and ∆V_ϕ (bottom row) for the small, distant, m = 2 kinematic kick simulation (Sect. 2.1), as a function of R, at four different times, at ϕ = 0°. In each panel, the blue line represents the contribution from the Ω − κ/2 wave, the purple line represents the Ω + κ/2 wave, and the black line is the sum of these two modes, which corresponds to the total velocity perturbation at that radius. The initial kick (t = 0 Gyr) decomposes into the two modes. As time evolves, the Ω − κ/2 mode remains dominant and shapes the overall perturbation pattern, while the Ω + κ/2 mode appears as smaller-scale radial oscillations on top of it.