Eulerian & Lagrangian The same flow, three ways. A steady stream drifts right while a travelling side-gust waggles it. Cyan is a smoke line from a fixed nozzle (a streakline). Violet traces where single parcels have actually been (pathlines). Faint white contours are the streamlines — the flow pattern frozen at this instant. All three disagree because the flow is unsteady. The journey: Euler watches fixed points, Lagrange rides with the parcel. Here u = (U, V₀ sin(ωt − kx)); parcels feel a Doppler-shifted wobble ω − kU, so slide the gust speed to U and every pathline snaps straight — the parcels surf the gust.