A representative droplet falls under gravity while quadratic air drag opposes its velocity relative to the cross-wind. Its 4-component state (height above pool, vertical speed, horizontal offset, horizontal speed) is integrated with 4th-order Runge–Kutta (RK4) at a fixed 3 ms sub-step:
k = ρ_air·Cd·A / (2m) A = πr², m = (4/3)πr³ρ_water
v_rel = (vy, vx − wind), s = |v_rel|
h' = −vy
vy' = g − k·vy·s
x' = vx
vx' = −k·(vx − wind)·s
Because drag grows with the square of speed while gravity stays constant, the droplet asymptotically approaches a terminal velocity where the two balance: v_t = √(8·r·ρ_water·g / 3·ρ_air·Cd). The velocity-vs-time graph bends toward this analytic line as the droplet falls, and the energy panel shows how much of m·g·H never becomes kinetic energy at impact — it is dissipated to the air as heat by drag, which is exactly the "drag energy loss" percentage.
- Stream — many droplets with the same physics and small randomized radius/offset, giving the visual waterfall its turbulent spread.
- v(t) graph — the tracked droplet's downward speed climbing toward the dashed terminal-velocity asymptote.
- Energy panel — potential energy (blue) converting into kinetic energy (cyan) plus drag loss (amber) as the droplet falls; larger droplets or thinner air (lower turbulence×) lose relatively less to drag and hit the pool closer to free-fall speed.