A hydraulic ram uses no motor — it pumps a small fraction of its own drive water uphill by repeatedly harnessing water hammer. Each cycle has three phases:
- Accelerate — the waste valve is open, so water accelerates down the drive pipe under head H: dv/dt = gH/L − kv²/L (gravity vs. pipe friction).
- Slam / surge — once v reaches the trip speed, the waste valve snaps shut. The abrupt deceleration launches a pressure wave described by the Joukowsky equation:
Δp = ρ·c·Δv (surge head Δh = c·Δv / g)
c = pressure-wave speed in the pipe (~1100 m/s)
pulse duration ≈ 2L/c (round-trip of the wave)
- Deliver — if the surge head (H + Δh) exceeds the delivery lift h, the check valve opens for that pulse and pushes a slug of water into the delivery line. The closer h sits to the pump's practical delivery ceiling, the smaller that slug gets — a bigger back-pressure eats into the same finite hammer spike, so less volume gets through per cycle. The valve then reopens and the cycle repeats, typically 20–100 times per minute.
Only a fraction of the supplied water is delivered — the rest is the "waste" flow that powers the hammer. Efficiency is compared with the classic D'Aubuisson formula:
D'Aubuisson: η = q·h / (Q·H)
The right-hand chart below sweeps delivery height h at the current H, trip speed and pipe length and re-runs the drive/delivery volume balance for each point — the fraction of drive water that reaches the delivery line falls monotonically as h grows, because the same finite water-hammer spike has to fight a bigger back-pressure and simply cannot force as much water past the check valve. Real-world relevance: hydraulic rams have pumped water for farms and villages without electricity since the 1770s, and the same Joukowsky surge visualised here is also what engineers must damp in long pressurised pipelines to avoid pipe-bursting transients.