Water Hammer Surge (Joukowsky Equation)

Nikolai Joukowsky derived this in 1898 after investigating burst mains in the Moscow water system, and the result is brutally simple: stop a moving column of liquid instantly and the pressure spike is ρaΔv, independent of pipe length or line pressure. Water at 2 m/s in steel pipe (a ≈ 1200 m/s) surges 1000 × 1200 × 2 = 2.4 MPa: about 350 psi on top of whatever the system already carried. That is why a slammed solenoid valve makes the pipes bang and why a 150 psi-rated fitting can fail on a 60 psi system.

The word "instantly" is doing real work. The surge only reaches full Joukowsky value if the valve closes faster than the wave's round trip, 2L/a: for a 300 m run in steel that is half a second, so most quarter-turn valves qualify. Close slower and the surge falls roughly in proportion. The engineering answers are all about slowing that change: geared or motorised valve actuators, soft-start and soft-stop pump drives, air chambers and bladder arrestors near quick-closing fixtures, and surge tanks on long transmission mains. And note the flip side: a pump tripping on power loss produces a downsurge that can pull the line below vapour pressure and cause column separation, whose rejoining slam is often worse than the original event.