Closing a valve at the end of a pipe stops the flow abruptly. The fluid's momentum has nowhere to go, so it compresses the fluid and stretches the pipe wall — the excess energy launches a pressure wave that races back up the pipe at the wave (celerity) speed a, set by the fluid's bulk modulus and the pipe's stiffness. For a fast ("instantaneous") closure the peak surge is the Joukowsky equation:
ΔP = ρ·a·ΔV
a: wave celerity (m/s)
ΔV: change in flow velocity (m/s)
ρ: fluid density (1000 kg/m³ for water)
The wave travels the pipe length L in L/a seconds, reflects off the open (reservoir) end, and returns — a full round trip takes the critical closure time tc,crit = 2L/a. If the valve closes faster than tc,crit, the full Joukowsky surge develops before any relief arrives ("rapid closure"). If it closes slower, the approximate Michaud rule scales the peak down: ΔP_eff ≈ ΔP·(tc,crit/tc).
The pipe strip shows the exact method-of-characteristics solution for an idealized frictionless instant closure: a rectangular pressure pulse that bounces between the closed valve and the open reservoir with period 4L/a, alternating +ΔP and −ΔP at the valve. The friction slider adds an exponential decay envelope so the oscillation dies out, as it does in a real pipe.
- Pipe strip — color shows local overpressure (red) / underpressure (blue) along the pipe at this instant.
- Strip chart — pressure measured at the valve over time, with the theoretical Joukowsky peak marked as reference lines.