One cycle of water hammer pressure wave

Water hammer

When the velocity of flowing water in a closed pipe is changed suddenly (valve closed quickly, pump stopped), the kinetic energy of the moving column is converted into pressure energy, producing a pressure wave that travels along the pipe at a speed $c$ (celerity) and is reflected at the ends. The pressure change is $\Delta p = \rho cV_0$. This repeated pressure rise and fall is called water hammer (it produces noise and vibration like hammering).

One cycle of the wave after sudden valve closure

Consider a pipe of length $L$ with a reservoir at the upstream end and a valve at the downstream end, initial velocity $V_0$ towards the valve. The valve is closed suddenly at $t = 0$. (Friction neglected; the wave speed $c$ is constant.)

Stage 1: $0 < t < L/c$ (wave travels from the valve to the reservoir)

 reservoir |<-------------- L -------------->| valve
           |=====[ V=V0 ]=====|====[ V=0 ]===X  closed
                     <---- c <----- pressure wave +dp

The layer at the valve is stopped at once and compressed; pressure rises by $\Delta p$ and the pipe expands. The wave front moves upstream at $c$; the water behind it ($V = 0$) is at high pressure; ahead of it, the water still moves at $V_0$. At $t = L/c$ the whole pipe is at $p_0 + \Delta p$ with $V = 0$.

Stage 2: $L/c < t < 2L/c$ (wave travels from the reservoir to the valve)

 reservoir |=====[ V=-V0 ]=====|====[ V=0, +dp ]===X
                     ----> c ----->  (reflected wave, pressure back to p0)

At the reservoir the pressure cannot stay above the reservoir pressure, so the water flows back into the reservoir with $V = -V_0$ and a wave $-\Delta p$ travels back towards the valve, restoring the pressure to normal. At $t = 2L/c$ the whole pipe is at normal pressure with $V = -V_0$ (flow towards the reservoir).

Stage 3: $2L/c < t < 3L/c$ (wave travels from the valve to the reservoir)

 reservoir |=====[ V=-V0, p0 ]=====|====[ V=0, p0-dp ]===X
                     <---- c <----- negative wave

The valve prevents any further backward flow, so $V$ drops to 0 at the valve and the pressure falls by $\Delta p$ below normal (possible column separation if the pressure falls below vapour pressure). The negative wave travels towards the reservoir. At $t = 3L/c$ the pipe is at $p_0 - \Delta p$, $V = 0$.

Stage 4: $3L/c < t < 4L/c$ (wave travels from the reservoir to the valve)

 reservoir |=====[ V=+V0, p0 ]=====|====[ V=0, p0-dp ]===X
                     ----> c ----->

The low pressure in the pipe draws water from the reservoir at $V = +V_0$; the wave restores normal pressure as it moves to the valve. At $t = 4L/c$ conditions are the same as at $t = 0$ just before closure (flow $V_0$ towards the valve with pressure $p_0$), and the cycle repeats. In reality friction damps the oscillations.

The period of one cycle is $T = 4L/c$.