---
title: "Water hammer causes and momentum equation"
subject: "Hydraulics"
chapter: 4
key: "semester_4/hydraulics:water-hammer-causes-momentum-equation"
source: https://www.bctengineer.com/q/semester_4/hydraulics/water-hammer-causes-momentum-equation
site: BCT Engineering
---

# Water hammer causes and momentum equation

Hydraulics · Chapter 4: Unsteady flow in pipes

- Asked in: 2072 Asoj, 3+5 marks
- Video: [Water hammer causes and effects](https://www.youtube.com/watch?v=Y1DOcxE50Lw) — 4MechEngineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/water-hammer-causes-momentum-equation
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-4

## Question

Explain the water hammer phenomenon and mention its causes. Derive the momentum equation for unsteady flow through pipe.

## Short answer

Water hammer is a rapid pressure fluctuation in a pipe when flow velocity suddenly changes, converting kinetic energy into pressure waves. Causes include sudden valve operation, pump start/stop, turbine load change, pipe rupture, and air/vapor pockets. The momentum equation is ∂V/∂t + V∂V/∂x + g∂H/∂x + fV|V|/(2D) = 0.

## Answer

#### Water hammer phenomenon
When the velocity of flow in a closed conduit is changed rapidly, the moving water column is suddenly slowed down or accelerated. Its kinetic energy is converted into pressure (strain) energy, and a pressure wave travels along the pipe at the wave speed $c = \sqrt{\dfrac{K/\rho}{1 + KD/(Et)}}$, reflecting at the reservoir and at the valve. The resulting rapid rise and fall of pressure, with a hammering noise, is **water hammer**. For instantaneous closure, $\Delta p = \rho cV_0$.

#### Causes
1. Sudden closing or opening of valves or gates (the commonest cause).
2. Starting or stopping of pumps, or a power failure to the pump.
3. Sudden change of load on a turbine (governor action closing the guide vanes or nozzle).
4. Rupture of a pipe, or a sudden change in the demand at the end of a pipe.
5. Air pockets trapped in the pipe being suddenly expelled, or vapour cavities collapsing when a separated water column rejoins.
6. Slamming of a check (non-return) valve when the flow reverses after a pump trips.

#### Momentum equation for unsteady flow through a pipe
Consider a short element of fluid of length $dx$ in a pipe of cross-section $A$ and perimeter $P$, inclined at an angle $\theta$ to the horizontal, with $x$ measured along the pipe. Forces along the pipe:
- Pressure: $pA - \left(p + \dfrac{\partial p}{\partial x}dx\right)A = -\dfrac{\partial p}{\partial x}A\,dx$
- Weight component: $-\rho gA\,dx\sin\theta$
- Wall shear: $-\tau_0P\,dx$

Newton's second law, with mass $\rho A\,dx$ and acceleration $\dfrac{DV}{Dt} = \dfrac{\partial V}{\partial t} + V\dfrac{\partial V}{\partial x}$:
$$
\rho A\,dx\left(\frac{\partial V}{\partial t} + V\frac{\partial V}{\partial x}\right) = -\frac{\partial p}{\partial x}A\,dx - \rho gA\sin\theta\,dx - \tau_0P\,dx
$$
Divide by $\rho A\,dx$. With the piezometric head $H = \dfrac{p}{\gamma} + z$ ($\dfrac{\partial z}{\partial x} = \sin\theta$), the pressure and weight terms combine to $-g\dfrac{\partial H}{\partial x}$. The wall shear $\tau_0 = \dfrac{f\rho V|V|}{8}$ and $\dfrac{P}{A} = \dfrac{4}{D}$ give $\dfrac{\tau_0P}{\rho A} = \dfrac{fV|V|}{2D}$:
$$
\boxed{\frac{\partial V}{\partial t} + V\frac{\partial V}{\partial x} + g\frac{\partial H}{\partial x} + \frac{fV|V|}{2D} = 0}
$$
This is Euler's (momentum) equation for unsteady flow in a pipe. In water hammer problems the convective term $V\,\partial V/\partial x$ is small compared with $\partial V/\partial t$ and is usually dropped.

**Use in water hammer.** With $V\,\partial V/\partial x$ and friction neglected, the equation reduces to $\dfrac{\partial V}{\partial t} + g\dfrac{\partial H}{\partial x} = 0$. Together with the continuity equation $\dfrac{\partial H}{\partial t} + \dfrac{c^2}{g}\dfrac{\partial V}{\partial x} = 0$, it gives the wave solution $\Delta H = \mp\dfrac{c}{g}\Delta V$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What physically happens when a valve in a pipe is closed suddenly?
   A. Friction increases and stops the flow gradually
   B. Kinetic energy of the moving water converts to pressure (strain) energy and a pressure wave travels along the pipe
   C. The pipe wall loses all stress
   D. The water cools and its density rises
   Answer: B. Rapid slowing of the water column converts kinetic energy to strain energy, producing a pressure wave that reflects at the ends. (Key point: Water hammer mechanism)

2. Which of the following is a recognised cause of water hammer?
   A. Steady flow at constant valve opening
   B. A slow, gradual change of reservoir level only
   C. Power failure to a pump or sudden turbine load change
   D. Laminar flow in a horizontal pipe
   Answer: C. Sudden valve operation, pump starting/stopping or power failure, and governor action on turbines all change velocity rapidly. (Key point: Causes of water hammer)

3. Which three forces act on the fluid element along the pipe in deriving the unsteady momentum equation?
   A. Pressure, weight component and wall shear
   B. Weight, centrifugal force and thrust
   C. Pressure, buoyancy and viscosity of air
   D. Surface tension, weight and inertia
   Answer: A. Net pressure force, the weight component along the slope and wall shear are balanced against mass times acceleration. (Key point: Forces in momentum equation)

4. In Newton's second law for the element, the acceleration is written as
   A. dV/dt + V dV/dx
   B. g dH/dx
   C. V dV/dx only
   D. dV/dt only
   Answer: A. The material acceleration DV/Dt is the local term plus the convective term. (Key point: Acceleration term)

## Related in Hydraulics

- [Derive continuity equation for unsteady pipe flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontinuity-equation-unsteady-pipe-flow) — Hydraulics, chapter 4
- [Define water hammer; continuity and momentum equations](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-define-continuity-momentum-equations) — Hydraulics, chapter 4
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- [Water hammer, Euler and continuity equations](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-euler-continuity-equations) — Hydraulics, chapter 4
- [One cycle of water hammer pressure wave](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-one-cycle-wave-sketch) — Hydraulics, chapter 4, asked 2×
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