---
title: "Define water hammer; continuity and momentum equations"
subject: "Hydraulics"
chapter: 4
key: "semester_4/hydraulics:water-hammer-define-continuity-momentum-equations"
source: https://www.bctengineer.com/q/semester_4/hydraulics/water-hammer-define-continuity-momentum-equations
site: BCT Engineering
---

# Define water hammer; continuity and momentum equations

Hydraulics · Chapter 4: Unsteady flow in pipes

- Asked in: 2070 Magh, 3 marks
- Video: [Numerical | Water Hammer in Pipes | Fluid Mechanics](https://www.youtube.com/watch?v=OjXI5EH1ABg) — The Content Paradox
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/water-hammer-define-continuity-momentum-equations
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-4

## Question

Define water hammer and write down continuity equation and momentum equation for unsteady flow in pipe.

## Short answer

Water hammer is a pressure surge from sudden velocity change. Momentum equation: ∂V/∂t + g ∂H/∂x + fV|V|/(2D)=0. Continuity equation: ∂H/∂t + (c²/g) ∂V/∂x =0, with c as wave celerity.

## Answer

#### Water hammer
Water hammer is the rise and fall of pressure in a pipe caused by a sudden change of flow velocity, such as the quick closing or opening of a valve or turbine gate, or a pump failure. The kinetic energy of the moving water changes into elastic (strain) energy of the water and the pipe wall. A pressure wave travels along the pipe at celerity $c$ and is reflected at the ends. It may burst or collapse the pipe and causes a hammering noise.

#### Equation of motion (momentum / Euler's equation)
Newton's second law for a water element in a pipe of diameter $D$, with $x$ along the pipe, velocity $V$ and piezometric head $H$:
$$
\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial H}{\partial x}+\frac{fV|V|}{2D}=0
$$
For water hammer, the convective term $V\,\partial V/\partial x$ is small compared with $\partial V/\partial t$ and is neglected:
$$
\frac{\partial V}{\partial t}+g\frac{\partial H}{\partial x}+\frac{fV|V|}{2D}=0
$$
Without friction, and for a horizontal pipe, this becomes the Euler equation $\dfrac{1}{\rho}\dfrac{\partial p}{\partial x}+\dfrac{\partial V}{\partial t}=0$.

#### Continuity equation
Mass conservation, including the compressibility of water (bulk modulus $K$) and the elasticity of the pipe:
$$
\frac{1}{\rho}\frac{dp}{dt}+c^2\frac{\partial V}{\partial x}=0\quad\text{or}\quad \frac{\partial H}{\partial t}+\frac{c^2}{g}\frac{\partial V}{\partial x}=0
$$
where $c=\sqrt{\dfrac{K/\rho}{1+KD/(Et)}}$ is the wave celerity.

These two equations (momentum and continuity) are solved together, e.g. by the method of characteristics, to find $H$ and $V$ along the pipe at any time.

## Check yourself

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

1. Water hammer is best defined as
   A. Air entrainment in pipes
   B. Pressure rise and fall in a pipe caused by sudden change of flow velocity
   C. Cavitation at a pump inlet only
   D. Steady friction loss in a pipe
   Answer: B. A sudden velocity change produces pressure waves travelling at celerity c. (Key point: Definition)

2. In the momentum equation for water hammer, which term is neglected and why?
   A. Friction, always
   B. g dH/dx, because pressure is constant
   C. dV/dt, because it is small
   D. V dV/dx, because it is small compared to dV/dt
   Answer: D. The convective term is much smaller than the local acceleration. (Key point: Convective term)

3. The continuity equation for water hammer takes the form
   A. dV/dt + g dH/dx = 0
   B. dH/dt + (c^2/g) dV/dx = 0
   C. dp/dx = 0
   D. Q = AV only
   Answer: B. It accounts for water compressibility and pipe elasticity through c. (Key point: Continuity equation)

## Related in Hydraulics

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- [Water hammer causes and momentum equation](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-causes-momentum-equation) — Hydraulics, chapter 4
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- [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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