---
title: "Governing equations, kinematic wave, Courant number"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:governing-eq-kinematic-courant"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/governing-eq-kinematic-courant
site: BCT Engineering
---

# Governing equations, kinematic wave, Courant number

Computational Techniques in Civil Engineering · Chapter 5: Finite difference method

- Asked in: 2078 Kartik, 2+2+1 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/governing-eq-kinematic-courant
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Write down the governing equations used for analyzing the movement of fluid. What are the kinematic wave approximations of these governing equations? Also define Courant number.

## Short answer

Fluid movement is governed by the Saint-Venant continuity and momentum equations. Kinematic wave approximation neglects local/convective acceleration and pressure terms, reducing momentum to S_f=S_0, giving unique Q=αA^β relation and wave celerity c=βV. Courant number C=cΔt/Δx; explicit schemes require C≤1 for stability.

## Answer

#### Governing equations
For 1D unsteady open-channel flow (Saint-Venant equations):

**Continuity (conservation of mass):**
$$
\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q
$$
**Momentum (conservation of momentum):**
$$
\underbrace{\frac{\partial Q}{\partial t}}_{\text{local acc.}}+\underbrace{\frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right)}_{\text{convective acc.}}+\underbrace{gA\frac{\partial y}{\partial x}}_{\text{pressure force}}-\underbrace{gAS_0}_{\text{gravity}}+\underbrace{gAS_f}_{\text{friction}}=0
$$
where $A$ = flow area, $Q$ = discharge, $y$ = flow depth, $q$ = lateral inflow per unit length, $S_0$ = bed slope, $S_f=\dfrac{n^2Q|Q|}{A^2R^{4/3}}$ = friction slope (Manning), $g$ = gravity. These are the Saint-Venant equations (1D unsteady open-channel flow).

In terms of velocity $V$ and depth $y$ for a wide rectangular channel:
$$
\frac{\partial y}{\partial t}+V\frac{\partial y}{\partial x}+y\frac{\partial V}{\partial x}=0,\qquad
\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g(S_0-S_f)
$$

#### Kinematic wave approximation
In many flood-routing problems on steep slopes the local acceleration, convective acceleration and pressure terms are small compared with the gravity and friction terms. Dropping them, the momentum equation becomes
$$
S_0-S_f=0\quad\Rightarrow\quad S_f=S_0
$$
so the flow is always at its normal-depth value for the local area and a unique relation exists:
$$
Q=\alpha_kA^{\beta},\qquad \beta=\frac53\ \text{(Manning)}
$$
Together with continuity, $\dfrac{\partial A}{\partial t}+\dfrac{\partial Q}{\partial x}=q$, this gives the kinematic wave equation
$$
\frac{\partial Q}{\partial t}+c\,\frac{\partial Q}{\partial x}=c\,q,\qquad c=\frac{dQ}{dA}=\beta V
$$
The wave moves downstream at speed $c$ without attenuation (no backwater effects).

#### Courant number
The Courant number is the ratio of the distance a wave travels in one time step to the grid spacing:
$$
C=\frac{c\,\Delta t}{\Delta x}
$$
For explicit schemes, stability requires $C\le1$. For the dynamic wave, $c=V\pm\sqrt{gy}$.

## Check yourself

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

1. What terms are neglected in the kinematic wave approximation of the momentum equation?
   A. Gravity and friction
   B. Local acceleration, convective acceleration and pressure terms
   C. Only the friction term
   D. Continuity
   Answer: B. Dropping them leaves S_0 - S_f = 0, so flow is at normal depth with a unique Q-A relation. (Key point: Kinematic approximation)

2. How does a kinematic wave move?
   A. Upstream with attenuation
   B. Both ways with backwater effects
   C. Downstream at celerity c = beta V without attenuation
   D. It does not move
   Answer: C. The kinematic wave equation Q_t + c Q_x = c q carries the wave downstream at c = dQ/dA = beta V with no attenuation. (Key point: Kinematic wave behaviour)

3. How is the Courant number defined and used?
   A. C = c dt/dx; explicit schemes need C <= 1
   B. C = dx/(c dt); explicit schemes need C >= 1
   C. C = c dx/dt; implicit schemes need C <= 1
   D. C = V/g; must equal 1
   Answer: A. C is the distance travelled by the wave in one step divided by dx, and explicit schemes need C <= 1 for stability. (Key point: Courant number)

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