---
title: "Derive kinematic wave approximation"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:kinematic-wave-approximation-derive"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/kinematic-wave-approximation-derive
site: BCT Engineering
---

# Derive kinematic wave approximation

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

- Asked in: 2070 Bhadra, 4 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/kinematic-wave-approximation-derive
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Derive the kinematic wave approximation for the movement of fluid.

## Short answer

Neglect local acceleration, convective acceleration and pressure gradient terms in the momentum equation, giving S_f = S_0 and locally uniform flow Q = α A^(5/3). Substituting this into continuity yields the kinematic wave equation ∂A/∂t + c ∂A/∂x = q with celerity c = dQ/dA ≈ (5/3)V, describing flood wave propagation without attenuation.

## Answer

#### Starting point
The momentum equation of the Saint-Venant system is
$$
\underbrace{\frac{\partial Q}{\partial t}}_{1}+\underbrace{\frac{\partial}{\partial x}\left(\frac{Q^2}{A}\right)}_{2}+\underbrace{gA\frac{\partial y}{\partial x}}_{3}-gA(S_0-S_f)=0
$$
Dividing by $gA$ gives the form with slopes: $S_f=S_0-\dfrac{\partial y}{\partial x}-\dfrac{V}{g}\dfrac{\partial V}{\partial x}-\dfrac1g\dfrac{\partial V}{\partial t}$, in which terms 1, 2 and 3 are the local acceleration, convective acceleration and pressure terms.

#### Approximation
For flood waves on steep channels these three terms are very small compared with $S_0$ and $S_f$. Neglecting them:
$$
S_0-S_f=0\ \Rightarrow\ S_f=S_0
$$
The flow is then locally uniform: the discharge depends only on the area at the same section. Using Manning's equation $Q=\dfrac{1}{n}\dfrac{A^{5/3}}{P^{2/3}}S_0^{1/2}$,
$$
Q=\alpha_kA^{\beta},\qquad \beta=\frac53,\ \ \alpha_k=\frac{S_0^{1/2}}{nP^{2/3}}
$$

#### Kinematic wave equation
Continuity: $\dfrac{\partial A}{\partial t}+\dfrac{\partial Q}{\partial x}=q$. From the relation above, $\dfrac{\partial Q}{\partial x}=\dfrac{dQ}{dA}\dfrac{\partial A}{\partial x}$, hence
$$
\frac{\partial A}{\partial t}+c\frac{\partial A}{\partial x}=q,\qquad\text{or}\qquad \frac{\partial Q}{\partial t}+c\frac{\partial Q}{\partial x}=c\,q,\qquad c=\frac{dQ}{dA}=\alpha_k\beta A^{\beta-1}=\beta V
$$
A flood wave therefore moves downstream at the kinematic celerity $c\approx\frac53V$ without changing shape (no attenuation or backwater effects).

## Check yourself

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

1. The kinematic wave approximation neglects:
   A. gravity and friction
   B. local acceleration, convective acceleration and pressure terms
   C. the continuity equation
   D. lateral inflow
   Answer: B. Those terms are small compared with S0 and Sf. (Key point: Neglected terms)

2. Neglecting them gives:
   A. Sf = 0
   B. Sf = S0, so Q depends only on A
   C. S0 = 0
   D. Q is constant
   Answer: B. Flow is locally uniform and Manning's law gives Q = alpha A^beta. (Key point: Result of approximation)

3. In Q = alpha_k A^beta, beta and celerity c are:
   A. beta = 3/5, c = 3V/5
   B. beta = 5/3, c = (5/3) V
   C. beta = 5/3, c = V
   D. beta = 1, c = V
   Answer: B. c = dQ/dA = beta V = 5V/3; the wave keeps its shape. (Key point: Beta and celerity)

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