---
title: "Speed of sound and Mach number"
subject: "Fluid Mechanics"
chapter: 9
key: "semester_4/fluid_mechanics_mechanical:mach-number-speed-of-sound-regimes"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/mach-number-speed-of-sound-regimes
site: BCT Engineering
---

# Speed of sound and Mach number

Fluid Mechanics · Chapter 9: Introduction to Compressible Flow

- Asked in: year not given, 6 marks
- Video: [Compressible Flow Lesson 01E: Speed of Sound and Mach Number](https://www.youtube.com/watch?v=QjSY1C6TLvY) — John Cimbala
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/mach-number-speed-of-sound-regimes
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-9

## Question

Define Mach number. Derive an expression for the velocity of a sound wave (small pressure disturbance) in a compressible fluid and show that for an ideal gas it is $a = \sqrt{kRT}$. Classify compressible flow according to Mach number, and state when a flow may be treated as incompressible.

## Short answer

Mach number Ma = V/a, the ratio of flow velocity to local speed of sound. For an ideal gas, the isentropic wave speed is a = sqrt(kRT). Flows are classified: Ma<0.3 incompressible, 0.3-0.8 subsonic, 0.8-1.2 transonic, 1.2-5 supersonic, >5 hypersonic. Flow may be treated as incompressible for Ma < 0.3 where density change is under about 5%.

## Answer

#### Mach number
$$
Ma = \frac{V}{a}
$$

where $V$ is the flow velocity and $a$ is the local speed of sound. It measures the importance of compressibility: the ratio of inertia force to elastic force.

#### Speed of sound
Consider a small pressure wave moving at speed $a$ into still fluid of density $\rho$ and pressure $p$. In a frame fixed to the wave, the fluid approaches at speed $a$ and leaves at $a - dV$, with properties $\rho + d\rho$ and $p + dp$.

```text
   still fluid         wave        disturbed fluid
   p, rho, V = 0  -->  |  a  |    p+dp, rho+drho, dV
```

**Continuity:** $\rho a = (\rho + d\rho)(a - dV)$. Neglecting the product $d\rho\,dV$:

$$
dV = a\frac{d\rho}{\rho}
$$

**Momentum:** $p - (p + dp) = \rho a\,[(a - dV) - a]$, so $dp = \rho a\,dV$.

Eliminating $dV$:

$$
a^2 = \frac{dp}{d\rho} \quad \Rightarrow \quad a = \sqrt{\frac{dp}{d\rho}} = \sqrt{\frac{E_v}{\rho}}
$$

where $E_v$ is the bulk modulus. The small wave is nearly reversible and adiabatic, so the process is isentropic: $p/\rho^k = $ constant, which gives $dp/d\rho = kp/\rho$. With $p = \rho RT$:

$$
a = \sqrt{kRT}
$$

For air, $k = 1.4$, $R = 287\ \text{J/kg K}$, so $a = 20.05\sqrt{T}$ m/s (343 m/s at 20 °C).

#### Classification of flow
| Mach number | Flow regime |
|---|---|
| $Ma < 0.3$ | Incompressible |
| $0.3 < Ma < 0.8$ | Subsonic compressible |
| $0.8 < Ma < 1.2$ | Transonic |
| $Ma = 1$ | Sonic |
| $1.2 < Ma < 5$ | Supersonic |
| $Ma > 5$ | Hypersonic |

#### Incompressible treatment
For $Ma < 0.3$ the density change in the flow is less than about 5%, so the flow can be treated as incompressible. Above this value compressibility effects must be included.

## Check yourself

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

1. Which result comes from continuity and momentum across a small pressure wave?
   A. a^2 = dp/d(rho)
   B. a^2 = d(rho)/dp
   C. a = dp/d(rho)
   D. a^2 = p rho
   Answer: A. Eliminating dV gives a^2 = dp/drho = E_v/rho. (Key point: Speed of sound derivation)

2. Why does a = sqrt(kRT) for an ideal gas?
   A. The wave is isothermal so dp/drho = p/rho
   B. The wave is isentropic so dp/drho = kp/rho, and p = rho RT
   C. The wave is isochoric
   D. The gas is incompressible
   Answer: B. A small wave is reversible and adiabatic (isentropic), so dp/drho = kp/rho = kRT. (Key point: Ideal gas speed of sound)

3. When can the flow be treated as incompressible?
   A. Ma < 0.3, density change under about 5%
   B. Ma < 1
   C. Ma > 0.3
   D. Ma < 0.8 always
   Answer: A. For Ma below 0.3 the density change is under about 5%, so compressibility can be ignored. (Key point: Incompressible limit)

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