Table of Contents
Speed of Sound and Its Applications
1. Introduction
The speed of sound is the distance travelled by a sound wave per unit time through a medium.
Sound travels through gases, liquids and solids, but its speed differs because the physical properties of these materials differ.
The speed of sound is important in acoustics, meteorology, aviation, engineering, seismology, medical technology and communication systems.
At approximately 20°C, the speed of sound in dry air is about 343 m/s under ordinary atmospheric conditions.
2. Definition and Basic Relationship
The speed of sound is calculated using:
[
v=\frac{d}{t}
]
Where:
- (v) = speed of sound in m/s
- (d) = distance travelled in metres
- (t) = time taken in seconds
For a periodic sound wave:
[
v=f\lambda
]
Where:
- (f) = frequency
- (\lambda) = wavelength
The speed of sound depends primarily on the medium’s elasticity and density.
3. Speed of Sound in Different Media
Sound generally travels faster in solids than in liquids and faster in liquids than in gases, although the exact comparison depends on the materials involved.
Typical approximate values are:
| Medium | Approximate speed of sound |
|---|---|
| Air at 0°C | 331 m/s |
| Air at 20°C | 343 m/s |
| Water | 1,480 m/s |
| Seawater | 1,500 m/s |
| Steel | 5,000 to 6,000 m/s |
| Aluminium | 6,300 m/s |
These values are approximate and depend on temperature, pressure, composition and material properties.
3.1 Why Sound Travels Faster in Solids
Sound speed depends on the relationship between elasticity and density.
Solids generally have high elastic stiffness, allowing disturbances to be transmitted rapidly between particles.
Although solids often have greater density than gases, their much higher elastic modulus generally produces a greater sound speed.
4. Speed of Sound in Gases
For an ideal gas, the speed of sound is given by:
[
v=\sqrt{\frac{\gamma P}{\rho}}
]
Where:
- (\gamma) = ratio of specific heats
- (P) = gas pressure
- (\rho) = gas density
Using the ideal gas relationship, the expression can also be written as:
[
v=\sqrt{\frac{\gamma RT}{M}}
]
Where:
- (R) = universal gas constant
- (T) = absolute temperature in kelvin
- (M) = molar mass of the gas
This equation shows that sound speed in an ideal gas depends on its absolute temperature, molar mass and ratio of specific heats.
5. Effect of Temperature on Sound Speed
The speed of sound in air increases as temperature increases.
An approximate relationship for dry air near ordinary atmospheric conditions is:
[
v\approx331.3+0.606T_C
]
Where (T_C) is the air temperature in degrees Celsius.
For example, at 20°C:
[
v\approx331.3+(0.606\times20)
]
[
v\approx343.4\text{ m/s}
]
The more general ideal-gas relationship is:
[
v\propto\sqrt{T}
]
Where (T) is absolute temperature.
Therefore, doubling the absolute temperature does not double the sound speed. It increases the speed by a factor of (\sqrt{2}), assuming the other gas properties remain constant.
6. Effect of Pressure and Density
For an ideal gas at a fixed temperature and composition, increasing pressure also increases density proportionally.
Because the speed of sound depends on the ratio (P/\rho), the speed is approximately independent of pressure when temperature remains constant.
However, pressure and density can affect sound speed in real gases under different conditions.
The key examination point is that temperature is a major factor affecting sound speed in ordinary air.
7. Speed of Sound in Liquids
The speed of sound in a liquid is given by:
[
v=\sqrt{\frac{B}{\rho}}
]
Where:
- (B) = bulk modulus of the liquid
- (\rho) = density of the liquid
The bulk modulus measures a liquid’s resistance to compression.
A higher bulk modulus generally produces a higher sound speed, while a higher density tends to reduce it if the bulk modulus remains unchanged.
8. Speed of Sound in Solids
For a long, thin rod undergoing longitudinal vibrations, an approximate expression is:
[
v=\sqrt{\frac{Y}{\rho}}
]
Where:
- (Y) = Young’s modulus
- (\rho) = density
For bulk longitudinal waves in an isotropic solid, the appropriate expression involves elastic constants and differs from the simple rod formula.
This distinction is important because the speed of sound in a rod is not always identical to the speed of longitudinal waves in an unrestricted solid.
9. Factors Affecting the Speed of Sound
9.1 Temperature
In gases, increasing temperature generally increases sound speed.
9.2 Nature of the Medium
Elasticity, density and molecular properties influence sound speed.
9.3 Humidity
At the same temperature and pressure, humid air generally has a slightly higher sound speed than dry air because water vapour has a lower molar mass than the average dry-air mixture.
9.4 Wind
Wind affects the speed of sound relative to the ground.
Sound travelling in the direction of the wind has a greater ground-relative speed, while sound travelling against the wind has a lower ground-relative speed.
9.5 Composition
Changes in the chemical composition of a gas can change its molar mass and ratio of specific heats, affecting sound speed.
10. Mach Number
Mach number is the ratio of an object’s speed to the local speed of sound.
[
M=\frac{u}{v}
]
Where:
- (M) = Mach number
- (u) = object’s speed relative to the surrounding medium
- (v) = local speed of sound
Aircraft speeds are commonly described using Mach numbers.
| Classification | Approximate Mach number |
|---|---|
| Subsonic | Below Mach 1 |
| Transonic | Around Mach 1 |
| Supersonic | Above Mach 1 |
| Hypersonic | Commonly Mach 5 and above |
These classifications are useful approximations. The exact aerodynamic regimes depend on flow conditions and the characteristics of the aircraft.
10.1 Sonic Speed
An object travels at the local speed of sound when its Mach number is 1.
10.2 Supersonic Speed
An object travels faster than sound when its Mach number is greater than 1.
10.3 Sonic Boom
A sonic boom is the loud sound associated with shock waves produced by an object travelling faster than sound.
The shock waves can combine into a characteristic pressure disturbance heard on the ground.
11. Measuring the Speed of Sound
The speed of sound can be measured using several methods.
11.1 Distance-Time Method
If the distance travelled by sound and the time taken are known:
[
v=\frac{d}{t}
]
11.2 Echo Method
A sound pulse is directed towards a reflecting surface. The sound travels to the surface and returns.
If the reflecting surface is at distance (d) and the total echo time is (t):
[
v=\frac{2d}{t}
]
The factor of 2 is necessary because sound travels both towards the surface and back.
11.3 Resonance Tube Method
A resonance tube can be used to determine the wavelength of sound.
If the frequency is known, sound speed can be calculated using:
[
v=f\lambda
]
11.4 Time-of-Flight Measurement
Electronic systems measure the time required for a sound pulse to travel between a transmitter and receiver.
This principle is used in ultrasonic sensors and industrial distance-measuring devices.
12. Applications of the Speed of Sound
12.1 SONAR
SONAR stands for Sound Navigation and Ranging.
It uses sound waves to detect underwater objects, measure depth and determine distances.
The distance to an object can be calculated from the time taken by a reflected sound pulse.
[
d=\frac{vt}{2}
]
12.2 Medical Ultrasound
Medical ultrasound uses high-frequency sound waves to produce images of internal body structures.
The speed of sound in tissues is used to calculate the depth and position of reflecting structures.
12.3 Industrial Ultrasonic Testing
Ultrasonic testing detects internal defects in metals and other materials.
The time taken by sound pulses to travel through a material and return from a boundary can reveal the position of cracks and flaws.
12.4 Seismology
Seismic waves travel through Earth’s interior at different speeds.
Scientists analyse these speeds to study Earth’s crust, mantle and core.
12.5 Aviation
Mach number helps describe aircraft speed and the aerodynamic effects of compressible airflow.
12.6 Meteorology
Sound propagation is influenced by atmospheric temperature, wind and humidity.
These factors affect the transmission of sound over long distances.
12.7 Acoustic Engineering
Knowledge of sound speed is important when designing auditoriums, recording studios, concert halls and sound-control systems.
13. Worked Numerical Examples
Example 1: Speed of Sound
A sound wave travels 1,020 m in 3 seconds. Calculate its speed.
[
v=\frac{d}{t}
]
[
v=\frac{1020}{3}=340\text{ m/s}
]
Answer: 340 m/s.
Example 2: Wavelength
A sound wave travels at 343 m/s and has a frequency of 490 Hz. Calculate its wavelength.
[
\lambda=\frac{v}{f}
]
[
\lambda=\frac{343}{490}=0.7\text{ m}
]
Answer: 0.7 m.
Example 3: Echo Distance
An echo returns after 2 seconds. If the speed of sound is 340 m/s, calculate the distance to the reflecting surface.
[
d=\frac{vt}{2}
]
[
d=\frac{340\times2}{2}=340\text{ m}
]
Answer: 340 m.
Example 4: Speed at 30°C
Calculate the approximate speed of sound in dry air at 30°C.
[
v\approx331.3+0.606T_C
]
[
v=331.3+(0.606\times30)
]
[
v=349.48\text{ m/s}
]
Answer: Approximately 349.5 m/s.
Example 5: Mach Number
An aircraft travels at 686 m/s where the local speed of sound is 343 m/s. Calculate its Mach number.
[
M=\frac{u}{v}
]
[
M=\frac{686}{343}=2
]
Answer: Mach 2, indicating supersonic speed.
14. Common Examination Distinctions
- Sound travels through solids, liquids and gases.
- Sound cannot travel through a vacuum.
- The speed of sound depends on the medium.
- Sound generally travels faster in solids than in liquids and gases.
- In air, sound speed increases with temperature.
- For an ideal gas, sound speed is proportional to the square root of absolute temperature.
- Echo calculations require the round-trip distance.
- Mach number compares object speed with local sound speed.
- Mach 1 represents the local speed of sound.
- A sonic boom is associated with shock waves produced by supersonic motion.
36 Key Takeaways for Exams
- Speed of sound is the distance travelled by a sound wave per unit time.
- The SI unit of sound speed is metre per second.
- Sound requires a material medium for propagation.
- Sound travels through gases, liquids and solids.
- Sound cannot travel through a vacuum.
- The wave equation is (v=f\lambda).
- Sound speed depends on the elastic and density properties of the medium.
- Sound generally travels faster in solids than in liquids and gases.
- The approximate speed of sound in dry air at 0°C is 331 m/s.
- The approximate speed of sound in air at 20°C is 343 m/s.
- The speed of sound in air increases with temperature.
- For an ideal gas, sound speed is proportional to the square root of absolute temperature.
- The speed of sound in an ideal gas is (v=\sqrt{\gamma RT/M}).
- The ratio of specific heats is represented by (\gamma).
- The speed of sound in a liquid depends on bulk modulus and density.
- The speed of longitudinal waves in a thin rod depends on Young’s modulus and density.
- Humid air generally has a slightly higher sound speed than dry air at the same temperature and pressure.
- Wind affects sound speed relative to the ground.
- Mach number is the ratio of object speed to local sound speed.
- Mach 1 represents the local speed of sound.
- Subsonic motion occurs below Mach 1.
- Supersonic motion occurs above Mach 1.
- Hypersonic flight is commonly associated with Mach 5 and above.
- A sonic boom is associated with shock waves produced by supersonic motion.
- The speed of sound can be measured using distance and time.
- Echo methods involve a round-trip journey of sound.
- The echo-distance formula is (d=vt/2).
- Resonance tubes can be used to determine sound wavelength.
- SONAR stands for Sound Navigation and Ranging.
- SONAR uses reflected sound waves to detect underwater objects.
- Medical ultrasound uses sound-wave reflections to form images.
- Ultrasonic testing can detect internal material defects.
- Seismic wave speeds provide information about Earth’s interior.
- Aircraft performance is often described using Mach number.
- Atmospheric conditions influence the long-distance propagation of sound.
- Sound-speed measurements have important applications in science, medicine and engineering.