Speed of Sound in Different Media

Formulas for Sound Speed Physics

The speed of sound is an essential concept in physics, helping us understand how vibrations travel through different substances. Sound waves, unlike electromagnetic waves such as light, cannot travel in a vacuum. They require a medium—such as air, water, or solid matter—because sound is the result of particles colliding with one another to transmit energy. In this extended article, we will explore the formulas governing the speed of sound in different media, discuss the underlying physics, provide real-world applications, and work through detailed examples. By the end, you will have a comprehensive understanding of how and why sound moves differently in gases, liquids, and solids.

What is the Speed of Sound?

The speed of sound is defined as the distance a sound wave travels per unit of time as it propagates through an elastic medium. It is usually measured in meters per second (m/s). The specific value depends on the medium’s properties, particularly its elasticity and density.

For example, in everyday life, we often notice that sound seems delayed over long distances in air. When watching fireworks, we see the flash before we hear the boom. This delay is because sound in air travels at roughly 343 m/s at 20°C, which is far slower than the speed of light. In contrast, when sound travels through steel or water, it arrives much faster because of the materials’ higher elastic properties.

The General Formula for the Speed of Sound

The physics behind sound propagation can be expressed with a general formula:

$$ v = \sqrt{\frac{E}{\rho}} $$

Where:

$ v $ = speed of sound (m/s)
$ E $ = elastic modulus of the medium (Pa)
$ \rho $ = density of the medium (kg/m³)

This relationship reveals the balance between elasticity and density. A highly elastic material will allow sound to move faster, while higher density tends to slow down propagation. Because solids are both highly elastic and dense, sound generally travels fastest in them. In contrast, gases are much less elastic, so sound is slowest in air and other gases.

Speed of Sound in Gases

In gases, the speed of sound is influenced not only by elasticity and density but also by thermodynamic properties such as temperature and pressure. The specific formula is:

$$ v = \sqrt{\frac{\gamma R T}{M}} $$

Where:

$ \gamma $ = ratio of specific heats ($ C_p/C_v $)
$ R $ = universal gas constant (8.314 J/mol·K)
$ T $ = temperature in Kelvin
$ M $ = molar mass of the gas (kg/mol)

Key Observations:

The speed of sound in gases increases with temperature. Hotter air transmits sound faster.
Gas type matters: helium has a much higher speed of sound than air due to its low molar mass.
Pressure has little direct effect on speed, since both pressure and density scale together.

Example 1: Speed of Sound in Air

Given conditions: $ \gamma = 1.4 $, $ R = 8.314 \, J/mol·K $, $ T = 293 \, K $ (20°C), $ M = 0.029 \, kg/mol $.

$$ v = \sqrt{\frac{1.4 \times 8.314 \times 293}{0.029}} $$

$$ v \approx 343 \, m/s $$

This matches the well-known value for air at room temperature.

Example 2: Speed of Sound in Helium

Conditions: $ \gamma = 1.66 $, $ M = 0.004 \, kg/mol $, $ T = 293 \, K $.

$$ v = \sqrt{\frac{1.66 \times 8.314 \times 293}{0.004}} $$

$$ v \approx 1005 \, m/s $$

This explains the “funny voice” effect when speaking after inhaling helium—sound waves travel faster in helium, altering the resonance frequencies of the vocal tract.

Speed of Sound in Liquids

In liquids, compressibility is measured by the bulk modulus $ B $. The formula becomes:

$$ v = \sqrt{\frac{B}{\rho}} $$

Water, being relatively incompressible, allows sound to travel much faster than in air. Typical values are around 1497 m/s at 25°C. The exact value depends on temperature, salinity (in seawater), and pressure.

Example 3: Speed of Sound in Water

Given: $ B = 2.2 \times 10^9 \, Pa $, $ \rho = 1000 \, kg/m^3 $.

$$ v = \sqrt{\frac{2.2 \times 10^9}{1000}} $$

$$ v = \sqrt{2.2 \times 10^6} \approx 1483 \, m/s $$

This calculation closely matches real-world measurements. It also explains why sonar is so effective underwater: sound can travel long distances without significant delay.

Speed of Sound in Solids

Solids have the strongest intermolecular forces, making them highly elastic. This allows sound to move fastest through solids. Depending on the type of wave, formulas differ:

Longitudinal waves: $ v = \sqrt{\frac{Y}{\rho}} $, where $ Y $ = Young’s modulus.
Transverse waves: $ v = \sqrt{\frac{G}{\rho}} $, where $ G $ = shear modulus.

Example 4: Speed of Sound in Steel

Given: $ Y = 2.0 \times 10^{11} \, Pa $, $ \rho = 7850 \, kg/m^3 $.

$$ v = \sqrt{\frac{2.0 \times 10^{11}}{7850}} $$

$$ v = \sqrt{2.55 \times 10^7} \approx 5050 \, m/s $$

This is why sound travels rapidly through railway tracks, often allowing people to detect an oncoming train by placing an ear on the steel rails before hearing it in the air.

Factors Affecting the Speed of Sound

While formulas provide exact relationships, several practical factors influence the actual speed:

1. Temperature

Higher temperatures increase molecular motion, which enhances energy transfer between particles, raising the speed of sound. In air, the formula can be simplified to:

$$ v \approx 331 + 0.6T $$

(where $ T $ is in °C)

2. Humidity

In air, higher humidity increases sound speed because water vapor is less dense than nitrogen and oxygen. This lowers the average molar mass of air.

3. Pressure

At constant temperature, pressure has little effect on the speed of sound in gases because both pressure and density change proportionally.

4. Medium Type

Solids > liquids > gases: this is the general order of increasing sound speed due to differences in elasticity and particle closeness.

Comparison of Sound Speeds in Different Media

Medium	Speed of Sound (m/s)
Air (20°C)	343
Helium (20°C)	1005
Water (25°C)	1497
Seawater	1530
Steel	5050
Glass	5640
Diamond	12000

Applications of Speed of Sound

The study of sound speed is not only theoretical but has broad applications in science and technology:

1. Aviation and Supersonic Flight

The speed of sound defines Mach numbers. Aircraft traveling faster than Mach 1 produce shockwaves, resulting in sonic booms. Engineers rely on precise calculations of sound speed at varying altitudes and temperatures to design efficient supersonic jets.

2. Underwater Navigation (Sonar)

Ships and submarines use sonar (Sound Navigation and Ranging) to map the ocean floor and detect objects. Because sound travels far and fast in water, it is more effective than radio waves underwater.

3. Medical Imaging (Ultrasound)

Ultrasound machines use the speed of sound in tissues to create detailed images. Since different tissues have slightly different sound velocities, reflections create contrasts used for diagnostic purposes.

4. Seismology

Geophysicists analyze sound and seismic waves to study Earth’s interior. By measuring the speed of waves through rock layers, scientists determine composition, depth, and even predict earthquakes.

5. Structural Engineering

Engineers measure sound speed in materials to assess quality and detect flaws. Non-destructive testing relies heavily on ultrasonic techniques that depend on accurate sound speed values.

Historical Perspective

The study of sound speed dates back centuries. In 1635, Pierre Gassendi attempted to measure the speed of sound by observing cannon fire at a distance. Later, Isaac Newton estimated it using air’s elasticity, but his value was slightly off due to not accounting for temperature effects. In 1738, Daniel Bernoulli and later Laplace corrected this by introducing the adiabatic process concept, leading to the modern formula for gases.

Everyday Examples

Understanding the speed of sound helps explain many everyday observations:

Thunder and lightning: The flash arrives instantly, while thunder is delayed, letting us estimate distance to a storm.
Echoes: Reflected sound waves depend on sound speed and distance.
Musical instruments: The pitch depends on sound speed in air and inside resonating cavities.
Underwater communication: Divers rely on sound because radio signals fail in water.

The physics formula for the speed of sound reveals how deeply material properties shape wave behavior. Whether in gases, liquids, or solids, sound’s journey is determined by elasticity, density, and temperature. From simple everyday experiences like hearing thunder to advanced fields such as sonar, ultrasound, and aerospace engineering, mastering the speed of sound is essential to both science and technology. By studying this phenomenon in detail, we gain insight into the invisible but powerful ways vibrations connect us to the physical world.