Speed of Sound in Media

Sound is a mechanical wave — it travels by making particles in a medium vibrate back and forth, passing energy from one particle to the next. Because sound needs matter to travel, the properties of that matter determine how quickly the wave propagates. Whether you are hearing thunder rumbling through air, a whale's call spreading through the ocean, or a tap on a metal rail carrying a warning down the track, the speed of sound depends fundamentally on what the medium is made of and what conditions it is under. This is why sound cannot travel through a vacuum. Two competing properties of a medium govern the speed of sound: its elasticity (or stiffness — how strongly it resists being compressed and then springs back) and its inertia (its density — how much mass must be set in motion). Stiffer media return to equilibrium faster, so the disturbance propagates more quickly. Denser media have more inertia, so they respond more sluggishly and the wave travels more slowly. These two effects combine in a clean formula that applies to any medium, from soft gases to rigid steel. Temperature adds another layer. In a gas, raising the temperature increases the average speed of molecules, which means disturbances are passed on more quickly — so sound travels faster in warmer air. This temperature dependence has real-world consequences: the speed of sound in air at 0 °C is about 331 m s⁻¹, rising to roughly 343 m s⁻¹ at a comfortable 20 °C. Understanding these relationships is essential in fields from architectural acoustics to sonar engineering.

Key takeaways

The Elasticity–Inertia Trade-off
General Formula: v = √(E/ρ)
Temperature Dependence in Air
Speed of Sound in Different States of Matter

The Elasticity–Inertia Trade-off

Think of a medium as a row of masses connected by springs. The stiffer the springs (higher elastic modulus), the faster each mass snaps back and pushes the next one. The heavier the masses (higher density), the more slowly each one responds. Sound speed is therefore proportional to the square root of stiffness divided by density. This is why steel (very stiff) carries sound at about 5 960 m s⁻¹ — nearly 17 times faster than air — even though steel is also much denser, because its stiffness increases far more than its density relative to air.

General Formula: v = √(E/ρ)

The general expression for the speed of sound in any medium is v = √(E/ρ), where E is the relevant elastic modulus and ρ is the density. For a fluid (liquid or gas), E is the bulk modulus B — the resistance to uniform compression. For a solid rod, E is Young's modulus. For air and other ideal gases, the bulk modulus under adiabatic compression equals γP, where γ is the adiabatic index (ratio of specific heats, ≈1.40 for diatomic air) and P is the pressure, giving v = √(γP/ρ).

Temperature Dependence in Air

For an ideal gas, pressure and density are both proportional to temperature, but their ratio P/ρ = RT/M depends only on the absolute temperature T (in kelvin), the molar gas constant R = 8.314 J mol⁻¹ K⁻¹, and the molar mass M. This means v = √(γRT/M). Because v is proportional to √T, a convenient linear approximation works well near room temperature: v ≈ 331 + 0.6θ m s⁻¹, where θ is the Celsius temperature. Every 1 °C rise adds roughly 0.6 m s⁻¹ to the speed of sound in air.

Speed of Sound in Different States of Matter

As a rule of thumb, sound travels fastest in solids, then liquids, then gases, because solids and liquids are far less compressible (higher bulk modulus) than gases. For example: air at 20 °C ≈ 343 m s⁻¹; water at 20 °C ≈ 1 482 m s⁻¹; steel ≈ 5 960 m s⁻¹. Interestingly, the very low density of hydrogen gas (M = 2 g mol⁻¹) makes sound travel through it at about 1 270 m s⁻¹ — much faster than in air — which is why helium makes your voice sound squeaky (same principle: higher speed, higher resonant frequencies).

The Key Equation

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

v — Speed of sound (m s⁻¹): The propagation speed of the pressure wave through the gas.
\gamma — Adiabatic index (dimensionless): Ratio of specific heat at constant pressure to specific heat at constant volume; ≈ 1.40 for diatomic air.
R — Molar gas constant (J mol⁻¹ K⁻¹): Universal constant equal to 8.314 J mol⁻¹ K⁻¹.
T — Absolute temperature (K): Temperature measured in kelvin; T(K) = θ(°C) + 273.15.
M — Molar mass (kg mol⁻¹): Mass of one mole of the gas molecules; for air M ≈ 0.029 kg mol⁻¹.

Worked Example

Problem. Calculate the speed of sound in air at 35 °C using (a) the exact ideal-gas formula and (b) the linear approximation. Take γ = 1.40, R = 8.314 J mol⁻¹ K⁻¹, M = 0.0290 kg mol⁻¹.

v \approx 352 \text{ m s}^{-1}

Answer. The speed of sound in air at 35 °C is approximately 352 m s⁻¹.

Figure

Diagram

Origins · 1 minErnst Chladni and how this was discovered

Ernst Chladni

1756–1827 · German

Ernst Chladni, often called the 'father of acoustics', was among the first to measure the speed of sound experimentally in solid rods and to study how sound speed varies with material. His 1802 treatise systematically documented how sound propagated through different media, laying essential groundwork for the mathematical theory of sound that Newton and Laplace had begun. Chladni is also famous for 'Chladni figures' — beautiful geometric patterns formed when sand on a vibrating metal plate migrates to the nodal lines. He demonstrated these to Napoleon Bonaparte, who was so impressed he offered a prize for the best mathematical explanation of the patterns.

Wave Speed, Frequency and Wavelength — The relationship v = fλ links the speed of sound directly to the frequency and wavelength of a sound wave, so understanding sound speed is essential for applying this equation.
Sound Wave Characteristics — Pitch, loudness, and timbre are all shaped by the properties of sound waves; knowing how speed depends on the medium completes the picture of how sound behaves.
The Doppler Effect — Doppler shift calculations require the speed of sound in the relevant medium, making this concept a direct prerequisite for analysing moving sources and observers.
Transverse and Longitudinal Waves — Sound travels as a longitudinal (compression) wave; understanding this mode of propagation explains why the bulk modulus — resistance to compression — governs its speed.