Speed of Sound in Gases, Liquids and Solids Explained

Sound is a longitudinal pressure wave, and its speed is set by a tug-of-war between the medium's elasticity (how hard it pushes back when squeezed) and its density (how much mass must be shoved). The rule v = √(elastic property/density) explains the famous order: sound crawls through gases (~343 m/s in air), hurries through liquids (~1480 m/s in water), and sprints through solids (~5000 m/s in steel).

These speeds are deeply practical: lightning-to-thunder seconds give storm distance, sonar maps the ocean floor by echo timing, and ultrasound dosing assumes tissue sound speed. Temperature, not loudness, changes the speed in air.

Key Concepts

• Gases: v = √(γP/ρ) = √(γRT/M) (the Newton–Laplace formula). γ = 1.4 for air (ratio of heat capacities), P = pressure (Pa), ρ = density (kg/m³), R = 8.314 J/mol·K, T = absolute temperature (K), M = molar mass (kg/mol). Compressons are too fast for heat to escape, so the process is adiabatic — Newton's original √(P/ρ) was corrected by Laplace with the γ factor.
• Liquids: v = √(B/ρ), where B is the bulk modulus (Pa) — resistance to uniform squeezing.
• Solids (thin rod): v = √(Y/ρ), where Y is Young's modulus (Pa) — resistance to stretching.
• Temperature rule: in gases v ∛ √T — sound travels faster on hot days. Pressure cancels out (P/ρ stays constant for an ideal gas at fixed T), so altitude alone doesn't change it.

Formulas

• Gas: v = √(γRT/M); liquid: v = √(B/ρ); solid rod: v = √(Y/ρ). All give m/s with SI inputs.
• Assumptions and limits: ideal-gas behaviour, small-amplitude (linear) sound, adiabatic compressions. Shock waves from explosions travel faster and need different physics; in humid air sound is slightly faster (water vapour is lighter than dry air).

Worked Example

Compare the speed of sound in air, water, and steel.

1. Air at 27°C (300 K): v = √(1.4 x 8.314 x 300 / 0.029). Numerator: 1.4 x 8.314 x 300 = 3491.9; divide by M = 0.029 kg/mol → 120410; square root → 347 m/s.
2. Water: B = 2.2 x 10⁸ Pa, ρ = 1000 kg/m³ → v = √(2.2 x 10⁸/1000) = √(2.2 x 10⁶) = 1483 m/s — over 4x faster than in air.
3. Steel: Y = 2.0 x 10¹¹ Pa, ρ = 7800 kg/m³ → v = √(2.0 x 10¹¹/7800) = √(2.56 x 10⁷) = 5064 m/s — nearly 15x faster than in air.
4. Storm check: thunder heard 6 s after lightning → distance ≈ 6 x 343 = 2058 m (~2 km). (Rule of thumb: 3 seconds per kilometre.)

Common Misconceptions

• "Denser media always carry sound slower." Wrong — stiffness usually wins. Steel is 6000x denser than air yet carries sound 15x faster, because its Young's modulus is astronomically larger.
• "Louder sounds travel faster." No — amplitude is nowhere in the formulas. A whisper and a shout cross the room together; loudness changes energy, not speed.
• "Sound speed drops with altitude because pressure drops." Pressure cancels out of v = √(γP/ρ) for ideal gases — what actually slows sound at altitude is the colder temperature.

Real-World Applications

• Sonar uses ~1480 m/s in seawater to map the ocean floor and find submarines.
• Medical ultrasound assumes ~1540 m/s in soft tissue to convert echo times into image depths.
• Thunder ranging: counting seconds between flash and bang locates storms.
• Railway workers once pressed ears to tracks — sound racing through steel at 5 km/s warned of trains long before air-borne sound arrived.

To recap: sound speed is √(elasticity/density) — √(γRT/M) in gases, √(B/ρ) in liquids, √(Y/ρ) in solids — about 347, 1483, and 5064 m/s for air, water, steel. Stiffness beats density; temperature rules in gases; loudness changes nothing.