Nature and Speed of Sound Waves

Sound is a longitudinal mechanical wave produced by a vibrating source, transmitted as alternating compressions and rarefactions in a medium; its speed is about 343 m/s in air at 20 °C, about 1,480 m/s in water and about 5,000 m/s in steel. In brief: sound cannot travel in a vacuum. Its speed depends on the properties of the medium, being greater in stiffer and less dense materials, and increases with temperature in air by about 0.6 m/s per °C. The approximate formula for air is v = 331 + 0.6T, with T in °C.

Thunder arrives after lightning, and sound travels faster in water than in air. This guide describes how sound is produced and transmitted, what determines its speed, compares the speed in different media in a table, works through calculations of distance by echo and thunder and corrects common mistakes.

What is sound?

A vibrating object, such as a loudspeaker cone or a guitar string, pushes on the neighboring particles of the air, creating a region of higher pressure (compression). When it moves back, it leaves a region of lower pressure (rarefaction). These pressure variations propagate outward as a longitudinal wave. Individual air molecules oscillate back and forth about fixed positions, and do not travel with the wave. When the wave reaches the ear, it makes the eardrum vibrate. Because sound needs particles to transmit it, it cannot travel in a vacuum: a ringing bell inside an evacuated jar becomes silent when the air is pumped out. The frequency of the sound is the frequency of the source, and the wave speed depends on the medium.

What determines the speed of sound?

The general formula is v = √(B ÷ ρ) for fluids (B is the bulk modulus, a measure of stiffness, and ρ the density), and v = √(E ÷ ρ) for a thin solid rod (E is Young’s modulus). Stiffer materials transmit disturbances more quickly, and denser ones more slowly. As a result, sound is usually fastest in solids, slower in liquids and slowest in gases, as the stiffness difference outweighs the density difference.

• Temperature: in air, the speed rises with temperature, approximately v = 331 + 0.6T (m/s, T in °C). More precisely, v = 20.05√T(K). The speed does not depend on pressure, since both the stiffness and density change in proportion.
• Humidity slightly increases the speed, because water vapor is lighter than the air it displaces.
• The speed does not depend on the frequency or amplitude of the sound, for normal sounds.

Speed of sound in different media

Worked example: thunder, echo and temperature

1. Thunder. The flash of lightning is seen, and the thunder is heard 6.0 s later. The light arrives almost instantly. Distance = 343 × 6.0 = 2,058 m, about 2 km. Rule of thumb: 3 s per km.
2. Echo in a well. A stone dropped into a well makes a sound that is heard 2.0 s after the release. The fall time t₁ and the return time of the sound t₂ add to 2.0 s: ½gt₁² = 343 × t₂. With t₂ = 2.0 − t₁, we have 4.9t₁² + 343t₁ − 686 = 0. The positive root is t₁ = [−343 + √(343² + 4 × 4.9 × 686)] ÷ (2 × 4.9) = [−343 + √131,095] ÷ 9.8 = (−343 + 362.1) ÷ 9.8 = 1.95 s. The depth is 4.9 × 1.95² = 18.6 m, and the sound takes 18.6 ÷ 343 = 0.054 s to return, so the fall time and the return time add up to about 2.0 s.
3. Temperature effect. On a cold day at −10 °C: v = 331 + 0.6 × (−10) = 325 m/s. On a hot day at 35 °C: v = 331 + 21 = 352 m/s.
4. Sonar. A ship’s sonar sends a pulse and receives the echo from the sea floor after 0.40 s. Depth = 1,500 × 0.40 ÷ 2 = 300 m.

How was the speed of sound measured?

Marin Mersenne and Pierre Gassendi measured the speed of sound by timing the delay between the flash and the sound of a cannon in the 1630s. A famous measurement was made in Lake Geneva in 1826 by Jean-Daniel Colladon and Charles Sturm, who struck an underwater bell and timed the sound at a distance of 13.5 km, to find a speed in water of about 1,435 m/s. Newton derived a formula in 1687, but it gave a value too low, and Laplace corrected it in 1816 by noting that the compressions in a sound wave are adiabatic rather than isothermal.

Where is the speed of sound used?

Sonar and ultrasound imaging use the time delay of echoes to measure distance. Seismologists use sound waves in rock to find oil, and aircraft designers consider the Mach number, the speed of an aircraft divided by the speed of sound. A speed above Mach 1 is supersonic, and the aircraft produces a shock wave heard as a sonic boom.

Which mistakes are common?

Mistake 1: Thinking that sound is faster in less dense media

Sound is faster in solids, in spite of their higher density, because they are much stiffer.

Mistake 2: Saying that louder sounds travel faster

The speed does not depend on the loudness or on the pitch.

Mistake 3: Using the speed in air for sound in water

Use the speed of the medium that the sound travels through.

Mistake 4: Forgetting that the delay with light is negligible

For lightning and thunder, light travel time can be ignored, but the sound time is the delay.