Speed of Sound

The speed of sound, or acoustic velocity, is the speed at which sound waves travel through a medium. It depends on the nature of the medium and, in a gas, only on its temperature: about 343 m/s in air at 20 °C.

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How to measure it in class

The FizziQ app turns the smartphone into an acoustic stopwatch and a frequency meter, the two instruments needed to measure the speed of sound in air.

Steps:

• Measure and record the air temperature in the room: it is the only atmospheric parameter the result depends on.
• Place two smartphones in acoustic stopwatch mode at a known distance d, measured with a tape measure, of at least 30 m so that the travel time is much larger than the triggering precision.
• Produce a short, loud sound (clapping two wooden boards together) near the first smartphone: the first stopwatch starts, the second stops when the wave arrives.
• Compute v = d/Δt, then repeat the measurement five to ten times and take the average to reduce the uncertainty on Δt.
• Compare the value obtained with the expected value v = 331 + 0.6·θ, where θ is the temperature in degrees Celsius recorded in the first step.
• Repeat the measurement outdoors at a clearly different temperature, and check that the difference goes in the direction predicted by the formula.

Scientific activities on this topic

One of the easiest experiments to carry out with a smartphone, and one of the most popular, is measuring the speed of sound. We have gathered the different ways to make this measurement in one article: seven ways to measure the speed of sound with a mobile phone.

The following activities, all aimed at measuring the speed of sound in air, can be done in class or as remote learning:

• Estimating the speed of sound by measuring travel time
• Measuring the resonance frequency of a cavity
• Measuring wavelength by interference
• Measuring the frequency of the sound emitted when opening a wine bottle
• Measuring the dominant frequency in a tube

Learn more

The speed of sound in air depends only on temperature

Unlike the speed of light in a vacuum, the speed of sound is not a universal constant: it depends on the medium it travels through. For air, one often reads that it depends “on temperature, pressure and density”. This is wrong for pressure. Theory gives v = √(γP/ρ), and for an ideal gas P/ρ = rT. Substituting, v = √(γrT): pressure and density disappear. At constant composition, the speed of sound in air therefore depends only on the absolute temperature T. Compressing air into a bottle without heating it does not change the speed of sound inside.

This is also why the speed of sound decreases with altitude: not because the air is less dense, but because it is colder. In the stratosphere, where the temperature rises again, the speed of sound rises too, even though the air keeps getting thinner.

In air at room temperature, the wave speed of sound is about 343 m/s at 20 °C. Between -20 °C and +40 °C, the relation v = 331 + 0.6·θ (θ in °C) gives an approximation accurate to better than 1%.

Why sound travels faster in solids

A common reflex is to think that a denser medium slows sound down, since the density ρ is in the denominator. Observation says the opposite: about 343 m/s in air, 1,480 m/s in water, 5,000 m/s in steel. The explanation lies in the general expression v = √(elastic modulus / ρ). Going from air to steel, the density is multiplied by about 6,500, but the elastic modulus is multiplied by about 1.5 million. Stiffness increases much faster than density, and the ratio increases. The correct statement is therefore: at comparable density, the stiffer medium carries sound faster.

A measurement four centuries old

Marin Mersenne published the first credible estimate in 1636, obtained by timing the echo of a gunshot: he found a value of the order of 450 m/s, still too large. Around 1656, Borelli and Viviani, at the Accademia del Cimento, measured the delay between the flash of a cannon and the detonation and obtained about 350 m/s, very close to the exact value. Newton proposed the first theoretical calculation in 1687, but he assumed isothermal propagation and found a result about 16% below the measurement. Laplace resolved the difficulty in 1816 by showing that sound compressions are too fast for heat to have time to diffuse: the transformation is adiabatic, which introduces the factor γ (ratio of heat capacities, γ ≈ 1.4 for air) and reconciles theory and experiment.

The sound barrier

The advent of the airplane in the early 20th century opened a new challenge: exceeding the speed of sound. Breaking the sound barrier in an aircraft proved difficult because of the complex formation of shock waves, the compression and heating of the air, transonic flow resistance, and the induced vibrations and turbulence. These phenomena posed major challenges in aerodynamics, structural design and control, requiring scientific and technological advances before aircraft could cross the sound barrier safely and successfully.

In the early 20th century, many aviation pioneers tried to build aircraft capable of supersonic flight, but these attempts generally ended in failure due to a lack of understanding of the fundamental principles. On October 14, 1947, American test pilot Chuck Yeager became the first human being to officially break the sound barrier aboard the experimental Bell X-1 aircraft. This achievement marked a turning point in aviation. The development of commercial supersonic aircraft led to Concorde, a Franco-British supersonic airliner that made its first flight in 1969. Concorde provided regular transatlantic supersonic flights for more than three decades.

Orders of magnitude

Air at 0 °C: 331 m/s. Air at 20 °C: 343 m/s. Air at 40 °C: 355 m/s. Helium at 20 °C: about 1,000 m/s, hence the high-pitched voice after inhaling it. Fresh water at 20 °C: about 1,480 m/s. Sea water: about 1,530 m/s. Concrete: about 3,100 m/s. Steel: about 5,000 m/s. Glass: about 5,300 m/s. For comparison, light in a vacuum travels at 3.0 × 10⁸ m/s, nearly a million times faster than sound in air.

Formula

In an ideal gas, the speed of sound follows from theory:

v = √(γP/ρ) = √(γrT)

where:

• v: speed of sound (m/s)
• γ: ratio of heat capacities of the gas, dimensionless (γ ≈ 1.40 for air)
• P: gas pressure (Pa)
• ρ: gas density (kg/m³)
• r: specific gas constant (r ≈ 287 J·kg⁻¹·K⁻¹ for air)
• T: absolute temperature (K)

Since P/ρ = rT, the pressure cancels out: for a given composition, only temperature matters.

In practice, in air around room temperature, the linear approximation is used:

v ≈ 331 + 0.6·θ

where:

• v: speed of sound in air (m/s)
• θ: air temperature (°C)

For any medium, the speed takes the general form:

v = √(M/ρ)

where:

• M: elastic modulus of the medium, which measures its stiffness (Pa)
• ρ: density of the medium (kg/m³)

Finally, the travel-time measurement gives directly:

v = d/Δt

where:

• d: distance between source and receiver (m)
• Δt: time taken by the sound to travel that distance (s)

Application examples

• Counting the seconds between lightning and thunder: 3 s corresponds to about 1 km, since 343 × 3 ≈ 1,030 m. Light, on this scale, arrives instantaneously.

• A boat’s sonar emits a pulse and receives the echo from the bottom 0.20 s later. The round trip is 1,500 × 0.20 = 300 m, so the depth is 150 m.

• An aircraft flying at Mach 2 at 11,000 m altitude, where the temperature is about -56 °C, moves at 2 × 295 ≈ 590 m/s. The same Mach 2 at ground level would correspond to 686 m/s: the Mach number alone is not enough to give a speed.

• In medical ultrasound imaging, sound propagates through soft tissue at about 1,540 m/s; this is the value the machine uses to convert echo times into depths.

• In an open-air orchestra, a musician 34 m from the conductor hears the beat 0.1 s late, that is a sixteenth note at a moderate tempo.

• The struck-rail method: when a steel rail is hit, a distant observer perceives two distinct impacts, one transmitted through the rail at about 5,000 m/s, the other through the air at 343 m/s.

FAQ

Q: Does the speed of sound depend on atmospheric pressure? A: No, not in air of a given composition. Theory gives v = √(γP/ρ), but for an ideal gas P/ρ = rT, so v = √(γrT): the pressure cancels out. A high-pressure day and a low-pressure day give the same speed of sound if the temperature is the same.

Q: Why does sound travel faster in steel than in air, even though steel is much denser? A: Because the speed depends on the ratio of stiffness to density, not on density alone. From air to steel, the density is multiplied by about 6,500, but the stiffness by more than a million. The ratio increases, so the speed does too.

Q: Does a loud sound travel faster than a quiet one? A: No. For ordinary intensities, the speed of sound depends neither on amplitude nor on frequency: a high note and a low note arrive at the same time, otherwise music would be unintelligible from a distance. This no longer holds for very intense shock waves, which travel faster than sound.

Q: Why does the voice become high-pitched after breathing helium? A: Helium is much lighter than air, so sound travels about three times faster in it. The resonance frequencies of the vocal tract, proportional to the speed of sound, rise accordingly. The vocal cords themselves vibrate at the same frequency.

Q: What distance is needed for the travel-time measurement? A: At least 30 m, and preferably 50 to 100 m. Over 30 m, the measured time is only 0.09 s; below that, the triggering uncertainty of the stopwatches becomes comparable to the measured duration and the result loses all meaning.

Related concepts

Wave Speed - Frequency - Wavelength - Doppler Effect - Sound Intensity - Echo