Speed of sound

Isaac Newton did the honest thing: he wrote down a formula for the speed of sound in air from first principles, plugged in the known pressure and density, and got an answer $16\%$ too low. For over a century nobody could explain the gap — until Pierre-Simon Laplace spotted the flaw in Newton's assumption, not his algebra. Sound compressions happen too fast for heat to escape; they are adiabatic, not isothermal, and that single correction closes the gap almost exactly. This article walks through Newton's formula, why it fails, Laplace's fix, and how the corrected speed of sound $v = \sqrt{\gamma p/\rho}$ responds to temperature, pressure, density and humidity — the numbers you need for Unit 10 of the Oscillations and Waves syllabus.

The core insight

Sound is a longitudinal wave whose speed in a gas is $v = \sqrt{\gamma p/\rho}$ , because rapid compressions are adiabatic rather than isothermal, so temperature (not pressure alone) is what changes it.

The core idea

Sound is a longitudinal mechanical wave: the medium's particles oscillate along the direction the wave travels, alternately compressing and rarefying. Because it depends entirely on the elastic and inertial properties of a material medium, sound cannot travel through vacuum — unlike light, which needs none.

Newton's isothermal assumption. Newton assumed each compression and rarefaction happens slowly enough that any heat generated has time to leak away, keeping temperature constant throughout the wave. For a gas obeying this isothermal assumption, the speed works out to

v_{\text{Newton}} = \sqrt{\frac{p}{\rho}}

where: $p$ is the gas pressure and $\rho$ is the gas density. At STP ( $p = 1.013\times10^5\ \text{Pa}$ , $\rho = 1.29\ \text{kg/m}^3$ ) this gives $v \approx 280\ \text{m/s}$ — about $16\%$ below the measured $332\ \text{m/s}$ at $0°\text{C}$ , a gap Newton himself could not resolve.

Laplace's adiabatic correction. Laplace realised the oscillations are far too fast for heat to escape a compressed region before the next one arrives — the process is adiabatic. Using the adiabatic gas relation $pV^\gamma = \text{constant}$ (taken as given, not derived, at this level) in place of Boyle's law raises the predicted speed by a factor of $\sqrt{\gamma}$ , giving the accepted formula for the speed of sound in a gas — derived below. The figure shows why the correction matters: in a real sound wave, compressed regions heat up slightly and rarefied regions cool slightly, exactly what "no time to exchange heat" implies.

Figure

Newton assumed heat escapes a compression instantly while Laplace recognised compressions and rarefactions happen too fast for heat exchange

How the speed changes with conditions. With temperature, $v \propto \sqrt{T}$ (absolute temperature) since $p/\rho \propto T$ for an ideal gas — sound in air speeds up by roughly $0.61\ \text{m/s}$ per degree Celsius near room temperature. With pressure alone, at fixed temperature, $p$ and $\rho$ scale together (Boyle's law), so $v$ is unchanged — a very common exam trap. With density, at fixed $p$ and $T$ , $v \propto 1/\sqrt{\rho}$ , so sound is slower in a denser gas such as $\text{CO}_2$ than in air. With humidity, moist air is less dense than dry air (water vapour molecules are lighter on average than the $\text{N}_2$ / $\text{O}_2$ they displace), so sound travels faster in humid air. Sound also travels faster in liquids ( $\approx 1480\ \text{m/s}$ in water) and faster still in solids ( $\approx 5000\ \text{m/s}$ in steel) than in air ( $\approx 340\ \text{m/s}$ ), because both resist compression far more strongly despite being denser. Finally, keep two distinct terms apart: supersonic describes an object moving faster than the local speed of sound, while ultrasonic describes a sound wave whose frequency exceeds about $20\ \text{kHz}$ , beyond human hearing — one is about speed of a body, the other about frequency of a wave.

Derivation ALaplace's corrected speed of sound

The goal is to show how replacing Newton's isothermal assumption with Laplace's adiabatic one converts $v = \sqrt{p/\rho}$ into the experimentally accurate $v = \sqrt{\gamma p/\rho}$ , using only the adiabatic gas law taken as given, so that the origin of the extra factor $\sqrt{\gamma}$ is clear rather than memorised.

Step 1. Start from the general result that the speed of a longitudinal wave in a fluid is $v = \sqrt{E/\rho}$ , where $E$ is the bulk modulus of the medium (a measure of how strongly it resists compression) and $\rho$ is its density. For a gas, the bulk modulus depends on how the compression happens — that is exactly where Newton and Laplace disagreed.

Step 2. Newton assumed an isothermal process, for which $pV = \text{constant}$ (Boyle's law). Differentiating, $p\,dV + V\,dp = 0$ , so $-dp/(dV/V) = p$ . The bulk modulus for an isothermal change is therefore $E_{\text{iso}} = p$ , giving Newton's $v = \sqrt{p/\rho}$ directly — this is where the too-low prediction comes from.

Step 3. Laplace instead treated the compression as adiabatic, governed by $pV^\gamma = \text{constant}$ (given, not derived here). Differentiating this relation gives $p\,\gamma V^{\gamma-1}\,dV + V^\gamma\,dp = 0$ , which rearranges to $-dp/(dV/V) = \gamma p$ . So the adiabatic bulk modulus is $E_{\text{adia}} = \gamma p$ — larger than the isothermal value by exactly the factor $\gamma$ .

Step 4. Substitute $E_{\text{adia}} = \gamma p$ into $v = \sqrt{E/\rho}$ to get the accepted formula, $v = \sqrt{\gamma p/\rho}$ , where $\gamma = C_p/C_v$ is the ratio of specific heats ( $\gamma \approx 1.4$ for air, a diatomic gas). Since $\sqrt{1.4} \approx 1.183$ , this raises Newton's $280\ \text{m/s}$ to about $331\ \text{m/s}$ , matching the observed $332\ \text{m/s}$ at $0°\text{C}$ and confirming that the adiabatic assumption, not the isothermal one, describes real sound waves.

Worked example

Calculate the speed of sound in air at $0°\text{C}$ using Laplace's formula, given $p = 1.013\times10^5\ \text{Pa}$ , $\rho = 1.29\ \text{kg/m}^3$ , $\gamma = 1.4$ , and then find the speed at $27°\text{C}$ .

Step 1. Apply $v = \sqrt{\gamma p/\rho}$ directly:

v = \sqrt{\frac{1.4 \times 1.013\times10^5}{1.29}} = \sqrt{\frac{1.418\times10^5}{1.29}} = \sqrt{1.0996\times10^5} \approx 331.6\ \text{m/s}

This matches the accepted experimental value of about $332\ \text{m/s}$ at $0°\text{C}$ , confirming Laplace's correction over Newton's isothermal formula.

Step 2. To find the speed at $27°\text{C}$ ( $300\ \text{K}$ ), use the temperature-scaling rule $v \propto \sqrt{T}$ with $0°\text{C} = 273\ \text{K}$ , taking $v_1 = 332\ \text{m/s}$ at $T_1 = 273\ \text{K}$ :

v_2 = v_1\sqrt{\frac{T_2}{T_1}} = 332\sqrt{\frac{300}{273}} = 332\sqrt{1.0989} = 332 \times 1.0483

Step 3. Complete the arithmetic: $v_2 \approx 348.0\ \text{m/s}$ . The speed of sound in air rises from about $332\ \text{m/s}$ at $0°\text{C}$ to about $348\ \text{m/s}$ at $27°\text{C}$ , consistent with the rule-of-thumb increase of roughly $0.6\ \text{m/s}$ per degree Celsius, and both values trace back to the single formula $v = \sqrt{\gamma p/\rho}$ combined with $p/\rho \propto T$ .

Common traps

Thinking sound travels through vacuum faintly rather than not at all — a material medium is a hard requirement, unlike for light.
Assuming higher pressure alone makes sound faster — at constant temperature, $p$ and $\rho$ scale together, so pressure by itself leaves $v$ unchanged; only temperature (or a different gas) changes it.
Mixing up supersonic (an object's speed exceeding sound's speed) with ultrasonic (a wave's frequency exceeding human hearing) — one is about speed of a body, the other about frequency of a wave.
Assuming air carries sound fastest among common media — sound is actually slowest in gases and fastest in solids, since solids resist compression far more strongly despite their higher density.

Ask the tutor: Want to see the isothermal-vs-adiabatic bulk modulus derivation worked out with real numbers, or try a few more speed-of-sound numericals across different temperatures and gases?

Why is the isothermal assumption wrong for sound but fine for a slow gas expansion?

Whether a process is isothermal or adiabatic depends on whether heat has time to flow before conditions change again. A slow expansion, like gas leaking from a punctured tyre over minutes, gives heat plenty of time to equalise with the surroundings, so it is close to isothermal. A sound wave oscillates hundreds of times per second — each compression and rarefaction lasts only milliseconds, far too brief for heat to conduct away. That is exactly why Newton's isothermal assumption, reasonable for many everyday gas processes, fails specifically for sound.

Does the speed of sound depend on the loudness or frequency of the sound?

No, not under normal conditions. The formula $v = \sqrt{\gamma p/\rho}$ depends only on the properties of the medium — pressure, density and $\gamma$ — not on the wave's amplitude (loudness) or frequency. A whisper and a shout travel through the same air at the same speed, and a low hum and a high whistle do too, as long as both stay well below the extreme intensities where the linear-wave assumptions break down. What frequency does change is pitch and whether the wave counts as audible, infrasonic or ultrasonic — not its speed.

Why does humid air carry sound faster if water vapour is heavier than air molecules?

The confusion is between the mass of a single water molecule and the density of the mixture. A water molecule ( $\text{H}_2\text{O}$ , molar mass $18$ ) is actually lighter than the $\text{N}_2$ (molar mass $28$ ) and $\text{O}_2$ (molar mass $32$ ) molecules it displaces in the air. So when water vapour molecules take the place of some nitrogen and oxygen molecules, the overall density of the air mixture drops slightly, even though the total pressure stays about the same. Since $v \propto 1/\sqrt{\rho}$ at fixed pressure, that lower density means humid air carries sound a little faster than dry air at the same temperature.

Speed of sound

The core idea

Worked example

Common traps

Before this

Where this leads