Speed of Sound Formula: v = √(γP/ρ), Laplace Correction

The speed of sound formula for a gas is $v = \sqrt{\frac{\gamma P}{\rho}}$ , where γ is the ratio of the specific heats of the gas, P its pressure and ρ its density. Putting in the values for air at 0 °C gives about 331 m/s. At room temperature sound moves through air at roughly 340 to 346 m/s.

Indian Class 11 books call this Laplace's correction to Newton's formula, and they usually ask for the velocity of sound formula in both versions. US courses more often use the simpler temperature rule $v \approx 331 + 0.6\,T_C$ m/s, with the temperature in degrees Celsius. Both are covered below, along with the general relation $v = f\lambda$ and the speed of sound in liquids and solids.

Quick facts

Gas (Laplace)
$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}$
Newton's form
$v = \sqrt{\frac{P}{\rho}}$ , which gives a value about 15 percent too low
Solid rod
$v = \sqrt{\frac{Y}{\rho}}$
In air at 0 °C
about 331 m/s
SI unit
metre per second (m/s)
Dimensional formula
[M⁰ L¹ T⁻¹]
Rough rule for air
v ≈ 331 + 0.6 T m/s, T in °C

Formula of velocity of sound, symbol by symbol

What goes into the Laplace formula
Symbol	Quantity	SI unit
$v$	Speed of the sound wave	m/s
$\gamma$	Ratio of specific heats, Cp / Cv (1.4 for air)	no unit
$P$	Pressure of the gas	Pa (N/m²)
$\rho$	Density of the gas	kg/m³
$R$	Gas constant, 8.314 J/(mol K)	J/(mol K)
$T$	Absolute temperature	K
$M$	Molar mass of the gas	kg/mol

Sound is a longitudinal wave of compressions and rarefactions. Its speed depends on how hard the medium pushes back when squeezed (its elasticity) and on how much mass has to be moved (its density). For a gas the elasticity is the bulk modulus, and that is where the pressure and γ come from.

Newton's formula and Laplace's correction

Newton's formula treats each compression as happening at constant temperature. For an isothermal change the bulk modulus equals the pressure P, which gives $v = \sqrt{\frac{P}{\rho}}$ . For air at 0 °C, with P = 1.013 × 10⁵ Pa and ρ = 1.293 kg/m³, that predicts about 280 m/s. The measured value is near 331 m/s.

Laplace's correction starts from the fact that the compressions are too fast for heat to flow in or out. The process is adiabatic, so the bulk modulus is γP, not P. That changes the speed by a factor $\sqrt{\gamma} = \sqrt{1.4} \approx 1.18$ :

$v = \sqrt{\frac{1.4 \times 1.013 \times 10^{5}}{1.293}} = \sqrt{1.097 \times 10^{5}} \approx 331\ \text{m/s}$

Using $P = \frac{\rho R T}{M}$ for an ideal gas turns the formula into $v = \sqrt{\frac{\gamma R T}{M}}$ . This shows what really matters: temperature and the kind of gas.

What changes the speed of sound

Temperature. $v \propto \sqrt{T}$ with T in kelvin. Warmer air carries sound faster.
Pressure. At a fixed temperature, changing the pressure changes the density in the same proportion, so P/ρ and the speed stay the same.
Humidity. Water vapour is lighter than dry air, so damp air is slightly less dense and sound travels a little faster in it.
Type of gas. Light gases such as helium carry sound much faster than air.
State of matter. Sound moves fastest in solids and slowest in gases, because solids are far stiffer.

Rough speeds of sound in different media
Medium	Approximate speed
Air at 20 °C	343 m/s
Water	about 1500 m/s
Steel	about 5000 to 6000 m/s

Worked examples using the speed of sound formula

Example 1: a warmer day

Sound travels at 331 m/s in air at 0 °C. Find its speed at 27 °C.

Use kelvin: T₁ = 273 K, T₂ = 300 K.

$v_2 = v_1\sqrt{\frac{T_2}{T_1}} = 331\sqrt{\frac{300}{273}} = 331 \times 1.048 \approx 347\ \text{m/s}$

The rough rule gives 331 + 0.6 × 27 = 347 m/s too.

Example 2: an echo off a cliff

A hiker claps and hears the echo from a cliff 1.5 s later. The speed of sound is 340 m/s. How far away is the cliff?

The sound goes there and back, so the distance one way is half the total path.

$d = \frac{v t}{2} = \frac{340 \times 1.5}{2} = 255\ \text{m}$

Example 3: sound along a steel rail

Steel has Young's modulus 2.0 × 10¹¹ Pa and density 7800 kg/m³. Find the speed of sound in a steel rail.

$v = \sqrt{\frac{Y}{\rho}} = \sqrt{\frac{2.0 \times 10^{11}}{7800}} = \sqrt{2.56 \times 10^{7}} \approx 5.1 \times 10^{3}\ \text{m/s}$

Sound in steel is about fifteen times faster than in air, which is why you can hear a distant train through the rail first.

This page's solver is built on the Laplace form. Enter γ, P in pascals and ρ in kg/m³ to get v in m/s.

Links to other wave formulas

Like every wave, sound obeys $v = f\lambda$ . A tuning fork at 512 Hz in air at 340 m/s sends out waves about 0.66 m long, as the wavelength formula page shows. The frequency of a note decides its pitch, while its amplitude and intensity decide how loud it is. Neither pitch nor loudness changes the speed.

Sound and light are very different waves. Light needs no medium and travels close to a million times faster, as the speed of light formula page explains. The whole set of wave and optics formulas is on the waves and optics hub.

Common slips

Putting the temperature in °C into $\sqrt{T}$ . The square root rule needs kelvin; only the linear rule uses °C.
Thinking higher pressure means faster sound. It does not, if the temperature is unchanged.
Forgetting to halve the path in echo problems.
Using Newton's formula when the question asks for the correct speed. Newton's version is the one that needs correcting.

Frequently asked questions

How do you calculate the speed of sound in a gas or a solid?

In a gas, v = √(γP/ρ), which can be written as v = √(γRT/M). In a solid rod, v = √(Y/ρ). For any sound wave, v = fλ also holds.

What is the speed of sound in air?

About 331 m/s at 0 °C and about 343 m/s at 20 °C. School problems often use 330 m/s or 340 m/s; use whichever the question gives.

Why is sound faster in solids than in gases?

Solids are much stiffer, so they push back far harder when compressed. The greater elasticity outweighs their greater density, so the speed is higher.

Does the speed of sound depend on pressure?

Not at constant temperature. Raising the pressure raises the density in the same ratio, so P/ρ and therefore v stay the same. Temperature is what matters.

What was wrong with Newton's formula for the velocity of sound?

It assumed the compressions were isothermal, which gives about 280 m/s in air. They are actually adiabatic, so the pressure must be multiplied by γ, giving about 331 m/s.