Short Answer

Laplace’s correction is an improvement made to Newton’s formula for the speed of sound in air. Newton assumed that sound travels under isothermal conditions, which gave a lower value than the real speed. Laplace corrected this by saying sound actually travels under adiabatic conditions, meaning no heat is exchanged during compression and expansion of air.

Laplace added a factor called γ (gamma), the ratio of specific heats of air, to Newton’s formula. The corrected formula became v = √(γP / ρ), which accurately matches the actual speed of sound.

Detailed Explanation :

Laplace’s correction

Laplace’s correction is a major scientific improvement made to Newton’s original formula for calculating the speed of sound in gases. Newton’s formula assumed that the compression and rarefaction of air during sound travel occur under isothermal conditions, meaning the temperature of the air remains constant while sound moves. Based on this assumption, Newton used the formula:

v = √(P / ρ)

Here,

v = speed of sound
P = pressure of air
ρ = density of air

However, Newton’s formula gave a speed of sound that was much lower than the actual observed value. Newton’s calculation predicted a speed of around 280 m/s, while experiments showed that the real speed at 0°C is about 331 m/s. This difference showed that something was missing in Newton’s reasoning.

Laplace’s idea

In 1816, French scientist Pierre-Simon Laplace found the mistake in Newton’s assumption. He explained that sound waves cause very rapid compressions and expansions in air. These changes happen so quickly that there is no time for heat to flow in or out of the gas. Therefore, the process is not isothermal, but adiabatic.

In an adiabatic process, temperature does not remain constant. Instead, the air cools or heats slightly during expansion and compression. This affects the elasticity of air and therefore changes the speed of sound.

Laplace corrected Newton’s formula by including the factor γ (gamma), which is the ratio of specific heats of the gas:

γ = Cp / Cv
where

Cp = specific heat at constant pressure
Cv = specific heat at constant volume

For air, γ ≈ 1.4.

Laplace’s corrected formula

By including this γ factor, Laplace modified Newton’s formula to:

v = √(γP / ρ)

This formula gave a speed much closer to the true value measured in experiments.

For example, at 0°C:

Using γ = 1.4
Using the known pressure and density of air

The corrected speed becomes about 331 m/s, which matches the actual speed of sound.

Why Laplace’s correction works

Laplace’s correction works because it considers the true nature of sound propagation:

Sound moves very fast
Air is compressed and expanded continuously
Temperature changes quickly during these processes
No heat exchange happens
Therefore, the process is adiabatic not isothermal

Adiabatic elasticity is greater than isothermal elasticity. This increases the speed of sound. Newton did not include this effect, so he underestimated the speed.

Effect of γ (gamma)

The ratio of specific heats, γ, plays a very important role:

For air, γ = 1.4, which increases the speed by a factor of √1.4 (about 1.18 times).
Without γ, Newton’s value is too low.
With γ, the formula fits real observations.

This shows that elasticity of air under sound propagation is higher than Newton assumed.

Importance of Laplace’s correction

Laplace’s correction has great importance in physics because:

It made the speed of sound formula accurate.
It explained the correct behaviour of gases during sound travel.
It improved the understanding of thermodynamics in wave motion.
It highlighted that sound propagation is an adiabatic process.
It helped scientists better understand how temperature affects the speed of sound.

Laplace’s correction is still used in the modern formula for the speed of sound in gases.

Comparison of Newton’s and Laplace’s results

Aspect	Newton	Laplace
Assumption	Isothermal	Adiabatic
Formula	v = √(P / ρ)	v = √(γP / ρ)
γ used?	No	Yes
Result	Lower speed	Accurate speed

Laplace’s corrected value matches experimental values, proving the correction correct.

Real-life evidence supporting Laplace’s correction

Experiments show speed of sound is higher than Newton predicted.
Thermodynamic studies confirm sound waves create adiabatic changes.
Modern applications like sonar and acoustics use Laplace’s corrected formula.

These observations validate Laplace’s work.

Conclusion

Laplace’s correction corrected Newton’s inaccurate formula for the speed of sound by recognizing that sound travels under adiabatic, not isothermal, conditions. By including the factor γ (ratio of specific heats), Laplace produced a formula that accurately represents the real speed of sound. This correction deepened scientific understanding of sound propagation and remains essential in physics even today.