Short Answer
Newton’s formula for the speed of sound states that the speed of sound in a gas is equal to the square root of the ratio of the elasticity of the gas to its density. Mathematically, Newton gave the formula as:
v = √(E / ρ),
where v is the speed of sound, E is the bulk modulus (elasticity), and ρ is the density of the gas.
According to Newton’s original formula, the speed of sound in air was calculated assuming isothermal conditions. However, the values predicted were lower than the actual speed. Later, Laplace corrected it by considering adiabatic conditions, giving more accurate results.
Detailed Explanation :
Newton’s formula for speed of sound
Newton attempted to calculate the speed of sound in air by treating sound as a mechanical wave that travels due to compression and rarefaction in a medium. To do this, he assumed that the elasticity of air is equal to its pressure, and that sound travels under isothermal conditions, meaning the temperature of the gas remains constant during compression and expansion.
Using these assumptions, Newton derived the formula:
v = √(E / ρ)
Here,
- v = speed of sound
- E = elasticity (bulk modulus) of the medium
- ρ (rho) = density of the medium
For air, Newton assumed E = P (pressure of air). Thus, the speed of sound became:
v = √(P / ρ)
Using normal atmospheric conditions, Newton calculated the value of speed of sound as approximately 280 m/s. However, experiments showed that the actual speed is around 332 m/s at 0°C. This difference occurred because Newton assumed isothermal conditions, which was incorrect for sound propagation.
Why Newton’s formula gave wrong value
Sound waves move very fast, and the compression and expansion of air happen too quickly for heat exchange with the surroundings. Therefore, the process is not isothermal. Instead, it is adiabatic, meaning no heat is exchanged.
Under adiabatic conditions, the elasticity of the gas becomes:
E = γP,
where γ (gamma) is the ratio of specific heats of the gas.
For air, γ ≈ 1.4.
Laplace corrected Newton’s formula by including γ:
v = √(γP / ρ)
This corrected formula matches the observed speed of sound much more accurately. At 0°C, it gives around 331 m/s, which is very close to the true value.
Understanding elasticity and density in the formula
Newton’s formula is based on two key properties of the medium:
- Elasticity (E):
This represents how strongly the medium resists being compressed. A highly elastic medium pushes back strongly, making sound travel faster. Solids have high elasticity, so sound travels fast in solids. - Density (ρ):
This represents how heavy the medium is. A dense medium has heavier particles that are slow to vibrate, making sound travel slower. Gases with low density allow sound to travel faster.
The formula v = √(E / ρ) clearly shows that speed increases with elasticity and decreases with density.
Why elasticity matters
When a sound wave passes through a medium, particles must compress and expand quickly. A medium that offers strong restoring force (high elasticity) responds quickly, helping sound travel fast. Solids, which have tightly bound particles, show very high elasticity and hence have high sound speeds.
Why density matters
If the medium’s density is high, its particles have more mass. Heavier particles take more time to move when a vibration is applied. This slows down the sound.
Thus:
- High elasticity → high speed
- High density → low speed
This relationship forms the core of Newton’s formula.
Historical importance of Newton’s formula
Newton’s work was extremely important because:
- It showed that sound is a mechanical wave.
- It linked sound speed to physical properties of the medium.
- It introduced the idea that pressure and density determine wave speed.
- It laid the foundation for future improvements by physicists like Laplace.
Even though his answer was not fully accurate, his formula helped scientists understand the physics of wave motion and develop more accurate models.
Laplace’s correction and final result
Laplace corrected Newton’s formula by adding the factor γ (ratio of specific heats):
v = √(γP / ρ)
This correction was crucial because sound propagation in air happens too fast to allow heat transfer. Laplace’s correction assumes adiabatic changes, which matches real-world behaviour.
At 0°C, using γ = 1.4, P = atmospheric pressure, and ρ = density of air, the corrected speed is approximately 331 m/s.
Conclusion
Newton’s formula for speed of sound, v = √(E / ρ), shows that sound speed depends on the elasticity and density of the medium. Although Newton assumed isothermal conditions and got a lower value, Laplace corrected the formula by including the factor γ, giving more accurate results. Newton’s work remains an important step in understanding how sound waves travel through different media.