Laplace Correction

What is Laplace Correction?

Laplace Correction adjusts Newton’s formula for calculating the speed of sound in air by considering the adiabatic process of compression and rarefaction in the propagation of sound waves, rather than an isothermal process.

What is the actual speed of the sound?

The speed of sound depends on the medium in which it moves. Elasticity and density are the two properties of the medium that have an impact on speed. The speed of sound in air is approximately 343 meters per second (1,235 km/h) at 20 degrees Celsius.

Why is Laplace Correction important?

It provides a more accurate calculation of the speed of sound in gases because it accounts for the fact that heat exchange during the propagation of sound waves is minimal and rapid, affecting the sound speed.

How does Laplace Correction affect the speed of sound?

Laplace Correction leads to a higher calculated speed of sound compared to Newton’s formula, which underestimates it by assuming constant temperature (isothermal conditions) during sound wave propagation.

Does Mach equal sound speed?

The term “Mach Speed” describes how quickly something is moving in relation to the speed of sound. This is equivalent to 1,235 kph, 768 mph, or 343 m/s at 68 °F NTP. When a plane exceeds the speed of sound and creates a sonic boom, it is said to be travelling at Mach 1. The speed at which an aeroplane exceeds the speed of sound is referred to as Mach 2.

Explain the Laplace Correction. What Justifies Laplace Correction?

An adjustment of the soundwave speed of the gas or air medium to acquire a precise value. The Laplace correction for sound waves refers to the modification Laplace made to Newton’s formula for sound waves by assuming that the compressions and rarefactions in the air are adiabatic processes.

What human-piloted jet is the fastest in the world?

Currently, the North American X-15 is the fastest human-piloted aircraft. Pilot William J. Johnson powered it to a top speed of Mach 6.70 (about 7,200 km/h) on October 3, 1967.

Use Newton’s Formula and Laplace Correction to Calculate the Speed of Sound at Standard Pressure and Temperature. Review the values.

• The Laplace Correction formula results in the following:

[Tex]ν=\sqrt \frac{\gamma P}{\rho} [/Tex]

Where,

γ = Adiabatic index = 1.4 ,

P = Atmospheric pressure = 1.013×10⁵ N/m²,

ρ = Density of Air = 1.293 kg/m³.

Substitute the value of γ, P and ρ in Laplace’s correction formula,

[Tex]ν=\sqrt \frac{1.4×1.013×10^5}{1.293} [/Tex]

∴ v = 332 m/s

• Newton’s formula provides the following results for sound velocity:

[Tex]ν=\sqrt \frac{P}{\rho}  [/Tex]

Where,

P = Atmospheric pressure = 1.013×10⁵ N/m²,

ρ = Density of Air = 1.293 kg/m³.

Substitute value of P and ρ in equation, we get

[Tex]ν=\sqrt \frac{1.013×10^5}{1.293} [/Tex]

∴ v = 280 m/s

By comparing the sound speed values derived using the Laplace Correction formula and the Newton’s formula, it is evident that the Laplace Correction formula’s value is in much better agreement with the sound speed in air than the Newton’s formula’s value is. Thus, the Laplace adjustment is also known as the Newton’s formula correction.

Consider a closed box of rigid walls so that the density of the air inside it is constant. On heating, the pressure of this enclosed air is increased from P₀ to P. It is now observed that sound travels 1.5 times faster than at pressure P₀ calculate P/P₀.

We have,

[Tex]ν_p=\sqrt \frac{\gamma P_0}{\rho} [/Tex]

[Tex]ν_{p0}=\sqrt \frac{\gamma P_0}{\rho} [/Tex]

ν_p = 1.5 ν_p0

[Tex]\sqrt \frac{\gamma P}{\rho}=1.5 \sqrt \frac{\gamma P_0}{\rho} [/Tex]

∴ P/ρ = 2.25 × P₀/ρ

∴ P = 2.25P₀

What are the factors affecting the speed of sound?

As sound waves travel through atmosphere, some factors related to air affect the speed of sound:

1. Effect of pressure on velocity of sound
2. Effect of temperature on speed of sound
3. Effect of humidity on speed of sound

What is the significance of the adiabatic index in Laplace Correction?

The adiabatic index γ represents the ratio of specific heats at constant pressure and volume. It is crucial in the Laplace formula as it influences how the speed of sound is calculated under adiabatic conditions, impacting the compression and expansion of the gas.

How does Laplace Correction compare to Newton’s formula?

Newton’s formula assumes isothermal conditions and calculates a lower speed of sound. Laplace’s correction, which assumes adiabatic conditions (no heat transfer), aligns better with experimental values by accounting for rapid thermal changes during sound wave propagation.

Laplace Correction is used to modify the speed of sound in the gas. Assuming that sound waves propagate in an isothermal state in air or gas, Newton calculated the formula for the speed of sound in a gaseous medium. It was found that the speed of sound in the air was just 280 m/s, disproving this presumption. Laplace came up with a theoretically and practically obvious correction as a result. It is therefore well recognized as a Laplace Modification to Newton’s formula. Let’s examine the Laplace correction formula’s concept.