Laplace Correction

Read

Discuss

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.

Laplace’s Correction

Since air has a very low thermal conductivity, it will quickly compress and rarefy, preventing heat from leaving or entering the system. As a result, there will be no change in the amount of heat applied, which denotes an adiabatic state. This is due to the Laplace Correction for sound waves in an air or gaseous medium.

Laplace demonstrated that adiabatic conditions are required for the propagation of sound waves. The fact that compression and rarefaction in the air will happen quickly indicates that an adiabatic situation exists when the change in applied heat is zero. This is because the thermal conductivity of air is so low. Heat won’t move into or out of the system as a result. This is referred to as Laplace correction for sound waves in an atmosphere or a gaseous medium. The Laplace Correction is used to modify the gas’s sound speed. Laplace created a theoretical change as well as one that is application-specific. As a result, Newton’s Formula is sometimes known as a Laplace Adjustment.

Laplace’s Correction Formula

The following formula gives the sound’s velocity:

$ν=\sqrt \frac{\gamma P}{\rho}$

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,

$ν=\sqrt \frac{1.4×1.013×10^5}{1.293}$

∴ v = 332 m/s

Newton’s Equation for the Speed of Sound

Newton investigated the propagation of sound waves in the atmosphere. He believed that the process of propagation was isothermal. Both the absorption and the emission of heat will occur during compression and rarefaction. The temperature stays the same as a result.

According to Boyle’s law,

PV = Constant

where,

P = Pressure,

V = Volume of gas.

With the above equation differentiated, we obtain-

PdV + VdP = 0

∴ PdV = – VdP

∴ P = – VdP / dV

∴ $P=\frac{dP}{-(\frac{dV}{V})}$

∴ P = B

So,

Bulk modulus of air

$B=\frac{dP}{-(\frac{dV}{V})}$

The sound wave’s velocity can be written as-

$ν=\sqrt \frac{B}{\rho}$

$ν=\sqrt \frac{P}{\rho}$ …(Equation 1)

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 1, we get

$ν=\sqrt \frac{1.013×10^5}{1.293}$

∴ v = 280 m/s

The value acquired here and the value from the experiment do not match. That is 332 m/s. It implies that Newton’s equation needs to be modified.

Derivation of Laplace Correction for Newton’s Formula

By assuming that no heat exchange takes place since compression and rarefaction happen so quickly, he modified Newton’s formula. The varying temperature causes the sound wave to move through the air in an adiabatic manner.

For an adiabatic process

PV^γ = Constant

Where,

γ = Adiabatic index

$\gamma=\frac{C_p}{C_v}$

Where,

C_p = With a constant pressure, specific heat

C_v = With a constant volume, specific heat

When we differentiate the both sides, we find-

V^γdP + PγV^γ-1dV = 0

Dividing both the sides by V^γ-1

dP + PγV^-1dV = 0

Pγ = – (VdP / dV)

$P\gamma=\frac{dP}{-(\frac{dV}{V})}$

So, Pγ = B

$ν=\sqrt \frac{\gamma P}{\rho}$

Factors affecting the speed of sound

Some air-related factors have an impact on the speed of sound as it travels through the atmosphere.

Effect of pressure on the velocity of sound
Effect of temperature on the speed of sound
Effect of humidity on the speed of sound

Effect of pressure on the velocity of sound

The equation for the sound speed in a gas, as determined above:

$ν=\sqrt \frac{\gamma P}{\rho}$

$ν=\sqrt \frac{\gamma RT}{m}$

Where,

R = Gas constant,

T = Temperature,

m = Mass.

Since both the medium’s pressure and density change over time, the ratio between the two quantities stays constant. As a result, at a certain temperature, (γP/ρ) stays the same.

As a result, pressure has no impact on the speed of sound waves.

Effect of temperature on the speed of sound

The velocity of the sound wave was calculated by applying Laplace’s adjustment to the formula:

From the Effect of pressure on the velocity of sound,

$ν=\sqrt \frac{\gamma RT}{m}$

ν α √T

The square root of the absolute temperature of gas determines its sound speed.

$\frac{ν_1}{ν_2}=\sqrt \frac{T_1}{T_2}$

As a result, as a medium’s temperature increases, sound waves move through it at varying rates.

If ν₀ and ν_t are the air’s sound velocities at 0° C and t° C, respectively:

$ν_t=ν_0(1+\frac{t}{273})^\frac{1}{2}$

$ν_t=ν_0+0.61t$

As a result, for every degree Celsius that the temperature rises, the speed of sound in the air increases by 0.61 m/s.

Effect of humidity on the speed of sound

The humidity of the air is based on the amount of water vapor present.

Let ρ_m and ρ_d be the densities of moist and dry air respectively. If ν_m and ν_d are the speeds of sound in moist air and dry air.

$ν_m=\sqrt \frac{\gamma P}{\rho_m}$

and

$ν_d=\sqrt \frac{\gamma P}{\rho_d}$

So,

$\frac{ν_m}{ν_d}=\sqrt\frac{\rho_d}{\rho_m}$

Moist air is always less dense than dry air.

Applications

Any issue that directly relates to a linear differential equation can be resolved using the Laplace equation. Differential equations were most frequently used in physics, mathematics, and engineering.
The Laplace equations describe steady-state conduction heat transmission in the absence of any heat sources or sinks.

FAQs on Laplace Correction

Question 1: What is the actual speed of the sound?

Answer:

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.

Question 2: Does Mach equal sound speed?

Answer:

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.

Question 3: Explain the Laplace Correction. What Justifies Laplace Correction?

Answer:

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.

Question 4: What human-piloted jet is the fastest in the world?

Answer:

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.

Question 5: Use Newton’s Formula and Laplace Correction to Calculate the Speed of Sound at Standard Pressure and Temperature. Review the values.

Answer:

The Laplace Correction formula results in the following:

$ν=\sqrt \frac{\gamma P}{\rho}$

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,

$ν=\sqrt \frac{1.4×1.013×10^5}{1.293}$

∴ v = 332 m/s

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

$ν=\sqrt \frac{P}{\rho}$

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

$ν=\sqrt \frac{1.013×10^5}{1.293}$

∴ 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.

Question 6: 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.

Answer:

We have,

$ν_p=\sqrt \frac{\gamma P_0}{\rho}$

$ν_{p0}=\sqrt \frac{\gamma P_0}{\rho}$

ν_p = 1.5 ν_p0

$\sqrt \frac{\gamma P}{\rho}=1.5 \sqrt \frac{\gamma P_0}{\rho}$

∴ P/ρ = 2.25 × P₀/ρ

∴ P = 2.25P₀

Question 7: What are the factors affecting the speed of sound?

Answer:

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

Effect of pressure on velocity of sound

Effect of temperature on speed of sound

Effect of humidity on speed of sound

My Personal Notes

Last Updated : 22 Aug, 2022

Related Articles

Laplace Correction

Laplace’s Correction

Laplace’s Correction Formula

Newton’s Equation for the Speed of Sound

Derivation of Laplace Correction for Newton’s Formula

Factors affecting the speed of sound

Effect of pressure on the velocity of sound

Effect of temperature on the speed of sound

Effect of humidity on the speed of sound

Applications

FAQs on Laplace Correction

CBSE Class 12 Computer Science

Complete Machine Learning & Data Science Program

Python Programming Foundation -Self Paced

Related Articles

Laplace Correction

Laplace’s Correction

Laplace’s Correction Formula

Newton’s Equation for the Speed of Sound

Derivation of Laplace Correction for Newton’s Formula

Factors affecting the speed of sound

Effect of pressure on the velocity of sound

Effect of temperature on the speed of sound

Effect of humidity on the speed of sound

Applications

FAQs on Laplace Correction

Please Login to comment...

CBSE Class 12 Computer Science

Complete Machine Learning & Data Science Program

Python Programming Foundation -Self Paced