Laplace Correction

Last Updated : 15 Apr, 2024

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.

Table of Content

Laplace’s Correction
Derivation of Laplace Correction for Newton’s Formula
Factors affecting the speed of sound
Applications of Laplace Correction

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

$ν=\sqrt \frac{\gamma P}{\rho}$

Check: Laplace Transform in Circuit Analysis

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.

Check: Inverse Laplace Transform

Applications of Laplace Correction

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.