Laplace Correction: Formula, Derivations, Examples

Laplace correction is used to correct the sound speed in a gas. Laplace devised a correction that was both theoretical and practical. As a result, it's commonly referred to as a Laplace correction to Newton's Formula.

What is Laplace Correction?

The propagation of sound waves occurs in adiabatic conditions, according to Laplace. 

Sound Waves Propagation

Since air's thermal conductivity is so low, compression and rarefaction in the air will occur quickly, resulting in heat flowing neither out of nor into the system, i.e. the change in heat applied will be zero, indicating an adiabatic condition. This is known as the Laplace Correction for sound waves in an air or gaseous medium. 

Newton's Formula for Sound Speed

In an elastic medium, the velocity of a longitudinal wave is:

ν = √\(\frac {B} {\rho}\)

The Bulk modulus is B, while the density of the medium is ρ. We may use this relationship to obtain the formula for sound speed through a medium because a sound wave is a longitudinal wave.

The transmission of sound through a medium, according to Newton, is an isothermal process. He reasoned that because the medium through which sound travels is in contact with the environment, any temperature variations will be neutralized by the environment.

Therefore,

ν = √\(\frac {B_iso} {\rho}\)

In the case of an isothermal process,

PV = const

When we separate the two sides of the equation, we get:

PdV + VdP = 0

∴ \(\frac {dP} {dV}\)V = -P

The elasticity of a medium's volume is,

B_iso = -V \(\frac {dP} {dV}\)

As a result, B_iso= P

Thus, according to Newton's formula, the speed of sound is:

ν = √\(\frac {P} {\rho}\)

Sound passes through the air at a certain speed,

The density of air at normal temperature and pressure is ρ=1.293kg/m³

Pressure in the Atmosphere P = 1.013 × 10⁵ Nm^–2

By changing the values,

However, this figure is over 16% lower than the real speed of sound in air, which is 320 ms^–1. This means that certain adjustments had to be made to the formula.

Speed of Sound in Air

The formula to compute the speed of sound using Laplace adjustment to Newton's formula at Normal Temperature and Pressure is:

ν = √\(\frac {\gamma P} {\rho}\)

Where,

The specific heat capacity ratio is γ = 1.4

Atmospheric pressure P = 1.1013 × 105 Nm^–2

Air has a density of ρ = 1.293kg m^–3

If we substitute the values in the formula above, we get:

The speed of sound in the air was almost identical to the speed of sound measured experimentally. As a result, Laplace's correction was incorporated into Newton's formula for calculating sound speed.

Newton's Formula with Laplace Correction 

He revised Newton's formula by assuming that there is no heat exchange because compression and rarefaction occur at such a quick rate. As a result, the temperature does not remain constant, and the transmission of sound waves through the air is an adiabatic process.

As a result, in the case of an adiabatic process,

PVγ = constant

Where γ is the specific heat capacity ratio and is equal to, γ = C_P/C_V

C_p: constant pressure specific heat

C_v: at constant volume, specific heat

By separating the two sides, we are able to-

γPV^γ–1dV + V^γdP = 0

V^γ–1 is used to divide both sides.

dP + γPV^–1dV = 0

Pγ = -V \(\frac {dP} {dV}\) = B

Velocity of Sound

The sound velocity is given by

When we replace B = γP in the above equation, we get:

ν = √\(\frac {\gamma P} {\rho}\)

Factors Affecting Speed of Sound

Any wave's speed is determined by the qualities of the medium through which it travels.

Pressure’s Effect 

The formula for the speed of sound in a gas, as calculated above.

The gas constant is R, the temperature is T, and the mass is mm.

The density of the medium changes as the pressure of the medium changes, therefore the ratio of the two values remains constant. As a result, at a constant temperature, (γP/ρ) remains constant.

As a result, pressure does not affect the speed of sound waves.

Temperature’s Effect 

The velocity of the sound wave was determined using Laplace's correction to the formula:

The square root of a gas's absolute temperature determines the speed of sound in that gas.

As a result, as the temperature of a medium rises, the speed of sound waves across that medium increases.

If υ₀ and υ_t are the sound velocities in the air at 0^oC and t ^oC, respectively, then:

As a result, for every 1^oC increase in temperature, the velocity of sound in air increases by 0.61m/s.

Density's Influence 

The formula for sound in a gaseous medium is as follows:

In a gas, the square root of the density is inversely proportional to the sound velocity.

This indicates that when the medium's density rises, the speed at which sound travels through it decreases.

Humidity's Influence 

For the same pressure, the speed of sound in moist air is larger than the speed of sound in dry air. We know that moist air has a lower density than dry air, so an increase in humidity equals more moist air. This means that when humidity rises, so does the speed of sound in the air.

Things to Remember based on Laplace Correction

• As per Newton, when sound waves propagate in air; compression and rarefaction occurs. And the processes are very slow. 
• According to Laplace, the compression and rarefaction processes take place very fast and no exchange of heat is happening. 
• Velocity of sound waves: Inertia and elasticity are the two properties of matter that determine the velocity of sound. Analytically, velocity of sound has been shown in a medium of elasticity E and density ρ. 
• Low velocity determines a denser medium and vice-versa. Thus, in traveling from air to water, a beam of sound gets bent away from the normal. However, the sound beam of light bends towards the normal.
• According to Newton, when sound propagates through air, temperature remains constant (i.e. the process is isothermal).