Laplace Correction

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Laplace correction gives correction to the speed of sound in the gas. The formula for the speed of sound in the gaseous medium was estimated by Newton, he assumed that the propagation of sound waves in air or gas is under isothermal condition. This assumption was found contradictory, the speed of sound in the air was found to be 280m/s which wasn’t accurate. Thus, Laplace came up with a correction to it which was evident theoretically as well as practically. Thus it is well known as a Laplace correction to Newton’s Formula. 


What Is Laplace Correction?

Laplace corrected that- the propagation of sound waves takes place under adiabatic conditions. The thermal conductivity of air is so less than the movement of compression and rarefaction in the air will be rapid, resulting in that the heat flows neither out of the system nor into the system, i.e., the change in heat applied will be zero, which is clearly an adiabatic condition. This is known as Laplace correction for sound waves in air or gaseous medium.


Laplace Correction for Newton’s Formula:

According to the theory of Newton, the compression and rarefaction in the air is an isothermal process and the speed of sound is given by:

$v=\sqrt{\dfrac{P}{\rho}} \approx 280m/s$

Where,

$P$ - Pressure in the medium

$\rho$ - Medium density


We know that the speed of sound in the air medium is around $332m/s$. The value obtained by Newton’s formula was $280m/s$ which is not accurate. To overcome this difficulty, Laplace conducted a few experiments and came up with a correction and hence derived Laplace Correction Formula.


Laplace carried out the same approach of Newton with a modification that the compression and rarefaction in the air is an adiabatic process as the thermal conductivity in the air will be very less and rapid. He assumed that the compression and rarefaction are completely insulated adiabatic processes, that the change in heat applied will be zero or constant.


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Laplace Correction Derivation:

Since we are assuming that the compression and rarefaction are adiabatic processes, then the gas equation for the adiabatic process is given by:

$\Rightarrow PV^{\gamma}$ = Constant ………(1)

Where,

$P$ - Pressure in the medium

$V$ - Volume of the gas

$\gamma$ - Adiabatic index

Now, differentiate equation (1) on both sides,

$\Rightarrow \gamma PV^{\gamma - 1} dV + V^{\gamma} dP = 0$

$\Rightarrow V^{\gamma-1} \left[\gamma P dV + V dP\right] = 0$

$\Rightarrow \gamma P dv + V dp = 0$

On rearranging the above equation,

$\Rightarrow \gamma P dV = -V dP$

$\Rightarrow \gamma P = \dfrac{-V dP}{dV}$ ……..(2)

$\dfrac{-V dP}{dV}$ is the ratio of volume stress to strain which is the definition of the Bulk modulus. Thus, substituting B in equation (2) we get:

$\Rightarrow \gamma P = B$ …….(3)


Now we know that the speed of sound is given by:

$v = \sqrt{\dfrac{B}{\rho}}$ …….(4)

Where,

$B$ - The bulk modulus

$\rho$ - The medium density

Substituting the value of bulk modulus from equation (3),

$v = \sqrt{\dfrac{\gamma P}{\rho}}$ …….(5)

Where,

$P$ - Pressure in the medium

$\rho$ -  Medium density

$\gamma$ - Adiabatic index

Equation (5) is known as the Laplace correction for speed of sound or the Laplace formula for the speed of sound. The Laplace correction in physics is in good agreement with the value of the speed of sound in the air at standard pressure and temperature.


Example:

1. Calculate the Speed of Sound Using Laplace Correction and Newton’s Formula at Standard Pressure and Temperature. Compare the Values.

Ans: 

i) The velocity of sound by Laplace correction formula is given by:

$\Rightarrow v = \sqrt{\dfrac{\gamma P}{\rho}}$

Where,

$P$ - Pressure in the medium $=1.101 \times 10^5 N/m^2$

$\rho$ =  Medium density $=1.293Kg/m^3$

$\rho$ - Adiabatic index $=1.4$

Substituting the corresponding values in the equation,

$\Rightarrow v = \sqrt{\dfrac{1.4 \times 1.101 \times 10^5}{1.293}} = 345m/s \approx 332 m/s$

ii)  The velocity of sound by Newton’s formula is given by:

$\Rightarrow v = \sqrt{\dfrac{P}{\rho}}$

Where,

$P$ - Pressure in the medium $=1.101 \times 10^5 N/m^2$

$\rho$ - Medium density = $1.293 kg/m^3$

Substituting the values in the equation,

$\Rightarrow v = \sqrt{\dfrac{1.101 \times 10^5}{1.293}} = 290 m/s$

By comparing the values of speed of sound by Laplace correction and Newton’s formula, it is evident that the value obtained with the Laplace correction formula is in good agreement with the value of sound in air in comparison to the value obtained by Newton’s formula. Hence it is known as Laplace correction to Newton’s formula.

FAQ (Frequently Asked Questions)

1. Define Laplace Correction. What is the Need for Laplace Correction?

Ans: A correction to the speed of soundwaves in the air or gaseous medium in order to get an accurate value. Laplace made modifications to Newton’s formula for sound waves by assuming that the compressions and rarefactions in the air are adiabatic processes, this correction is known as Laplace correction for sound waves.


2. What is the Laplace Corrected Formula? And How is it Different From Newton’s Formula?

Ans: Laplace correction formula for sound waves is given by,


⇒ v = √𝛾P/⍴ 

Where,

P - Pressure in the medium

⍴ - Medium density

𝛾 - Adiabatic index


Newton’s formula for sound waves,

 v = √P/⍴ 

Where,

P - Pressure in the medium

⍴ - Medium density

3. What Do You Mean by Compression and Rarefaction?

Ans:

Compression:  The region of the wave where the particles will be closest to each other.

Rarefaction: The region of the wave where the particles will be far away from each other.