Laplace Correction: Concept, Formula & Applications

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. The sound speed of a gas is corrected using Laplace Correction. Laplace devised a theoretical as well as a practical adjustment. As a result, Newton's Formula is usually referred to as a Laplace adjustment.

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=Pρ−−√≈280m/sv=Pρ≈280m/s

Where,

P - Pressure in the medium

ρ - 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.

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:

⇒PVγ = Constant………(1)Where,

P

P - Pressure in the medium

V - Volume of the gas

γ - Adiabatic index

Now, differentiate equation (1) on both sides,

⇒γPVγ−1dV+VγdP=0

⇒γPVγ−1dV+VγdP=0⇒Vγ−1

[γPdV+VdP]=0⇒Vγ−1[γPdV+VdP]=0⇒γPdv+Vdp=0

⇒γPdv+Vdp=0 On rearranging the above equation,

⇒γPdV=−VdP⇒γPdV=−VdP⇒γP=−PDV⇒γP=−VdPdV……..(2)−VdPdV

−VdPdV is−√v=γPρ…….(5) ium

ρ -  Medium density

γ - 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:

\[\Rightarrow v=\sqrt{\frac{\gamma p}{\rho }}\]

Where,

P - Pressure in the medium=1.101×105N/m2

ρ =  Medium density=1.293Kg/m3

ρ - Adiabatic index=1.4

Substituting the corresponding values in the equation,

\[\Rightarrow v=\sqrt{\frac{1.4\times 1.101\times10^{5} }{1.293}}=345m/s\approx 332m/s\]

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

\[\Rightarrow v=\sqrt{\frac{p}{\rho }}\]

where,

P - Pressure in the medium=1.101×105N/m2

ρ - Medium density =1.293kg/m3

Substituting the values in the equation,

\[\Rightarrow v=\sqrt{\frac{1.101\times10^{5} }{1.293}}=290m/s\]

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.

Things to Keep in Mind about Laplace Correction

• When sound waves travel through the air, compression and rarefaction occur, according to Newton. Furthermore, the processes are quite slow.

• According to Laplace, the compression and rarefaction processes occur relatively quickly with little heat exchange.

• Sound wave velocities: The two qualities of matter that determine the velocity of sound are inertia and elasticity. The velocity of sound has been calculated in a medium with elasticity E and density.

• A denser medium is determined by a lower velocity, and vice versa. As a result, when sound travels from air to water, it bends away from its regular path. The sound beam of light, on the other hand, bends towards the normal.

• When sound travels through air, according to Newton, the temperature remains constant (i.e. the process is isothermal).

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