Velocity of sound in gases | Laplace’s correction

Newton’s formula for the velocity of sound in gases

Newton was of the opinion that longitudinal waves travel in a gaseous medium, the changes taking place in the medium are isothermal in nature. Thus, according to Newton, the temperature of the gaseous medium remains constant, when sound travels through it. At the regions of compression, where the heat is produced, the heat is conducted away to the surrounding medium and at the regions of rarefaction, where cooling is produced, the heat is conducted in form the surrounding medium.

Velocity of sound in gases

Thus, in the equation
V= √(B/ρ) ….(1)
B represents isothermal bulk modulus of the gas whose value is equal to the initial pressure (p) of the gas. Therefore, according to Newton, the speed of sound in gasses is given by:
V= √(P/ρ) ….(2)
Let us use the equation (2) to estimate the value of the speed of sound in air at N.T.P. We know that for air at N.T.P.
P= 76 cm of mercury column = 76×13..6×980 dynes/cm² = 1.01 × 10⁵ N/m²
And ρ = density of air = 1.293 × 10^-3 g cm^-3 = 1.293 Kg m^-3
Therefore Speed of sound in air at N.T.P. v = √[(1.01 × 105)/1.293]=278 m/s
However, from experiments, the velocity of sound in air at N.T.P. is estimated to be 332 ms^-1, which is quite different from the value obtained by Newton’s formula. Hence Newton’s formula for the velocity of sound in gas is not accepted.

Laplace’s correction

(1)Laplace, in 1816, discovered the error in Newton’s formula and modified it satisfactorily. Laplace pointed out that it was wrong to assume that changes taking place in a gaseous medium are isothermal in nature, when sound waves travel through it.
(2)Actually, when sound waves travel in a gaseous medium then at any point in the medium the states of compression and rarefaction occur alternately. At the moment of compression some heat is produced at the point, while at the moment of rarefaction some cooling is produced at the point. The compression and rarefaction occur so quickly that heat produced during compression cannot go out into the surroundings, and the heat disappeared during rarefaction cannot come in from the surroundings. Moreover, the exchange of heat does not occur because gases are bad conductors of heat. Therefore, the temperature at any point in the medium rises at the moment of compression and falls during rarefactions. Thus, during the propagation of sound waves in gases, temperature changes take place continuously i.e., process of propagation of sound waves in gases is adiabatic in nature and B in Newton’s formula should actually represent the adiabatic bulk modulus of the gas whose value is equal to ϒ times the initial pressure of the gas where:
ϒ = Cp/Cv=specific heat of gas at constant pressure/ specific heat of gas at constant volume

(3)Thus, according to Laplace, the formula for the speed of sound in gases is given by:
v =√( ϒ P/ρ)

For air, the value of ϒ is 1.41. Hence according to Laplace’s formula, the speed of sound in air at N.T.P. will be:
v = √[(1.01 × 1.01× 10⁵)/1.293]=332.5 m/s
Therefore, we find that theoretical value for the speed of sound in air at N.T.P. is now in close agreement with the experimental value.
Hence Laplace’s formula for the speed of sound in gases is correct.