Laplace's Correction to Newton's Formula

Assuming isothermal conditions to prevail when sound travels through air, Newton has applied Boyle's law to the changes in pressure and volume. In a region of compression, there is a slight increase in temperature and in a region of rarefaction, there is a slight decrease in temperature. These changes in pressure occur rapidly and air is a poor conductor of heat thus, equalization of temperature among the different regions was improbable, according to Laplace.. He was of the view that the changes in temperature occur under adiabatic conditions, i.e., no heat enters the gas from outside or leaves it from inside. The heat developed in the compressed layers remains fully confined to those layers and has no time to get dissipated into the entire body of the gas. Similarly, the cold caused in the rarefied layers cannot be compensated for, by flow of heat into it from other layers.
Thus Boyle's law does not apply in this case.
           See Newton's Formula For Velocity Of Sound 
The relation between pressure and volume of a gas under adiabatic conditions is given by

where   i.e., the ratio of the principle specific heats of the gas at constant pressure and constant volume respectively.
Let the pressure change by an amount dP, producing a change in volume by dV. Then

Taking out from the 2^nd factor in the above expression,

But from binomial approximation,




Canceling P on both sides and neglecting the term containing P.V because it is too small, we get



But the LHS in the above equation represents the bulk modulus, B.

Substituting in equation (1-23), we get

This is known as Newton-Laplace formula for the velocity of sound in a gas.



This is in close agreement with the experimental value.

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