State Newton's formulas for velocity of sound in air. Point out the error and hence discuss Laplace's correction.

Explanation

Newton’s formula for velocity of sound in gases : Newton’s gave an empirical relation to calculate velocity of sound in gas
`v=sqrt((B)/(rho)),B=`Bulk modulus
`rho`=density of gas
He assumed that changes in pressure and volume of a gas when sound waves are propagated through it are isothermal. Using isothermal coefficient of elasticity i.e., `B_(1)`, formula becomes
`v=sqrt((B_(i))/(rho))`
Here `B_(i)=(dP)/(dV//V)`
But there was error in Newton’s formula. He put forward a number of arguments but none of them was satisfactory.
Laplace correction, Laplace pointed out that the pressure variations in the propagation of sound waves are so fast that there is little time for the heat flow to maintain constant temperature. Therefore, these variation are adiabatic not isothermal.
(i) Velocity of sound in a gas is quite large.
(ii) A gas is a bad conductor of heat

Using coefficient of adiabatic elasticity ie., `B_(a)=v=sqrt((B_(a))/(rho))`
Calculation of `B_(a)`, Consider certain mass of gas Let P be initial pressure and V be initial volume of gas
`PV^(gamma)=`constant
where `gamma=C_(P)//C_(V)`=ratio of two principal specific heats of gas.
`PV^(gamma)`=constant
where `gamma=C_(P)//C_(V)=`ratio of two principal specific heats of gas.
Differentiating (i), we get
`P(gammaV^(gamma-1)dV)+V^(gamma)dP=0,gammaPV^(gamma-1)dV=-V^(gamma)dP`
`gammaP=-(V^(gamma))/(V^(gamma-1))((dP)/(dV))=-(dP)/(dV//V)=B_(a), B_(a)=gammaP`
Correct formula is `V=sqrt((gammaP)/(rho))`
The value of v depends on nature of the gas.