Describe Newton's formula for velocity of sound waves in air.

Explanation

Newton's assumed that when sound propagates in ari, the formation of compression and rarefaction takes place in a very slow manner so that the process is isothermal in nature. It is found that, the heat produced during compression (pressure increases, volume decreases), and heat lost during rarefaction (Pressure decreases, volume increases) occure over a period of time such in a wavy that the temperature of the medium remains constant. Hene, by treating the air molecules to form an ideal gas, the changes in pressure and volume obey Boyle's law, Mathematically
`PV="chonstant" " "..(1)`
Differentiating equation (1) we get
`PdV+VdP=0`
(or) `P-V(dP)/(dV)=B_(T)" "...(2)`
Where `B_(T)` is an isothermal bulk modulus of air.
`v= sqrt((B)/(rho))" "...(3)`
Substituting equatio (2) in equation (3), the speed of sound in air is
`v_(T)= sqrt((B_(T))/(rho))= sqrt((P)/(rho))`
Since P is the pressure of air whose value at NTP (Norma Temperature and Pressure) is 76 cm of mercury, we have
`P=(0.76xx13.6xx10^(3)xx9.8)Nm^(-2)`
`rho=1.239 kg m^(-3)`. Here `rho` is density of ari. Then the speed of sound in air at normal temperature and pressure (NTP) is
`v_(T) = sqrt(((0.76xx13.6xx10^(3)xx9.8))/(1.293))`
`=270.80 ms^(-1)`
`=280 ms^(-1)` (theoretical value) But the speed of sound in air at `0^(@)C` is experimentally observed as `332 ms^(-1)` that is close upto 16 % more than theoretical value.