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

Explanation

Newton assumed that when sound propogates in air, the formation of compression and rarefaction takes place in very slow manner so that the process is isothermal in nature. That is, the heat produced during compression (pressure increases, volume (decreases), and heat lost during rarefaction (pressure decreases, volume increases) occur over a period of time such that the temperature of the medium remains constant. Therefore, by treating the air molecules to form an ideal gas, the changes in pressure and volume obey Boyle's law, Mathematically
PV= Constant ................(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.
Substituting equation (2) in equation `v= sqrt(B/p)`, the speed of sound in air is
`v_(T) = sqrt(B_(T)/rho) = sqrt(P/rho)`............(3)
Since P is the pressure of air whose value at NTP (Normal Temperature and Pressure) is 76 cm of mrercury, we have
`P=(0.76 xx 13.6 xx 10^(3) xx 9.8) Nm^(-2)`
`rho = 1.293 kg m^(-3)`, here `rho` is density of air
Then the speed of sound in air at Normal Temperature and Pressure (NTP) is
`v_(T) = sqrt((0.76 xx 13.6 xx 10^(3) xx 9.8)/(1.293))`
`=279.80 ms^(-1) ~~ 280 ms^(-1)` (theoretical value)
But the speed of sound in air at `0^(@)` C is experimentally observed as `332 ms^(-1)` which is close upto `16%` more than theoretical value (Percentage error is `(332- 280)/332 xx 100% = 15.6%)`
This error is not small.
Laplace assumed that when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefraction occur very fast. Hence the exchange of heat produced due to compression and colling effect due to rarefraction do not take place, because, air( medium) is a bad conductor of heat. Since, temperature is no longer considered as a constant here, sound propogation is an adiabatic process. By adiabatic consideratios, the gas obey's Poisson law (not Boyle's law as Newton assumed), which is
`PV^(Y)` = constant ............(4)
where, `lambda =C_(p)/C_(v)`, which is the ratio between specific heat at constant pressure and specific heat at constant volume.
Differentiating equation (4) on both the sides, we get
`V^(t)dP + P(lambdaV^(t-1) dV)=0`
or `lambda P=-V(dP)/(dV) B_(A)`............(5)
where, `B_(A)` is the adiabatic bulk modulus of air. Now, substituting equation (5) in equation
`v= sqrt(B/rho)`, the speed of sound in air is
`v_(A) = sqrt(B_(A)/rho) = sqrt((lambda P)/rho) = sqrt(lambda v_(T))`...........(6)
`V_(A) = 331 ms^(-1)`.