Write the equation of speed of sound wave of Newton and explain the correction by Laplace.

Explanation

Speed of sound in ideal gas according to Newton,
`v = sqrt ((p)/(rho)) " "…(1)`
It was pointed out by Laplace 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. These variations therefore are adiabatic and not isothermal.
For adiabatic processes the ideal gas satisfies the relation,
`PV ^(gamma) =` constant
`therefore Delta (PV ^(gamma)) =0`
`therefore P gamma V ^(gamma-1) Delta V + V ^(gamma) Delta P =0`
`therefore gamma Delta VP =-V Delta P `
`therefore gamma P =- (Delta P )/((Delta V )/(Delta V ))`
`therefore gamma P =B " "...(2)`
where `gamma` is the ration of two specific heats,` (C _(P))/(C _(V))` (By using equation (2) in equation (1),
`v = sqrt ((gamma P )/( rho ))" "...(3)`
This modification of Newton.s formula is referred to as the Laplace correction. For air,
`gamma = 7/5 -1.4`
Now using equation (3), estimatte the speed of sound in air at STP, we get a value `331.3 ms ^(-1),` which agrees with the measured speed.