Use the formula `v= sqrt((gamma p)/(rho))` to explain why the speed of sound in air
(a) is independent of pressure.
(b) increases with temperature.
(c ) increases with humidity

Explanation

(a) Given`:v = sqrt ((lamda P)/(rho))`
(where `lamda =` specific heat ratio `= (C _(P))/(C _(V))`
P = pressure of air
`rho=` density of air)
Taking air as an ideal gas, we have its equation of state,
`PV = mu RT= (M)/(M _(0)) RT`
(Where `mu=` no. of moles of molecules
M = mass of air,
`M _(0)=` molecular weight of air)
`therefore P ((V)/(M)) = (RT)/(M _(0))`
`therefore (P)/(rho) = (RT)/(M _(0))`
= constant at given temperature ...(1)
`therefore v = sqrt ((gamma P)/(rho))=` constant
`(because (P)/(rho)=` constnat and `gamma` is already constnat )
At constant temperature, changes in pressure of air will not effect velocity of sound in air. It means that at constnat temperature, velocity of sound in air does not depend on its pressure.
(b) We have,`v = sqrt ((gammaP)/(rho)) = sqrt ((gamma RT)/(M _(0)))`
(From equation (1))
`implies v prop sqrtT (because gamma, R, M _(0)` are constants)
`implies As` temperature increases, velocity of sound in air increases.
(c ) We have, `v = sqrt ((gamma P)/(rho)) implies v prop (1)/(sqrtrho)`
As amount of moisture, of humidity (amount of water vapour) increases, its density decreases and so velocity of sound in moist air increases as per above relation.