Use the formula v = √γP/ρ to explain why the speed of sound in air

(c) increases with humidity.

Solution

Let v_m and v_d be the speed of sound in moist air and dry air respectively.

Let ρ_m and ρ_d be the densities of the moist air and dry air respectively.

Using space the space relation comma space straight nu space equals space square root of γP over straight rho end root space Speed space of space sound space in space moist space air space is comma space straight v subscript straight m space equals space square root of γρ over straight rho subscript straight m end root space space space space space space space space space space space space space... space left parenthesis straight i right parenthesis thin space Speed space of space sound space in space dry space air space is comma straight v subscript straight d space equals space square root of γρ over straight rho subscript straight d end root space space space space space space space space space space space space space... space left parenthesis ii right parenthesis On space dividing space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have space straight v subscript straight m over straight v subscript straight d space equals space square root of γρ over straight rho subscript straight m straight x straight rho subscript straight d over γρ end root space space equals space straight rho subscript straight d over straight rho subscript straight m

However, the presence of water vapour reduces the density of air.
i.e., ρ_d <ρ_m

∴ v_{m >}v_d

Hence, the speed of sound in moist air is greater than it is in dry air.
Thus, in gaseous medium, the speed of sound increases with humidity.

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