Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air is independent of pressure.

#### Solution 1

Take the relation:

`v = sqrt((gamma P)/rho)` ....(i)

where

Density, `rho = "Mass"/"Volume" = M/V`

M = Molecular weight of the gas

V = Volume of the gas

Hence, equation (i) reduces to

`v = sqrt((gamma "PV")/"M")` .....(ii)

Now from the ideal gas equation for n = 1:

PV = RT

For constant T, PV = Constant

Since both M and γ are constants, v = Constant

Hence, at a constant temperature, the speed of sound in a gaseous medium is independent of the change in the pressure of the gas.

#### Solution 2

We are given that `v = sqrt((gamma p)/rho)`

We know PV = nRT (For n moles of ideal gas)

`=> "PV" = "m"/"M" "RT"`

where m is the total mass and M is the molecular mass of the gas

`:. P = "m"/"M" * "RT"/"M"`

`= (rho"RT")/"M"`

`=> "P"/rho = "RT"/"M"`

For a gas at constant temperature `"P"/rho = "constant"`

∴ As P increase, rho also increase and vice versa. This implies that `v = sqrt((gamma P)/rho)` = constant, i.e velocity is independent of pressure of the gas.