Question

Use the formula υ = √ (γ P/p) to explain why the speed of sound in air (a) is independent of pressure, (b) increases with temperature, (c) increases with humidity

Answer 1

The speed of sound in air is given as,

v= γP ρ

where, γis the ratio of the specific heat, Pis the pressure of the air and ρis the density of the air.

The universal gas equation is given as,

PV=nRT PV= m M RT

where, mis the total mass and Mis the molecular mass of gas.

Rewrite the above equation as,

P= m MV RT = ρRT M [ Since m=ρV ] P ρ = RT M

a)

For a gas at constant pressure, the ratio of pressure to density is constant. With the increase in pressure, the density of the gas increases simultaneously. The speed of sound is directly proportional to the root of the ratio of pressure to density.

As the ratio is constant, the speed of sound is independent of the pressure of gas.

b)

The speed of sound is directly proportional to the square root of the ratio of pressure to density. The ratio of pressure to density is directly proportional to the temperature.

The speed of sound is directly proportional to the square root of temperature. Therefore, the speed of sound increases with the increase in the temperature.

c)

The effective density of air decreases with the increase in humidity. The velocity of air is inversely proportional to the square root of density. Therefore, with the increase in humidity, the speed of sound also increases.