Use the formula v=γPρ to explain why the speed of sound in air is independent of pressure. - Physics

Numerical

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.

Concept: The Speed of a Travelling Wave
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NCERT Class 11 Physics
Chapter 15 Waves
Exercises | Q 4.1 | Page 387