Question

Recall that the definition of the speed of sound is c^2=(dpressure)/(ddensity). Show that c =sqrt(kRT) for an ideal gas. (Hint: assume isentropic relations: p/p^k=constant).

Answer 1

Speed of sound² = change in pressure / change in density

c = (dP / dρ)^1/2  ....... Eq1

P/ρ^K = constant

We can write

ln P - k ln ρ = constant

Now Differentiate the above equation with respect to ρ

(1/P) dP/dρ - (k/ρ) = 0

dP/dρ = (kP/ρ) ...... Eq2

From eq1 and eq2

c = (kP/ρ)^1/2

For ideal gas equation

P/ρ = RT

c = (kRT)^1/2

For an ideal gas =