Question

The fundamental expression for the speed of sound in a gas is not given by which of the following equations?

(a) a^2=\(\frac {dp}{d\rho }\)

(b) a=\(\sqrt {(\frac {\partial p}{\partial \rho })s}\)

(c) a=\(\sqrt {\frac {\gamma p}{\rho }}\)

(d) a=\(\sqrt {\gamma ^2RT}\)

I had been asked this question in examination.

This intriguing question comes from The Basic Normal Shock Equations topic in section Normal Shock Waves of Aerodynamics

Answer 1

Right option is (d) a=\(\sqrt {\gamma ^2RT}\)

Easiest explanation: The flow through the sound wave is isentropic, giving a=-ρ\(\frac {da}{d\rho }\). Also, a^2=\(\frac {dp}{d\rho }\) and since the change is isentropic, subscript sis used i.e. a=\(\sqrt {(\frac {\partial p}{\partial \rho })_s}\). Moreover, using the isentropic relations for pressure and density we get a=\(\sqrt {\frac {\gamma p}{\rho }}\) and by using the equation of state, we get the temperature relation with the speed of sound in medium i.e. a=\(\sqrt {\gamma RT}\).