Question

How is local velocity of sound related to the Riemann invariants?

(a) a = \(\frac {γ – 1}{4}\)(J+ + J–)

(b) u = \(\frac {γ – 1}{4}\)(J+ J–)

(c) a = \(\frac {γ – 1}{4}\)(J+ – J–)

(d) a = γ – 1(J+ – J–)

I had been asked this question by my school principal while I was bunking the class.

My question is from Finite Waves in portion Unsteady Wave Motion of Aerodynamics

Answer 1

Right choice is (c) a = \(\frac {γ – 1}{4}\)(J+ – J–)

For explanation: The two Riemann equations along characteristic lines are given by:

J+ = u + \(\frac {2a}{γ – 1}\) = constant

J– = u – \(\frac {2a}{γ – 1}\) = constant

Where J+ is along C+ characteristic line and J– is along C– characteristic line. When we solve these two equations, we obtain the local velocity of sound which is given by:

a = \(\frac {γ – 1}{4}\)(J+ – J–)