As a relativistic prediction, it is widely accepted that the frequency of a light wave receding from the surface of a massive plant is reduced due to its motion upwards against the gravitational field. I wonder if the same happens for a sound wave moving upwards against the gravitational field.
We know that the air density, as well as its temperature, reduces in high altitudes of the atmosphere. Moreover, the speed of sound in the air can be calculated as follows:
$\gamma$: A constant depending on the kind of gas. For air $\gamma=1.4$.
$R=8.314$ j/mol.K: The universal gas constant.
$M$: The molecular mass. For air $M=0.029$ Kg/mol.
$T$: The temperature in Kelvins.
Can anybody calculate the change in a sound wave frequency as it is emitted on the earth (sea level) and received at an altitude $h$? Moreover, please explain what happens to the energy of this sound wave, i.e., is there any relation between the energy and frequency of a sound wave similar to $E=nh\nu$ we know for the light waves?
P.S. I know that there is no sound in outer space, then consider the inner one, please!