Does gravity slow down sound (does sound travel faster downwards than upwards)?

Question

I have read this question:

So assuming the heavier planet has the same atmosphere as Earth, and that we can treat the atmosphere as an ideal gas, the speed of sound on the heavier planet would be the same as the speed of sound on Earth if we measure at the same temperature. That means sounds will be the same on the heavy planet and on Earth.

Sound and gravity

Now this is not what I am asking and none of the current topics on this site talk about the speed of sound upwards (climbing out of the gravitational field) versus the speed of sound downwards (traveling into the gravitational field deeper).

As far as I understand, everything, that does have rest mass, is affected by gravity so that whenever something massive tries to climb out of the gravitational field, it loses speed. The only exception is massless particles like photons, they can only lose energy.

Now sound is a phenomenon, but it is based on a medium (air here on Earth), and this medium happens to constitute of massive particles, atoms, made up of electrons and quarks, both massive.

Now as the soundwave travels outwards, trying to climb out of the gravitational field, it should theoretically lose speed. That being said, sound itself is not (made of) the massive particles of the medium, but only their relative motion (waving). So to me it is not naively simple to determine whether soundwaves would slow down upwards (relative to when they move downwards) here on Earth for example.

Question:

1. Does gravity slow down sound (does sound travel faster downwards than upwards)?

Answer 1

I think one thing to consider here is that the speed of sound varies as the inverse square root of density. Since a gaseous planet is generally more dense the further down (closer to the center) you go due to the pressure from above, one would naively assume the speed of sound actually decreases for a downward traveling wave, and increases for upward travelers, assuming a uniform chemical composition. Interesting question.

It is unclear to me how the bulk modulus of elasticity, the other factor affecting the speed of sound, would vary in this atmosphere. I would think that as the gas is compressed more and more as we look closer to the core, the bulk modulus would increase, meaning an applied pressure would result in less change in volume closer to the planets center (think degeneracy pressure). Assuming bulk modulus increases as does density, one would have to figure out which of these effects is dominant. This question: Relation between Young's modulus and density? may shed some light on that.

In any case I don't think the force of gravity itself will have any direct, noticeable affect because, as you point out, sound itself is just a pressure wave, and the individual particles don't move far enough to be affected by a gravitational gradient.