# Speed of sound in solid — temperature dependence

Let $v$ denote the speed of sound in a fixed solid, at a fixed temperature $T$. This will depend on properties of the solid (such as the bulk modulus and density).

Given an increase in $T$, does $v$ necessarily increase?

For calculation purposes, we should assume a "small" change in temperature. For an ideal gas, $v$ increases as the square root of $T$, so perhaps the same is true in the limit of compressing the gas into a solid.

In solids, I came up with two opposing naive/vague thoughts:
1) If there is an "index of refraction" for sound in materials, then I would expect an increase in $T$ to decrease this index, hence increasing $v$.
2) If I can view the propagation of sound in terms of a coherent collection of phonons, I would expect an increase in $T$ to hurt this coherency, hence decreasing $v$.

Edit: To clarify, I am searching for a theoretical explanation.

• I'm not sure if there's a direct connection besides how $T$ varies $\rho$ and $B$. I don't think point (2) is relevant because sound in a gas is by no means a coherent collection of phonons (or at least I think so, see this question). We can just restrict to considering 'classical' sound. – knzhou Jun 22 '16 at 20:26
• Your question about the temperature dependence of the speed of sound is one which can be researched using the internet, but you do not seem to have made any effort to do so before asking on PSE. The impression I get is that you want to discuss your ideas. – sammy gerbil Jun 22 '16 at 23:44
• You're wrong -- can you please provide the obvious references then? I have researched the Internet, and gave my attempt at a solution, hence why I asked this question. I know how StackExchange works, and I'm sorry you got the wrong impression. – Chris Gerig Jun 23 '16 at 2:17
• Chris, I apologise. I did not see any indication in your question that you had searched. I did not find anything direct, only (after less than 5 mins searching) the following indirect answer (by nahira) based on the temperature dependence of the elastic moduli and density. It gives a numerical answer, not a theoretical one : physicsforums.com/threads/… – sammy gerbil Jun 24 '16 at 1:33