[Physics] Do longitudinal waves travel more quickly in denser gases

Question

I understand that sound waves are longitudinal, and they travel faster in water than in air. However, would this imply that longitudinal waves travel more quickly in denser gases?

Answer 1

No, it is not true that sound travels faster in denser media. In fact it travels slower.

In the adiabatic approximation we assume that the portions of the gas vibrate so fast that it is not able to exchange heat with the surroundings. The the longitudinal displacements can be shown to satisfy the wave equation $$\frac{\partial^2 u(x,t)}{\partial t^2}=\frac{\gamma P_0}{\rho_0}\frac{\partial^2 u(x,t)}{\partial x^2}.$$

As you can read, the velocity of this wave is $$v=\sqrt{\frac{\gamma P_0}{\rho_0}},$$ where $\gamma$ is the adiabatic constant, $P_0$ is the gas pressure and $\rho_0$ is the gas density.

If you speak right after inhaling Helium you will sound at higher pitch. The reason is that the speed of sound in Helium is higher than in air. Air is denser than Helium.

For liquids the speed of sound is given by $$v=\sqrt{\frac{B}{\rho_0}},$$ where $B$ is the volumetric elasticity, defined as $$B=-\frac{\Delta P}{\Delta V/V},$$ i.e. it gives the relative change in volume when we change pressure. Again we see the speed of sound decreases when we increase density.