# Do longitudinal waves travel more quickly in denser gases?

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?

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.

• Not so obvious. Clearly, $P$ is a function of $\rho$ which may increase faster than linear in a dense gas. May 17, 2016 at 15:20

It depends: 1) On whether you mean mass density or particle density, 2) on the temperature and type of gas. The main observation is that speed of sound is approximately independent of particle density, and mostly a function of temperature.

1) The speed of sound (squared) is given by the compressibility at constant entropy per particle $$c_s^2 = \left. \frac{\partial P}{\partial \rho}\right|_s \, ,$$ where $\rho=mn$ is mass density and $n$ is particle density. In a dilute gas $P=nT$ and $$c_s^2 = \frac{5T}{3m}$$ where the factor $5/3$ comes from the condition $s=const$ (the isothermal speed of sound squared is T/m). There is no dependence on $n$, so the speed does not change with density, but there is a trivial $1/m$, so the speed is higher in gases made of light atoms (like H and He).

2) You might think that at higher density gases become more incompressible and the speed of sound increases. The real situation is a little more complicated. The next correction can be found from the virial expansion $$P=nT( 1 + b_2(T) z + b_3(T)z^2 + \ldots )$$ where $z=n\lambda^3$ is the fugacity, $\lambda=[2\pi/(mT)]^{1/2}$ is the de Broglie wavelength, and $b_2$ is the second virial coefficient. The speed of sound is $$c_s^2=\frac{5T}{3m} \left[ 1 - z \left( b_2(T)+\frac{8}{15}T b_2^\prime(T) +\frac{4}{15}T^2 b_2^{\prime\prime}(T) \right)\right]$$ A simple excluded volume effect $b_2\sim a^3/\lambda^3$ reduces the speed of sound with density. I think this is the main behavior at high $T$. At lower $T$ we have repulsive forces (like the Pauli exclusion prnciple) which give $b_2<0$. In this regime $c_s$ grows with density.

3) Note that in real air, humidity plays a more important roles than density.