What's the frequency response of air, with respect to sound propagation? It's a very simple question but I could not find an answer anywhere on the internet.
I'm interested not only on the air magnitude response, but especially on the phase response. Is it a linear phase system? If it's not, how bad is the phase distortion with frequency?
 A: Dissipation of energy
There are 3 types of losses that occur when sound waves propagate through a gas.  These losses are caused by


*

*viscosity

*heat conduction

*molecular exchange


Each process is characterized by a phenomenological relaxation time $\tau$, with modified wave equation
\begin{equation}
\left( 1 + \tau \frac{\partial}{\partial t}\right) 
\nabla^2 p = \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} \; .
\end{equation}
Lossy Helmholtz equation
When the time-dependence is harmonic ($e^{i\omega t}$), this equation becomes the so-called lossy Helmholtz equation
\begin{equation}
\left( \nabla^2 + \cal{K}^2 \right) p = 0 \; ,
\end{equation} 
where $\cal{K}$ is a complex wavenumber: 
$$\cal{K} = \frac{\omega}{c} \frac{1}{\sqrt{1+i\omega\tau}} = k(\omega) - i \alpha(\omega) \; .$$
The implications are that


*

*The $\omega$ dependence of $k$ means that the propagation is dispersive.  

*The existence of a finite $\alpha$ means that the wave decays like $e^{-\alpha x}$ as it propagates.


Results for typical gases
The value of $\alpha/f^2$ is approximately constant for typical gases, and depends on the type of gas (it is about 0.5 for Helium and 2.0 for Oxygen). For monatomic gases like Argon and Helium, the losses are dominated by (1) and (2) (so-called classical losses).  Classical losses are not adequate to accurately describe polyatomic gases, like Oxygen.
Frequency response
The original equation asks for the frequency response, although this is not normally the way to characterize dissipation.  However, based on the previous result, one observes that at a distance of 1m, the decrease in sound pressure level (SPL) is
$$ p = p_0 e^{-\alpha - i k} $$
When $\omega \tau \ll 1$, we can write 
$$\alpha = \frac{\omega^2 \tau}{2c} \; .$$
Thus, a measure of the frequency response is 
$$ \frac{p}{p_0} = e^{-\omega^2 \tau/c} \; , $$
where I have dropped an uninteresting phase factor.
