How to obtain mean Power Spectrum from two audio channels? I have a source of noise in a duct, on either side of the noise source I have a microphone, which register the pressure, so that I get two pressure signals: $p_1$ and $p_2$. I want to get the acoustic Sound Pressure Level (SPL) spectrum which can be estimated using a periodogram approach, more specifically using the welch procedure. This function is readily available in Matlab as 

pwelch


or in python using 

scipy.signal.welch


The Question

The question is, since I have two channels of audio, how do I combine them to get a representation of the total SPL spectrum? Or is the mean SPL more appropriate? 

Some ideas:

I know that for two sound sources giving rise to $p_1$ and $p_2$ then we have two cases:

Correlated sound: $&lt;p_{tot}^2&gt; = \frac{1}{2}(p_1^2+p_2^2)$
Uncorrelated sound: $&lt;p_{tot}^2&gt; = \frac{1}{2}(p_1+p_2)^2$  

In my case I'm guessing I should consider the time series $p_1$ and $p_2$ to be correlated since they come from the same source but are measured on separate sides, and this would imply that the total SPL spectrum would be given by averaging the resulting spectra given by welch on the individual channels. Thoughts?
The sound pressure level (and derived metrics such as the SPL spectrum) are dependent on the position of the listener, since it is a measure of the pressure of the wavefront moving through space. As such it doesn't make all that much sense to try to combine SPL's from different positions into a single value, except perhaps if the listeners were exactly the same distance from a single omnidirectional audio source and there was no possibility of reflections.

The equations you have posted are for two audio sources, not two listeners, so I am not sure they are relevant.

The only value of any use I can imagine you being able to calculate from two listeners would be the peak SPL / spectrum, this would simply be the maximum value of both channels for any given sample.