audio - How do Phons relate to loudness?

Let's say I have the very same signal stored in 2 .wav files. And lets say I process each file using a function, and it is established that one file is 20 phons larger than the other. Lets ignore how that function calculates this.

Does this mean that one file is approximately 4 times louder than the other?

What formula can I use to relate the difference in phons for two signal to the difference in their loudness?

-10 phons = 0.5 
0 phons = 1
10 phons = 2

Answer

Your equations are correct (as a first order approximation).

The phon is an attempt to establish a perceptual loudness measurement. Perceived loudness is rather complicated since it needs to take into account what happens in the human auditory system and in the human brain.

Here is is how it works: First you measure the sound energy at the listener's position. The relevant physical quantity is the variation in air pressure, called "sound pressure" and measured in Pascal or Pa like any other pressure. This is then referenced to 20 micro Pascal and turned into dB like this $$ x = 20\cdot log_{10}\frac{p}{20\mu Pa} $$ where p is the measured sound pressure and x is the sound pressure level in dBSPL. This is a direct measurement of the physical sound energy.

The human auditory system has different sensitivity at different frequencies. It's most sensitive around 2kHz-4kHz, a lot less sensitive at 100 Hz and at 30 kHz you don't hear anything at all, regardless of how much sound energy is there. The "phon" takes this into account. It's based on measurements that compare the perceived loudness of sine waves at different frequencies. The results are often referred to as "equal loudness contours" (see for example http://en.wikipedia.org/wiki/Equal-loudness_contour).

A loudness of X phons means "as loud as a 1 kHz sine wave at X dBSPL". For a 1 kHz sine waves phons and dBSPL are the same thing. For sine waves of other frequencies phons and dBSPL are related through the equal loudness contours. For more complicated signals calculating phons is a bit more tricky.

Phon is not a direct loudness measurement. Perceived loudness, as most human modalities, follows roughly the Weber-Fechner law (e.g. http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law). As a first order approximation it takes ten times the energy to double the perceived loudness (all other things being equal). An increase in 10 phone doubles the loudness and an increase in 20 phone does, indeed, increase the perceived loudness by a factor of 4.

To calculate phon (relative or absolute) from a wav file, the file MUST be calibrated in units of Pascal (sound pressure) since the equal loudness contours are non linear and change with absolute level.