Why is acoustic intensity inversely proportional to density of the medium? The definition of sound (acoustic) intensity is given by
$$ I = {p^2 \over {\rho c}} \;\;\;\;  \text{or} \;\;\;\;  I = {p^2 \over {2\rho c}}$$
I've seen both definitions in different textbooks and am not sure which equation is more accurate. But in either case, the relation of intensity and density of air is:
$$ I \propto {1 \over \rho} $$
This contradicts my intuition.
I would hear no sound in a vacuum because there is no molecules in the medium to scatter the vibration from the source. If I start adding some molecules in the medium (low density), the source will vibrate these molecules in the medium, acoustic energy can be transferred and sound may be heard if there are just enough molecules hitting by eardrum (and within the audible frequency range). If I add a whole lot more molecules in the medium (high density), many more molecules will vibrate and hit my eardrum, and I would therefore hear a louder sound (or detect a higher acoustic intensity).
So why the equation suggests the opposite relation? Could it be that the instantaneous pressure is also a function of density, or $ p(t, \rho) $, in a way that pressure is proportional to density of higher order to offset the drop of $ 1 \over \rho $ have on intensity?
 A: Your intuition is in the right direction. Nevertheless, you are only considering the amplitude and the intensity is the power per unit area. So, you need to consider that when a material is more dense you need a higher increment in pressure to get the same increment in particle speed.
If we write it mathematically, we get the following relationship between the pressure and particle velocity
$$v = \frac{p}{\rho c}\, $$
where $\rho c$ is the characteristic acoustic impedance of the medium. Thus,
$$I = \frac{1}{2}\operatorname{Re}\{p v^*\}\, ,$$
or
$$I = \frac{1}{2}\frac{|p|^2}{\rho c}\, .$$
A: A candid way to visualise that $density / amplitude$  relationship is the following image:

*

*Imagine the medium as a 1 or 2 or 3 dimensional lattice of marbles linked to their nearest neighbors by springs (a N-dimensional array of harmonic oscillators, if you wish).


*The density (of the simplistic medium) is related to the mass of the marbles, consequently, for a given (fixed) amount of energy transferred to the medium, the amplitude (of the oscillations) will diminish for different (increasing) values of the mass, to keep kinetic energy constant.
