Why the "20" in calculating db ratio? In wikipedia, I read the formula to convert a power ratio to db. This formula is :

10*Math.log10(power_out / power_in);


But, for the sound pressure, I read the formula is :

20*Math.log10(soundLevel_out / soundLevel_in);


I read in some places a reason : it is because the sound pressure, like the voltage, have to be squared in the ratio and this squared becomes a 20 instead of a 10. Ok, but why ?
When I want to double the sound "feeling", I was told I need to double my power amplifier so, I understand it's a question of power (with no squaring) and not a question of voltage, but it's wrong.

Who can help me to understand where does this square comes from ?
According to this source, the sound power received by an aperture is proportional to the pressure squared, i.e.:
$$
P = \frac{A p^2}{\rho c} \cos \theta,
$$
where:


$P$ is the received power
$A$ is the surface area of the receiving aperture 
$p$ is the pressure
$\rho$ is the density of medium
$\theta$ is the angle between the propagation direction and the surface normal