What is physics behind of explosion under Atmospheric pressure? An explosion is a rapid increase in volume and release of energy in an extreme manner.
A blast wave in fluid dynamics is the pressure and flow resulting from the deposition of a large amount of energy in a small very localised volume. 
The equation for a Friedlander waveform describes the pressure of the blast wave as a function of time:
$$P(t)=P_oe^{-\frac {t}{t^*}}(1-\frac {t}{t^*})$$
where P$_o$ is the peak pressure and t$^*$ is the time at which the pressure first crosses the horizontal axis (before the negative phase).
My Question:
Why does the pressure temporarily drop below ambient after the wave passes?
 A: If you start with a finite amount of gas in the inner sphere and then deposit a massive amount of energy, the molecules of the gas begin moving rapidly outwards and piling up, creating the blast wave. However, the rate at which the gas is moving outwards may not be balanced by the amount of gas molecules being created by the explosive. If this is the case, then the pressure must decrease below ambient as the molecules are pushed outwards with the blast wave. 
You can see this in videos of blast waves. The initial wave continues to move outwards, but the smoke/dirt/debris caused by the explosive will move outwards initially, then inwards as the lower pressure region sucks it back in towards the center. There is actually considerably banging that goes on where the low pressure behind the blast wave moves inwards and outwards until it relaxes back to atmospheric pressure. 
Here is a great video that shows the blast and resulting banging as the pressure relaxes. 
A: If I understand correctly what you're asking, you are referring to the vacuum created from an explosion. Think about it like this: An explosion is rapidly expanding gas from a central point (the explosive material). As that gas expands, it pushes all of the material (the gas of the atmosphere) away from that central point. The front of the gas (shockwave) is what pushes the atmosphere. This leaves a vacuum behind the shockwave. I believe that is the negative pressure to which you are referring. This is best illustrated in the old cold-war era nuclear test footage that shows the bending of the palm trees. When the explosion goes off, the shock front pushes out in a generally 360 degree pattern in 3-dimensional space. This will cause the palm tree to bend away from the explosion. Then, shortly after, you will see the trees bend towards the explosion. This is due to the atmosphere rushing back in to fill the vacuum created by the shock front. These different phases of the explosion are called the positive phase - when the pressure moves away from the blast - and the other is cleverly named, the negative phase - when the surrounding fluid (the atmosphere in this case) rushes back in to fill the vacuum created by the positive pressure phase. 
I hope that was the drop in pressure to which you were referring.  
A: Is this "vaccum" or "underpressure" vawe a pure mechanical thing that would occur also in a long air cylinder if we produce a shock wave with a plunger or piston?
In the case of an explosion I understand that shortly after the explosion the hot air bubble in the center starts to move upwards by convection, because it is less dense than the cooler aire around it. I understand that this produce a wind towards the explossion and possibly a "negative pressure" at some distance from the explosion near the ground, but I think it does not explain the underpresure spherical wave that seems to expand in 3D from the explosion (according to some videos in which a spherical condenstaion cloud is seen expanding spherically around the explosion in the first instants).
