How can I apply special relativity for sound? What will happen if someone moving in a cart at speed of light shout(at same frequency) 
Can we expect a sonic boom in that case or something else will happen?
 A: So there is subsonic, transsonic, and supersonic--which are moving through a medium slower, around, and faster than the speed of sound in that medium. There is also hypersonic--which has various definitions in the aerospace industry, but physics-wise it means your a dissociating the atoms in the medium. Meteors and (re)entry spacecraft achieve this.
You are asking about ultra-relativistic. At these energies, the binding energy in the solids and even the atomic binding energies don't really matter, and you can consider the cart as a dense neutral plasma interacting with the atmosphere.
I think it's difficult to answer what exactly will happen. For a single piece of the car (an iron nucleus, for example), it will collide and create an air shower.
Note the kinetic energy of a small 1000 kg car traveling at minimum ionizing speed ($\gamma\approx 4$) which isn't even that close to the speed of light, then energy loss in the air will be:
$$ mc^2(\gamma-1) \approx 65 {\rm Teratons TNT}$$
It's just not reasonable to discuss sonic booms and sound propagation at this point.
