Sound dampening? As in vibrating water? I guess just making the speakers wet could be sound dampening...
Sound damping with some sort of adhesive weight-adding material is probably what you're after.
The black stuff around the speaker is already giving a good bit of damping. You can buy Dynamat kits that have two 12" x 12" squares to put around the speakers for $25-50, but it doesn't really make a whole lot of difference unless you're sending deep bass frequencies to the door speakers (usually caused by not having a subwoofer and/or sending a full-range signal to the speakers, including sub-100-Hz frequences that makes them shake).
If your Sport doesn't have the angled adapter for the stock 6x8 speakers (it might be attached to the back of the factory speakers if you took them off), you can just remove the factory material and add on some Dynamat if you're hearing some buzz from bass frequencies. To apply, just remove everything (speaker, factory foam, move all wires and screws out of the way), clean the surface, remove the adhesive backing from the damping material, stick it on, and maybe use a hair dryer or heat gun to make it stick real well and conform to the surface, then use a knife or something to make the hole for the speaker and punch the screw holes back in, and mount the speaker and everything again. You can even put the factory foam back over the damping material for added benefit.
The rear speakers are mounted to brackets, so there's not a lot of use in adding damping material, but any leftover scrap pieces from the front speaker cutouts can be stuck on the brackets close to the speaker to absorb whatever vibration there is.
It's a good buzz killer if you listen to a lot of bass thumping stuff, otherwise the factory front adapters make the doors pretty rigid, most of the buzzing comes from the plastic door panels. Sticking some damping material on those can help kill that, too.
