One "version" of the Brown Sound, which in my opinion had many variations up until 1991, when it was no longer Brown (as these ears hear it), sounds like the beginning and end of this clip. The middle 50% isn't bad either, but the more prime examples I feel are at the start and at the end.
http://www.funnymoneyband.com/sw/tubesclip1.mp3
Supposedly it's just a guy using a Pearly Gates through a cranked Plexi, but I can't recall exactly.
My favorite versions of the Brown Sound are on Van Halen II, and Women & Children First, which is about as warm & mushy a tone you can have while still spitting out attitude. 1984 is probably my favorite "polished" version of the tone.