Mark, here's a few ideas
17 Ideas on How to Build a Soundproof Box for Air Compressor
I can't say I agree with all of the choices there, just some ideas - also, here's a thread that went on during the 8 years or so I moderated construction on this site
John Sayers' Recording Studio Design Forum •
View topic - from shed to studio?
My screen name there is knightfly.
You'll need to join (free) in order to see any of the several sketches I drew for the OP; unfortunately, pics from the OP didn't survive. The techniques remain the same, you could just scale them down to fit your compressor.
I no longer have license to some of the software I used to use (not cheap and not doing that any more), but generally you should start with knowing what frequency components your compressor generates the most; can be done with a free android app called Spectrum Analyzer - next would be finding which materials/thicknesses, etc, will give best attenuation of those frequencies. If EITHER side of a double wall (aka Mass-Air-Mass, same concept as Mass-Spring-Mass, since air is a spring) is resonant at the offending frequency, there'll be nearly NO attenuation.
Don't bother to respond to that thread, the OP is long gone and I don't frequent there regularly any more either; got too burned out and too little progress on MY stuff.
Oh, if anybody tries to sell you "soundproofing" foam, RUN AWAY - most of it is NOT fire resistant, and insulation like Roxul's safe-n-sound works MUCH better.
If you like, PM or call me and I can dispel a few more myths... Steve