A "cloche" I imagine is similar to one of those ghastly telephone hoods common in the 70s and 80s. The worst ones were a kind of perspex bubble that resonated horribly, though the better ones used fibreglass held in place by perforated metal sheet. An "enclosure" is far more general: for example, gas turbines in warships are surrounded by an "acoustic enclosure" which does a decent job of attenuating the noise by some tens of dBs.
This must be a very large inductor to allow the free circulation of people around it.