"On the air" OR "On air"

Solution 1:

There are subtle differences between the two constructs in other contexts, however, real stations have used both interchangeably for warning lights. KAKM, in Anchorage, had "on the air" at one door to the soundstage, and "on air" at the other, back in the late 1970's. The transmitter light, however, said "on air" as did the monitor. (This author spent time in that studio.)

On the air as a phrase is almost always referring to the current state of being broadcast, and rarely is used as an adjective about the potential of being broadcast. E.G.: Fred is on the air. (Meaning: Fred is currently being broadcast as the speaker says it.)

On air, sometimes on-air, can be used to convey the state of being currently broadcast, but also is often used for the potential to be broadcast, or the expectation of being broadcast. E.G.: John is on-air talent. (John can be expected to be broadcast.) Fred is on air right now. (Same as Fred is on the air.)

Solution 2:

Both are logically and grammatically valid. I grew up in the SE United States with a couple of relatives who worked in the broadcast industry. The phrase I always heard, and continue to use, is "on the air." There is some difference between the frequency of use of the two terms in American English and British English. For what it's worth (Nothing), I was a fan of the show and I remember the signs changing.