Welcome to the Apple Support Communities
If you have a non-compatible MacBook Air and you want to mirror the MacBook Air onto the Apple TV, you can use a third-party application instead of AirPlay Mirroring. In this case, you can use AirParrot > http://www.airparrot.com It has got a good reputation and it works great
About AirPlay and Airplay Mirroring
AirPlay Mirroring requires a second-generation Apple TV or later, and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). For non-qualifying Macs you can try using Air Parrot.
Several Apple Articles Regarding AirPlay
Thanks to the $15 Beamer, AirPlay streaming is still possible on Macs that do not support Airplay and mirroring.
Another solution is the Air Parrot.