Basically you have 3 options:
- Apple AirPort Express
- Bluetooth Adapter
- Chromecast Audio
Apple AirPort Express
You only need an Apple AirPort Express and a minijack cable.
Connect the AirPort Express to your excisting home Wi-Fi network. During AirPort Express configuration select "Connect to a wireless network" to provide internet connectivity.
Connect your Libratone Beat speaker and your Apple AirPort Express through the analog minijack cable. Speaker minijack input is found down on the backside of your Beat speaker and is protected by a rubber dustplug.
Now you can play your Libratone Beat speaker through AirPlay by addressing the AirPort Express in the AirPlay menu in Control Center.
Apple Airport Express are sold by Apple at around USD/EUR 100.
NOTE: Use AirPort Express 1st generation. Airport Express 2nd generation and later has a slightly higher minimum volume level which can be annoying if you wish to play silent music.
Searching the internet you can find a lot of different Bluetooth adapters that can be used for this setup.
One of these adapters is the Philips AEA 2000 Adapter which is sold at around USD/EUR 40.
An alternative to Apple Airport Express and a Bluetooth adapter is the new Chromecast Audio from Google.
Remember to check if you favorite application is supported by Chromecast Audio.
Chromecast Audio are sold at around USD/EUR 35.