Why not connect to the internet stream from one computer and then use PulseAudio to provide the audio to the network?
You'll need to install paprefs for this, on the "server" (the computer connecting to the internet stream) and on each "client" (computer receiving audio data from the server) you wish to use.
paprefs is a GUI program, but its options are a little opaque. The manpage is, as is too often the case, useless for figuring out the options. It does provide a Web address, http://0pointer.de/lennart/projects/paprefs/ , which has a link to "Documentation" but contains no actual documentation. I will attempt to walk you through the configuration process, but my setup is different from yours so I may err. Hopefully someone will see it and correct it if I do.
First, run paprefs on the server machine. Under "Network Access" tab, select:
"Enable network access to local sound devices"
"Allow other machines on the LAN to discover local sound devices", and
"Don't require authentication"
There are two more options on that tab regarding DLNA/UPnp media streaming. I don't know what that is so I have left it alone.
Under "Multicast/RTP" tab, select:
"Enable Multicast/RTP sender" and
"Create separate audio device for Multicast/RTP"
That should do it for the server.
For each client, run paprefs and:
Under "Network Access" tab, select:
"Make discoverable PulseAudio network sound devices available locally"
Under "Multicast/RTP" tab, select:
"Enable Multicast/RTP receiver"
This should suffice for client configuration.
When it's time to listen to your broadcast, connect the server to the internet stream in the usual way. Then open System→Preferences→Sound and select "RTP Multicast" from the "Output" tab.
After that, on each client, open System→Preferences→Sound and look in the "Input" tab. It should contain an entry for an RTP Multicast from (server's hostname). I am not sure of the exact wording as I am not in a position to test it right at the moment. In any case, select the multicast, adjust the volume, and you should be hearing the broadcast (and any other audio) coming from the server.
As far as I know, there is no way to bind particular applications to particular inputs/outputs. If anyone knows of such a way, I would very much like to know about it. I have a question on the subject at How to clone audio output?.
While these may not be the options you land on, I'd like to offer you two solutions: One from the 90s (maybe late 80s) and the other from the 1870s (yeah, late nineteenth century technology):
You could buy some wireless headphones. You can get fairly good enclosed sets for under £50. If quality isn't too much of an issue, halve that number.
Now we cast back a wee bit further, we hit the granddad of all remote listening: the audio extension cable. These are as cheap as anything and if you have a soldering iron you can buy a stereo spool for almost nothing, add on some pennies for the connectors and wire-up an extended port to your bed.
Both these solutions have next-to-no lag. Obviously the second has less lag but it's a much more hackish solution. I'd just buy some remote headphones and be done with it.
Time (that you might spend looking for the perfect solution) isn't worthless, after all.
Best Answer
As there is a proprietary implementation of fransferring of audio via a RDP connection on Windows (Audio Output Virtual Channel Extension) but this feature is not available from X11RDP server (or VNC as a sidenote), where only basic remote protocols such as screen, keyboard, or mouse are present.
To be able to get the sound output of the server sent to a client we therefore need an additional protocol. Provided we had installed pulseaudio on our Wyse OS this can be done by using pulseaudio's inbuilt network features. These are all nicely documented and listed here:
Pulse Audio Network Setup
The transfer methods may be set up by command line, or with a GUI using paprefs
.
See also answers to these questions: