Hi there Richard,
The obvious thing for me to suggest would be relocating the speakers to see if this makes a difference but, as you have said your space is limited, I guess this isn't an option for you. It could be worth checking your speaker cables. If they are badly arranged or coiled, they can act like a radio receiver and if there is a poor contact between the jacks and their inputs this can also cause it.
Does that mean that this is NOT likely to be doing any harm and it's just something to live with ?
The reality is that when a PC shuts down it stops it's services, such as sound, video, internet connection, before actually turning off the CPU, HDD's etc. The sound is most probably due to the fact that the audio drivers have stopped but there is still a connection to the amp/speakers (Your white noise description is very apt). Although it is annoying I doubt there will be anything that you can do to actually stop it from happening and, so long as the level isn't too loud, it won't be causing any problems to your hardware, only your ears.
To be honest I come from a music/sound/video orientated background and have been building PC's for over 10 years now. I have always turned off my amp/speakers etc., before shutting down the PC. A force of habit I have gained over many years, as even high tech amps and studio recording equipment can suffer from this.
I hope this helps.
