Based upon the information you've provided I can think of two things that may create the "medium frequency whirring" sound.
1. Low Power Steering Fluid
When your power steering fluid is low the fluid cannot absorb sound and what you hear is the pump possibly going bad as a result. Sometimes, when the fluid is just low there will be audible component based upon the lack of sound absorption properties that are the fluid's innate properties.
2. Power Steering Pump Failure
As above, the pump can make the sound you describe when going into a failed state. I'm unsure of the sound source though.
I would defer to @handyhowie chiming in based upon the newer information you have provided or another resource on the site that might be more knowledgeable.
I will say I am sure that the fluid could be low or the pump could be failing. Rather vanilla, I know....but I know those are at least two possibilities.