Idle Whistling (BMW E39)

[Skids]

Literally. My car's engine whistles at idle and has done for some time.
It started when there was a fault that would drain the battery over the course of two days. It only whistles when the engine is cold and the longer I leave the car the worse the whistling is when first started. In an attempt to cure the battery drain I changed the climate control resistor and I removed a redundant system, but the damn thing carried on whistling.
I took it to a specialist and he immediately said that its the alternator whistling because its making up for a poorly charged battery. Which seems to fit in with the evidence, as the whistling stops after a while of driving, when the engine is warm and the battery would've charged.
I've done charging and current drain checks and found that the alternator is pumping out enough voltage and when the car is locked and shut down the current draw is well below specified limits.
I took the battery out and fully charged it. Put it back in and there was no whistling.
Problem now is that I can leave it for two weeks without using it and there is plenty of charge in the battery to start, no battery drain but it still whistles.
What is it?

[Graham-W]

Tighten the alternator belt or, if it's the self tensioning type, replace the spring loaded tensioner.
Sometimes just replacing the alternator belt will cure it.

[TWR]

Hi, It's a diaphram, as you look at the unit from the front, look at the Right side & lower of the Rocker cover.