A bad bearing in the water pump could be one possibility.
Low coolant? Cooling system airbound?
As the engine revolutions increase, so does the noise from the engine as it is working harder as opposed to being still idling. Also, could be a leak in your exhaust system that could be the cause of the additional noise.
Could be your timing belt.
I would think that you have a leak in your exhaust system.
No. The idler pulley only maintains tension on the serpentine belt. A faulty idler pulley can cause noise and a loose tensioner arm can cause belt slipping but other than having a similar name, it has nothing in common with engine idling.
This might be the cooling fan which in today's cars is electric and runs only when needed, not continuously like the 50's and 60's vehicles. When you sit idling, with no cooling air passing over the engine, the coolant temperature rises and causes the fan to switch on. This could also happen during acceleration.
It is the fan blower for the engine cooling system (Radiator)
if it is in fact the starter, it is probably engaging the flywheel. starter may be faulty.
Overheating or air trapped in cooling system.
Could be your cooling fan. And depending on what year your car is, you could have two cooling fans. My cavalier did the same thing. The fan would try to go on when the car heated up, but couldn't. I heard a distinct humming noise with it. It didn't cause any problems until I got stuck on the highway in the summer and my car almost overheated. Take it to a service station.
Noises are impossible to correctly diagnoise unless you hear them. TAke it to a good diagnoistition.
Check the idler pully.