Whining or slip noise when accelerating

My 1999 auto HSE, starts fine and all are ok for a short distance. If I drive longer then there is a whining noise which i hear basically under the main console which sounds like a belt is sliping or a bearing gets "tired". It only sounds when I push the accelerator and when I lift my foot, it dies away slowly depending on the speed (when I am slow it goes very fast). If I leave the car for an hour or so, when I start it is again ok, until it runs for a while. Also it does not do it when the car is stopped. Only when it is moving. Even if I stop breaking and I press the accelarator it does not do it. Or even when I have it in N.
Having read relevant articles, should I worry about the transfer box? Is there anything else I should start looking at first?
Many thanks in advance for your input.

Panos.

i had a whining noise in my 4.6hse when ever i accelerated hard, i cured it by leaving mrs tufty at home

if you look on your dashboard there is a small grill like opening,behind this there is a little fan,this gets noisey due to crap build up on it,or a dry bearing.you have to get some forced air to blow the fan clean,or take the dash out if it is the bearing.

Many thanks,
leaving Mrs at home did not help, so I shall look into Gemsdad suggestion
Best regards,

Panos.