So the x-type is making a new noise. Here's what happens: I start the car in the morning or after an extended period of cooling down. Then I start driving and at first everything is normal. After driving about 300 ft, I hear a high pitch whistling noise. For a few seconds it gets louder and then goes away completely. It only happens for this short time and after an initial short period of driving.
I made a audio clip which should be posted here. The clip covers my initial start, backing out the driveway and then putting into gear and driving off. The sound is heard from about 2:13 to 2:20 in the video.
Hope someone has some ideas - thanks!

 

One other thing, people might ask where the sound is coming from. It's hard to say, but the best I can say is from the passenger front door area. It doesn't sound like it's coming from the engine bay nor the rear of the car. But, again, it's hard to pinpoint. The audio was made from my cell phone sitting on the passenger seat.

 

Wait for others to chime in, but it sounds like the prop shaft. I see you are in New Hampshire and I am assuming you are experiencing below freezing temperatures. I am also guessing that the sound comes into play around 30-40 mph?

 

The prop shaft bearing is no doubt a possible cause, and it has been cold. Last night it got down to 15F, and this morning the noise was there again. Same characteristics, it doesn't happen until the speed gets up to around 30 mph and only lasts a short time (a few seconds). Then I don't hear it anymore until I do a cold start again.
Another concern of mine would be the transmission. Maybe something about the initial shift into higher (4th or 5th) gears are causing metal-to-metal contact until sufficient lubrication is achieved. It's too bad there is no easy way to check the fluid level! I am wondering if adding a small amount (maybe 1 cup) of transmission fluid or conditioner would be something I could try.
My next "experiment" will be to set it into 3rd on the shift console on and see if I hear the noise. That could help differentiate between the bearing and transmission as possible causes.

 

I dont know....it doesnt sound like metal-to-metal. To me, it sounds like some sort of bearing whine. I dont know about the prop shaft bearing - if its the case, shouldnt the sound pitch go higher if you accelerate? I could be wrong...

To me, it sounds like the type of sound the bad bearing on climate fan would make if the fan is revving down.

 

If the noise is in the 28-32mph range it is almost a certainty it is the propshaft carrier bearing.

 

It was a cop chasing you..Slow down dude!

Seriously..It is most certainly the Center Support bearing on the propeller shaft....The damn thing can only be bought as a complete shaft and bearing assy...I don't think you can just get the bearing.

 

Thanks to those who replied, I appreciate your suggestions. I greased the prop shaft bearing about a year ago, so maybe it's time to try that again. I will also look to see if there are any signs of metal-on-metal wear anywhere near the driveshaft.
Today was another cold morning, and this time I ran with the transmission set in 3rd gear. But, the same noise appeared; I did another audio which is posted below and you can hear it start at about 00:20. It seems to go up with speed and then disappear altogether. To me it seems high in pitch for a bearing, but I'm at a loss as to what else it could possibly be.

 

Can't offer a solution but I too had this sound in my 2007 Jag x-type 3.0L Sportwagon. I live in Vermont on the Canadian border so our climate is very similar. It occurred only in below freezing temperatures. The colder it was the longer/louder it was. -20F was probably as cold as I drove the car. It did happen around 30 MPH and I felt it could be controlled a bit by throttle response. At -20 below it lingered longer and then would go away. However after maybe 10 miles at 50 MPH it might happen again upon deceleration but not as loud. Extremely difficult to pinpoint where it was coming from. Come Spring the noise left and only returned the next Fall! 2 years in a row! The Jag had 59K miles on it when it 1st started and 76K miles when I sold it. The noise had no adverse effect on anything. Seasonal Gremlins!

 

Aha - the famous cold weather whistle - a unique feature of the X!!!

just to add further reassurance - I bought my x type in the fall of 2010 with just under 200,000km and still have it now with just over 300,000km. I have had that whistle in cold weather for most of that time. I read somewhere it was not a sign of any trouble ahead - just an X type quirk. 100k km later, that appears to have been good advice. So don't worry - if you can live with the whistle, the whistle can live with you.