I have noticed that when I am cruising for example 35mph (a bit over 1500RPM) the engine speed varies a bit in a rhytmic fashion. The cycle is something like 2-3 seconds. Not much, perhaps 100RPM, but it sounds irritating. Is this normal, and if not, what to look for?

 

Your Public Profile shows that you have a 2007 XK. At that age, I'd suspect brittle hoses and a vacuum leak. Have you checked for any codes? There are lots of possible causes.

 

Is there an audible change in pitch, or are you noticing the change on the gauge cluster without an associated aural input?

 

Mine does it too on occasion as it warms up from a cold drive away, say 3-4 minutes into the trip around 35 mph. Just enough of a tone change to look at the tach and see the needle wandering maybe 50 rpm. I think it's just a blip in the warm up cycle where the o2's aren't warmed up to fully reliable yet but the puter's reacting to them.

 

If it's an air leak the LTFTs at idle (hot engine, parked) should show it.

 

When mine is cold-started and for the next while sort of sounds like a single misfire every three to five seconds or so, but it goes away within probably two minutes.

 

My XJ350 had the issue and it was transfer box and expensive. Common issue on the '04 4.2 model, but may be you have something similar. I hope not.

 

I think you mean the torque converter, not transfer case - there is no transfer case in the 2004 XJ8. My 2004 XJ8 had the symptom you are speaking of. But the oscillation in rpm disappeared after the engine and transmission were properly warmed up so I did nothing about it - a minor issue in any case. I do not think this is the same thing that the OP is describing.