I don’t move my tongue when I whistle though, but maybe that’s just me…
Fair enough, and sorry for the strong reaction. But breaking the breath vibrato habit, once acquired, is one of the hardest things to do.
Bernoulli’s principle says that narrowing the bottleneck speeds up the fluid. If you’re fluting (fluiting?), the bottleneck is your embouchure
Actually, Mr Guru Sir, I thought your reaction was restrained - mild, even. I’m also a bit puzzled - Gordon, you say you’re talking about whistle playing in general rather that just for Irish music … Um, I’m not aware of classical whistle as a routine serious instrument (despite certain people’s fantastic and virtuoso efforts as previously notified on this board 
), and it’s not a usual instrument for jazz … In fact, it’s almost exclusively used for trad. For my money, I don’t care what sort of trad it is, Irish, English, Welsh, whatever - breath vibrato is a no no on whistle. It sounds all James Galway-ish. 
Lots of good stuff already said.
BrewerPaul hit the nail on the head.
Ask any brass instrument teacher, any clarinet teacher, etc “how do I improve my tone” and they will answer “long tones”.
If you read about how flute players, opera singers, etc develop tone you’ll note that a lot of attention is given to the size and shape of the oral cavity, the openess of the throat, and “support”.
When people talk about support they’ll often mention pushing the tone up from the diaphragm, from the bottom, rather than from the upper chest.
About the size/shape of the oral cavity, note the differing positions of the tongue when you say a long “oh” , a long “uh”, a long “oo”, a long “ee”. Feel the tongue rising higher and higher? Seems the best tone on many instruments comes from when the tongue is as low as possible, in other words the oral cavity has the greatest possible volume. The tongue is a restrictor and the best tone seems to come with the most open and free airstream, that which comes from an open throat and oral cavity.
Another issue: I’ve noticed especially on my low D whistles that the tone and tuning are effected by how far the mouthpiece is shoved in my mouth. The clearest tone, best octaves, and best tuning all happen when the whistle is shoved as far as possible in my mouth. Too bad that that feels odd to me, and when I’m not paying attention the whistle moves out so that the very tip is in my mouth. Tuning-wise, what happens specifically is that “B” becomes flat unless the whistle is farther up in my mouth.
About using a “diaphragm vibrato”, I would suggest at least doing some practicing on low notes while using a slow, broad diaphragm vibrato as it will demonstrate how nice a tone you can get from your whistle due to the fact that the tone is being supported from the diaphragm.
Oh, I forgot one thing:
About the strength of the airstream on any given note on the whistle, the fact is that any note on a whistle is only in tune when it is played at one specific strength. So a whistle player, if he wishes to play in tune, has the strength for each note dictated to him by the whistle, and has no choice in the matter.
For example, most Generation whistles have a rather flat 2nd octave so to be played in tune the low octave must be blown quite gently and the 2nd octave must be blown strongly.
My Burke Low D has, to my way of blowing, a 2nd octave which tends to a slight sharpness so on that I’ve got used to blowing the low octave strongly and backing off on the 2nd octave.
On either sort of whistle, if I tried to impose my own notions of how the things should be blown I wouldn’t play in tune.
thats all super good advice, long tones espectially, and playing around with the whistle in your mouth to find what sounds you like, but also listen to whistle players you like, I find that as your ear listens to that sound more, its easier to bring it out when you play and its easier to find the sound you are looking for as you play around
good luck!
This is only sort of true. It’s true if you can’t tune your instrument, but for those whistles that are tunable, you have a range of pressures to play with. To be in tune with A=440, you could either push in the headpiece and blow softly, or pull it out and blow harder. Choose the scheme which suits the situation, i.e. if you’re trying to play softly, sharpen up the whistle and back off the pressure, v.v. for a fierce session with a box player. You’ll find that the whistle responds entirely differently depending on which way you go. Cheers,
Rob