Flute Vibrato
Vibrato is that shimmery variation in pitch that warms up the tone of the flute. It is similar to the effect produced by a violinist's hand as it moves back and forth.
As Sir James Galway wrote in his book 
Vibrato is another subject on which experts violently disagree, and all of them are right. If one thing rather than another expresses the individual, it is vibrato. [...] There is no last word on what vibrato should be used, or how it should be varied to suit the musicality and individuality.
Why Vibrato?
Vibrato gives life to the sound, allows for intensity, and helps a player to focus. Occasionally the music calls for stillness, and then a note or two or a phrase can be played without vibrato to advantage. But as a rule music without vibrato is a pretty lifeless thing.
On the other hand, if the vibrato never varies, neither does the life of the music, and the result is rather tiresome and tedious. The human body has a range of intensity of life, from sleeping peacefully to running the hundred yards' sprint. Music needs this too; but the intensity should always be intended and under control. Therefore, the ultimate target is to master a whole variety of speeds of vibrato, on every note and at every volume level.
What Exactly is Vibrato?
Vibrato is the pulse of the sound brought about by the rapid alteration of more or less forceful pressure of the breath. Where it comes from, that is, what part of the breathing equipment actually operates this alternating pressure, is not agreed upon by all flutists. The received wisdom has it that the muscles of the diaphragm are responsible; others maintain that the throat is responsible, while the diaphragm merely quivers in sympathy. Still another position is that proper vibrato is produced by a combination of pulsations by the diaphragm and throat.
These two extremes are commonly referred to as “stomach vibrato” and “throat vibrato”, and they are the only methods possible for producing a vibrato sound on the flute. Other techniques, like the movement of the jaw sometimes employed by saxophonists, are impossible or impractical for flutists.
It is important to observe that flute vibrato is a more complex phenomenon than the vibrato produced, for instance, by string players. In fact, while a violinist's vibrato mostly consists of a periodic variation in the pitch (frequency) of a note, a flutist's vibrato also introduces large fluctuations in the volume (intensity) of the note.
How Do You Learn Vibrato?
In some flutists vibrato occurs naturally as a result of having good breath support. Sometimes, though, vibrato can be a difficult concept for the beginning flutist, because being an internal process it is hard to grasp.
It has been said that vibrato is something a performer should feel, not something to be learned. For those who feel it this may be true; but for the large majority who can't do it and want to know how to do it, it is something that should be studied and correctly learned.
Focused and in-depth study on vibrato can be found in volume 4 (Intonation and Vibrato) of Trevor Wye's Practice Books for the Flute, published by Novello. We highly recommend purchasing this book for serious work on flute technique, but here are some basic exercises to get you started.
First of all, it is absolutely essential to be able to play a perfectly straight note without any bumps or pulses, with the air coming directly from your lungs in an open, relaxed pipeline. It is of primary importance that the air meets no constriction from your lungs to your lip opening, and that you are comfortable with relaxing your throat.
Once you can do that, take a full breath and play a long, controlled note that you like. You should choose a note that is easy to produce, like low A or low G. Now increase the volume of the note in short (but not too short, say one per second) rhythmic bursts, basically saying “ha ha ha...” into the stream. You can make the note as loud and as soft as you want, as long as you keep playing the same note. Be careful not to use your vocal chords, that is, do not produce any sound using your throat. Also make sure you are not moving your cheeks or creating the pulses with your mouth.
After some days of practice you should start working on the same exercise with a metronome. Start slowly, say at 60 BPM, and pulse your long tone once per beat. Just take it easy, and try to do a little every day. Don't overdo it. Although this exercise can be frustrating at the beginning, if you keep at it it will eventually work. Once you have a fair degree of success, you should try to gradually (never, ever, rush through your exercises!) speed the exercise up until you get a smooth “wa-wa-wa” sound. After some time you should be able to start applying the vibrato in a slow étude. When you do this, remember to pay close attention to the dynamics, and feel free to slow the piece down if you think this would help.
Another excellent thing that you can do is to play duets with someone who already does vibrato well, and then imitate them. Listen to recordings of flute players that you like, and keep at your basic exercises. Eventually what you're hearing and what you're doing will start to match up.
One Vibrato, or Many?
The three registers of the flute require different amount of physical effort to obtain similar sounding vibratos. What sounds beautiful for an upper-register vibrato may sound awful in the lower register. By varying the speed at which you change your airflow, you can increase and decrease the speed of the vibrato. As we have already said, the goal is flexibility. You want to develop a wide range of vibrato speeds as well as intensities so that you can apply them to different music and situations. Of course, this takes practice and supervision. As always, the wisdom of a seasoned teacher can be invaluable to a beginner.
Vibrato and Tone Quality
Everyone has strong ideas based on musical taste regarding optimal flute sound. A student should not sound exactly like his teacher or other professionals; instead, every student should develop all the musical controls necessary to make musical choices.
Too often vibrato is just used as a way to disguise a bad sound. Remember that vibrato can change the character of the tone, but not the heart of it. Thus, in order to become a good player, you will need to constantly check the quality of your tone to make sure you are producing the best possible sound. On the long run, this will work wonders on your tone.
Use of Vibrato in Classical Music
Folk music instrumentalists and singers rarely or never use vibrato. In fact, this technique tends only to be used for the so-called “art music”, or for the performance of transcriptions by musician from a classically-trained background.
Although vibrato is usually thought of as an effect added onto the note itself, in some cases it is so fully a part of the style of the music that it can be very difficult for some performers to play without it. Still, the use of vibrato in classical music is a matter of some dispute. For much of the 20th century it was used almost continuously in the performance of pieces from all eras from the baroque onwards. A drastic change in approach took place with the rise of “historically informed performance” from the 1970s onwards.
For example, an interesting observation is that, occasionally, composers up to the baroque period indicated vibrato with a wavy line in the sheet music, which strongly suggests the effect was not desired for the rest of the piece. However, there is no actual proof that musicians performed without vibrato in the baroque era.
At any rate, vibrato is not a modern invention. It began as an ornament, usually produced by the fingers, and only occasionally by the breath. Vibrato as we know it today — a more or less continuous pulsation or shimmer in the tone — originated in the late nineteenth century in Paris. Paul Taffanel and oboist Fernand Gillet were two of the instigators. This may seem surprising in view of the statement in the famous method by Taffanel & Gaubert:
There should be no vibrato or any form of quaver, an artifice used by inferior instrumentalists and musicians. It is with the tone that the player conveys the music to the listener. Vibrato distorts the natural character of the instrument and spoils the interpretation, fatiguing quickly the sensitive ear. It is a serious error and show unpardonable lack of taste to use these vulgar methods to interpret the great composers.
The advent of vibrato in France, around 1905, was the fuel for a great debate. Because it was new, it was often not done very well and was used indiscriminately, and so it got a bad name. Furthermore, flutists had sought for too long, not without difficulty, to find good tone in all registers that was pure, stable and flexible, not to conceive of this perfection as the height of their art.
Vibrato was later brought to the United States by flutists Georges Barrère and Georges Laurent, and by 1940 it had become an accepted part of American orchestral woodwind performance.