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This is probably the most simple exercise you are ever going to hear about, but perhaps the most beneficial, as well. Your current idea of a long tone may be 6 seconds long at the most. You should keep practicing those, but this exercise is different. In this exercise, simply sustain a note as long as you can, as quietly as you can, as evenly as possible. This needs to be really quiet, almost like a hum. In a practice session, try to do this exercise at least once in each range.
Extended long tones are great for developing breath control. Most breathing exercises focus on the intake of air while doing nothing for the control of exhalation, when the sound is actually being produced. To produce an absolutely even sound at an extremely soft dynamic requires an extremely high level of breath control. At a mezzo forte, a trombonist's sound may be clear and quite steady, but that is because it takes very little breath control to produce such a sound. If the student takes it down to as soft as they can play without the sound breaking up entirely, they will hear small aberrations and wobbliness in the sound they could not hear otherwise. By trying their very best to smooth these out through control of the breath, they will be greatly increasing their breath control.
Extended long tones are an excellent way to get the body prepared for the type of breathing required for playing the trombone. Normal breathing throughout the day is more or less even, shallow, and constant. On a wind instrument, inhalation is relatively short and deep and exhalation long and deep. A few extended long tones at the beginning of a practice session can get the body acclimated to the type of breathing required for playing a wind instrument. Since the body has had an abundance of air during the day, it feels no great need to breathe deeply. If a player truly plays the long tone for as long as they possibly can, they will find themselves almost gasping for air after they finish each tone. The breath following an extended long tone will feel so wonderful, the trombonist will want to repeat that feeling of deep breathing throughout the practice session. The player will also become more aware of how much air they can squeeze out of their lungs when they need to extend a phrase and have no chance to breathe.
The hardest part of any extended long tone is the beginning of the tone. Having filled your lungs to the brim with air, it takes a great deal of control to keep that air from exploding out of your lungs when you start your note. Developing this control will train your body and mind to have absolute control over the amount of air being expended. With this control in place, a player can perform with greater dynamic nuance.
Embouchure muscles are strengthened like any other muscle - through breaking down of the muscle through use, then resting and rebuilding. When a trombonist has only a limited time to practice but wants to break down the muscles of the embouchure as they would in a longer practice session, they can do this very quickly by playing extended long tones. Extended long tones require great embouchure dexterity and will tire you out very quickly, especially if played in the high range for any length of time.
Extended long tones also help to train the embouchure to vibrate under conditions where minimal air is present. Getting notes to speak is never difficult when a large quantity of air is being blown. During extended long tones, the absolute minimum of air required to keep the lips vibrating is used. This is quiet beyond a pianissimo. A player who frequently plays extended long tones at the quietest levels will have no trouble playing pianissimo passages with ease. They will have developed both confidence and sensitivity for playing under such conditions.
I would recommend extended long tones to any wind musician wanting to develop breath control, deeper breathing, dynamic nuance, and embouchure strength.
-Zachary Crawford