This is THE easiest question to answer. No gray areas with this one.
Sound IS vibrations.
Air in the sound IS caused from too big an aperture for the note/volume we are playing.
What happens is that some of the air gets through the hole without touching any lip tissue and doesn’t get put into vibration. So it sizzles right through the sound.
Air in the tone goes hand in hand with every bad thing that can happen. Not being able to play soft cleanly or at all, poor range, poor endurance, excessive mouthpiece pressure…….
All BECAUSE the aperture is too big.
Soft playing makes the aperture smaller.
Here are some causes of an airy sound:
A shallow mouthpiece causes this because you bottom out and it spreads the center of the chops.
Fatigue causes it because the chops don’t stay together and they spread.
Moving the lower lip out of the mouthpiece when you breathe also causes this.
Shows in pictures how this happens and how to fix it.
More air usage HIDES it because you are now making the air compress to get through the lips, this is why we never really hear airy tone when playing loud but often do when playing soft.
Different teachers use different things to get through this. I use several but some are easier to explain in a post than others.
Lip buzzing a few minutes a day can help focus the aperture.
Mouthpiece buzzing can also help.
Soft (at a whisper) playing will force you to work on the focus that you currently lack.
Other more physical approaches (embouchure change, lip curl…) will upset your daily playing for a while and should only be tried as a LAST resort after giving the other ideas a few months to work.