A steadyrest will work but what is happening is that the piece is being caused to vibrate in phase. That should make sense to him if he is musical. So you can posibly adjust the lathe speed up or down untill the cut dosen’t cause it to vibrate in phase. Also learning to use a skew properly will enable you to turn much thinner stock. When turning it down close to the final dia. with a gouge I will place my thumb on the flute close to the tip and rap my fingers over the turning. Simmilar position for useing the skew as well. Loosen the tailstock a tiny bit after you turn it down some. Choosing the right piece of wood can also be key. some pieces will have stresses that are released after you remove some stock. Not much you can do about that.
