Why would the hot water side of the bath tub faucet cause the water line to vibrate when turned partially on?

Answer

If you have the old style seats and washers type shower valve, the water is trying to squeeze between the seat and the washer. If it was air, it would whistle. Water hammers. Usually high pressure contributes to this effect. The only way to solve this problem is to replace your shower valve with a cartridge type shower valve. Or,reduce the pressure in your house.

Or do like my wife told me to do. Our pipes in the bathroom sink have been vibrating for months now. We researched everything possible. She said " try turning up the valve a little". Guess what? it worked. Do the 'KISS' method.(Keep It Simple Stupid)