Question

A string clamped at both ends vibrates in its fundamental mode. Is there any position (except the ends) on the string which can be touched without disturbing the motion? What if the string vibrates in its first overtone?

Answer 1

No, because in the fundamental mode all points except the ends are vibrating with different amplitudes. Touching any point other than the ends will disturb the motion. When the string vibrates in the first overtone there is a node also at the midpoint which is stationary. This midpoint can be touched without disturbing the motion.