Q.A string is fixed at the two ends and is vibrating in its fundamental mode. It is known that the two ends will be at rest. Apart from these, is there any position on the string which can be touched so as not to disturb the motion of the string? What will be the answer to this question if the string is vibrating in its first and second overtones?

[!TLDR] In the fundamental mode (1 loop) there is no internal node besides the two ends, so no other point can be touched without disturbing it; the first overtone (2 loops) has one extra node at the centre; the second overtone (3 loops) has two extra nodes, at l/3 and 2l/3. [!ANSWER] Fundamental: none besides the two ends. First overtone: the midpoint. Second overtone: the two points at one-third and two-thirds of the string's length.

A point on a vibrating string can be touched without disturbing the string's motion only if it is already a NODE (a point of permanently zero displacement, section 6.5.2) -- touching any other (moving) point would interfere with its ongoing oscillation. In the FUNDAMENTAL mode (section 6.7.5), the string forms a single loop, with nodes existing ONLY at the two fixed ends -- there is no additional internal node, so there is NO other position that can be touched without disturbing the vibration. In the FIRST OVERTONE (second harmonic, two loops), an additional node exists exactly at the string's MIDPOINT (used, in fact, to set up this very mode by holding the centre still while plucking off-centre, section 6.7.5) -- so the midpoint alone can be touched without disturbing this mode. In the SECOND OVERTONE (third harmonic, three equal loops), there are two additional internal nodes, evenly spaced at one-third and two-thirds of the string's length from either end -- so exactly these two points can be touched without disturbing the vibration in this mode. [!ANSWER] Fundamental: none besides the two ends. First overtone: the midpoint. Second overtone: the two points at one-third and two-thirds of the string's length.

Identify the internal node positions for each mode from the string's standing-wave pattern (section 6.7.5) -- only a node can be touched without disturbing the motion.

Assuming an antinode (the point of LARGEST amplitude) could safely be touched, when in fact touching any point other than a true node will disturb the vibration, since every non-node point genuinely has nonzero displacement.