Final answer:
The amplitude at a distance of l/3 from one end of a string vibrating in its third overtone is zero because this point corresponds to a node in the standing wave pattern. Neither of the given options is correct.
Step-by-step explanation:
The question relates to the vibration of a string in its third overtone, also known as the fourth harmonic. When a string fixed at both ends vibrates in its third overtone, the standing wave pattern has four antinodes where the amplitude is maximum. To find the amplitude at distance l/3 from one end, we need to understand that the string has a length of one and a half wavelengths in the third overtone.
The position l/3 will be at an antinode or a node depending on the mode of vibration. Since the third overtone has four antinodes, the points at l/3 and 2l/3 will be nodes where the amplitude is zero. Therefore, none of the given options (a, b, c, d) correctly describes the amplitude, as the amplitude at l/3 is zero in the third overtone.
Remember that the string's vibrational patterns follow the principles of standing waves, with certain points along the string (nodes) always remaining at zero displacement, and points in between (antinodes) reaching the maximum amplitude.