Final answer:
For the fundamental overtone in a pipe closed at one end and open at the other, the displacement node is at the closed end and the antinode is at the open end. The length of the tube is one-fourth of the wavelength. For the fundamental, there are no other nodes along the pipe.
Step-by-step explanation:
To determine the locations of the displacement nodes for the fundamental overtone in a pipe that is closed at the left end and open at the right end, we must first understand the concept of standing waves. A standing wave has nodes, points of minimum amplitude, and antinodes, points of maximum amplitude. For a tube closed at one end and open at the other, the fundamental vibration creates a quarter-wave resonance. The node occurs at the closed end, and the antinode occurs at the open end.
The length of the tube is then equal to one-fourth of the wavelength of the fundamental overtone. If the wavelength (λ) is known, we can express the length (L) of the tube as L = λ/4, which would also be the distance from the closed end to the first antinode. However, since there's only one node at the closed end, there are no other nodes to locate along the length of the tube for the fundamental frequency.
For higher overtones, or harmonics, nodes and antinodes will occur at different locations, with nodes still forming at the closed end and antinodes at the open end. For instance, the first overtone will have a node halfway along the tube and an antinode at both the open and closed ends, corresponding to a wavelength equal to three times the length of the tube.