In the fundamental standing wave on a string fixed at both ends, which statement is true?

• A. There is a node at the center only.
• B. There are nodes at the ends only.
• C. There are three nodes equally spaced.
• D. There is a node at each end and an antinode in the middle.

Answer: (D)

In standing waves on a string fixed at both ends, the ends must have zero displacement, so there are nodes at both ends. For the fundamental mode, the entire length of the string spans exactly half a wavelength, which creates one location of maximum displacement in the middle—an antinode. So the pattern has a node at each end and an antinode in the center. The other statements don’t fit this lowest mode: a node only at the center would ignore the end nodes; nodes only at the ends ignores the middle antinode; three nodes equally spaced describes the second harmonic, not the fundamental. Hence the correct description is a node at each end and an antinode in the middle.