For both SL and HL, standing waves on strings and in pipes contain fixed nodes and antinodes. A string fixed at both ends and a pipe open at both ends support all harmonics, while a pipe closed at one end supports only odd harmonics.
The Patterns and Their Boundaries
A standing wave forms when two identical waves travelling in opposite directions superpose. At a displacement node, particles remain stationary; at a displacement antinode, particles oscillate with maximum amplitude.
The boundary conditions determine which patterns fit:
| System | Boundary conditions | Allowed patterns |
|---|---|---|
| String fixed at both ends | Displacement node at each end | , so |
| Pipe open at both ends | Displacement antinode at each end | , so |
| Pipe closed at one end | Displacement node at closed end; displacement antinode at open end | , so |
Here, . For a fixed string or open pipe, the first three patterns contain one, two, and three half-wavelengths respectively. Their frequencies are .
For a closed pipe, the first pattern contains one quarter-wavelength. The next allowed patterns contain three and five quarter-wavelengths, producing . Even harmonics cannot satisfy the different conditions at the two ends.
A common misconception is that an open pipe has a node at its open end. This is incorrect for air displacement: an open end is a displacement antinode. Pressure patterns are reversed, so a displacement antinode corresponds to a pressure node.
Exam Technique
When asked to sketch a standing-wave pattern, label every node and antinode and apply the end conditions before finding . Use the IB term first harmonic, not “fundamental” or “overtone,” and state explicitly whether the pipe is open-open or closed-open.