How to Draw Standing Waves with n = 3 and n = 5 in a Pipe with One Open End

Introduction to Standing Waves in a Pipe with One Open End

Standing waves in a pipe with one open end and one closed end form a classic pattern in physics and music acoustics. These pipes, called closed at one end or open‑closed pipes, support only odd harmonics and specific resonant wavelengths. For a given harmonic number n, the wavelength is λ = 4L / n, where L is the pipe length, and only odd integer values of n (1, 3, 5, …) are allowed. This restriction creates a characteristic node at the closed end and an antinode at the open end. Drawing these shapes helps visualize pressure and displacement nodes and antinodes, and it connects directly to real instruments and experimental setups. Below is a clear, repeatable method to draw the n = 3 and n = 5 patterns accurately.

Key Definitions and Boundary Conditions

Closed End vs Open End Behavior

A closed end must be a displacement node because air molecules cannot move past the sealed wall. At the same time, it is a pressure antinode because pressure can build up there. An open end behaves as a displacement antinode, where air moves with maximum amplitude, and a pressure node, where pressure matches the outside atmosphere. These rules hold for all normal modes in a pipe closed at one end and are the foundation for sketching any standing wave pattern.

Allowed Harmonics and Wavelength Formula

Only odd multiples of the fundamental frequency are supported. The harmonic number n takes values 1, 3, 5, 7, …, and the corresponding wavelength is given by λ = 4L / n. The fundamental (n = 1) has wavelength λ₁ = 4L, the first overtone (n = 3) has λ₃ = 4L / 3, and the second overtone (n = 5) has λ₅ = 4L / 5. This formula ensures the correct number of quarter‑wavelength fits inside the pipe, matching the boundary conditions.

How to Draw the n = 3 Standing Wave

For n = 3, the pipe contains one‑half of a wavelength more than the fundamental, fitting three quarter‑wavelengths into the length L. Start by marking the closed end on the left as a displacement node, then place an antinode at the open end on the right. Add a second node at L/3, a second antinode at 2L/3, and finally the third node cannot exist inside the pipe because the pattern ends at the open end. The displacement wave looks like one and a half sine waves, with the center of the pipe at an antinode. To sketch it, draw a smooth half‑sine from node to antinode, then a half‑sine back to node, then another half‑sine to the open end antinode. Pressure nodes and antinodes are shifted by a quarter wavelength from displacement features, so pressure antinode sits at the closed end and pressure node at the open end.

Step‑by‑Step Displacement Sketch for n = 3

1. Draw a horizontal axis for position, labeling 0 at the closed end and L at the open end.
2. At x = 0, mark a node (displacement zero).
3. At x = L/3, mark another node.
4. At x = 2L/3, mark an antinode (maximum displacement).
5. At x = L, mark an antinode at the open end.
6. Connect the points with a smooth curve resembling one and a half sine lobes, ensuring the curve passes through nodes and peaks at antinodes.

How to Draw the n = 5 Standing Wave

For n = 5, the pipe holds five quarter‑wavelengths, producing a more oscillatory pattern. The closed end remains a displacement node and an antinode in pressure, while the open end remains a displacement antinode and a pressure node. You will have additional nodes at L/5 and 2L/5, and additional antinodes at 3L/5 and 4L/5. The pattern looks like two and a half sine waves fitting within the pipe. When drawing, keep the wavelength λ₅ = 4L/5 in mind and ensure the curve crosses zero displacement at each node and peaks at each antinode.

Step‑by‑Step Displacement Sketch for n = 5

1. Draw a horizontal axis from 0 to L, marking the closed end at 0 and the open end at L.
2. Place displacement nodes at x = 0, L/5, 2L/5, and note that the pattern ends at the open end.
3. Place displacement antinodes at x = L/2? Not exactly; use 3L/5 and 4L/5, consistent with λ = 4L/5.
4. Connect the points with a smooth curve forming two and a half sine cycles, maintaining correct node and antinode positions.
5. Label the open end as a displacement antinode and remember the phase relationship between displacement and pressure.

Visualization Tips and Common Pitfalls

• Always place a node at the closed end and an antinode at the open end for displacement; reverse for pressure.
• Do not draw symmetric sine waves that assume both ends are open or both ends are closed; the boundary conditions differ.
• Use quarter‑wavelength segments to check your drawing: each segment should span L/n for odd n.
• Label axes clearly with positions such as L/3, L/5 and mark nodes and antinodes explicitly.
• Remember that increasing n increases the number of nodes and antinodes but shortens the wavelength proportionally to 1/n.

Relationship to Real Instruments and Experiments

Clarinet-like instruments behave approximately like pipes closed at one end, so their resonant frequencies follow the odd‑harmonic series. Observing nodal and antinode positions helps explain timbre differences and why only certain overtones are strong. In laboratory setups, microphones and sensors can map pressure variations, confirming that pressure antinodes align with displacement nodes. Understanding how to draw these patterns supports both interpretation of experimental data and design of acoustic devices.

Practice Checklist and Quick Reference

Use this checklist when sketching any closed‑at‑one‑end standing wave:

• Confirm the pipe type and boundary conditions.
• Write down the allowed n values (only odd integers).
• Calculate wavelength λ = 4L / n.
• Mark displacement node at closed end and antinode at open end.
• Divide the length into n quarter‑wavelength segments.
• Add intermediate nodes and antinodes at correct fractions of L.
• Draw a smooth curve through the points, matching sine wave shape.
• Optionally annotate with pressure nodes and antinodes.