Physics HL · Chapter 15: Standing Waves and Resonance
15.3 Standing Waves in Pipes
Use open/closed end conditions to derive allowed modes in air columns and handle odd-harmonic restrictions correctly.
Estimated time: 46 minutes
Boundary Conditions in Air Columns
In pipe problems, always define which physical quantity your node/antinode statement refers to. For displacement-based diagrams used in this chapter, open ends behave as displacement antinodes and closed ends as displacement nodes. Once this choice is fixed, the same geometric fitting logic used for strings applies directly.
An open-open pipe has antinode-antinode boundaries, while a closed-closed pipe has node-node boundaries. Those two cases share the same wavelength family because both impose symmetric boundary pairs. An open-closed pipe is asymmetric and therefore produces a different allowed sequence.
Open-Open and Closed-Closed Pipes
lambda_n = rac{2L}{n},quad f_n = rac{nv}{2L},quad n=1,2,3,ldots
Same mathematical family as fixed-fixed strings because both boundaries are of the same type.
Mode sketches still matter here: they show where particle displacement is zero and where it is maximal. In wind-instrument contexts, these mode shapes determine which resonances are strongly excited for a given excitation mechanism. The formula alone does not tell you coupling strength or sound quality.
Open-Closed Pipes and Odd Harmonics
lambda_n = rac{4L}{n},quad f_n = rac{nv}{4L},quad n=1,3,5,ldots
Only odd harmonic numbers satisfy one-node one-antinode boundary conditions.
The odd-only rule is a high-frequency exam trap. The next mode after n=1 is n=3, not n=2. So consecutive resonance frequencies differ by 2f_1, not f_1. If you use integer-step harmonic counting from symmetric-boundary cases, your answers will be systematically wrong.
Resonance-tube experiments often measure successive loud lengths. The spacing between adjacent resonance lengths in a fixed-frequency setup is lambda/2, even in one-open-one-closed cases, because moving from one allowed mode to the next odd mode adds half a wavelength to the air-column length.
Note
In many practical contexts you also track pressure nodes/antinodes, which invert relative to displacement. Keep the variable explicit to avoid diagram contradictions.
Simulation: Pipe Modes for Open and Closed Ends
Switch boundary types, sweep harmonic indices, and compare allowed wavelength and frequency families.
Mode number
3
Allowed λ
1.267 m
Resonant f
268.421 Hz
Period
3.73e-3 s
Sound speed
340 m/s
Use this lab as a model-switching tool: start from boundary conditions, identify allowed modes, and then link harmonic equations to measurable frequencies and damping-limited resonance response.
Test Yourself
An open-closed pipe has length 0.85 m and sound speed 340 m/s. Enter the first-harmonic frequency in Hz.
Hint: For open-closed first mode, lambda_1 = 4L.