Stationary Waves Part 2: Open–Open and Open–Closed Systems
Specification: OCR A H556 | Section: 4.4.4 Stationary waves | This page: a focused follow-on from Part 1, concentrating on air columns, allowed harmonics, phase behaviour and exam traps
- compare open–open and open–closed systems without re-covering all of Part 1
- recognise the node/antinode boundary conditions at each end of a pipe
- derive the allowed wavelengths and frequencies in both systems
- understand phase relationships in the loops of each stationary wave pattern
- spot common OCR exam traps, especially about odd harmonics and end conditions
Big idea: once you know what each end of the tube must be — node or antinode — the whole harmonic pattern becomes predictable.
Boundary conditions: what each end must do
You already met nodes and antinodes in Part 1. The main job here is to use them correctly in air columns.
Open end
An open end is a displacement antinode. Air can vibrate most freely there.
Closed end
A closed end is a displacement node. The air at the boundary cannot move back and forth.
Why this matters
These end conditions decide which harmonics are allowed and which are impossible.
For air columns, it is safest to say open end = antinode and closed end = node. That is the language most useful for stationary wave diagrams and harmonic questions.
Open–open systems
An open pipe has an antinode at both ends. So the simplest possible pattern fits half a wavelength inside the pipe.
Higher harmonics fit whole numbers of half-wavelengths into the length of the pipe.
So an open–open pipe supports all harmonics: 1st, 2nd, 3rd, 4th, and so on.
Open–open behaves like a string fixed at both ends in the mathematics of allowed harmonics, even though the physical end conditions are antinode–antinode rather than node–node.
Open–closed systems
An open–closed pipe must have a node at one end and an antinode at the other. So the smallest possible pattern is a quarter wavelength.
The next allowed pattern must still keep one end as a node and the other as an antinode. That means only odd multiples of a quarter wavelength fit.
f = v/4L, 3v/4L, 5v/4L, …
So an open–closed pipe supports odd harmonics only: 1st, 3rd, 5th, 7th …
Students often say “the second harmonic of a closed pipe”. In standard OCR A treatment, there is no second harmonic for an open–closed pipe. The next allowed mode after the fundamental is the third harmonic.
Open–open vs open–closed
| Feature | Open–open pipe | Open–closed pipe |
|---|---|---|
| End conditions | Antinode at both ends | Node at closed end, antinode at open end |
| Fundamental pattern | Half wavelength in the pipe | Quarter wavelength in the pipe |
| Fundamental wavelength | λ1 = 2L | λ1 = 4L |
| Fundamental frequency | f1 = v / 2L | f1 = v / 4L |
| Allowed harmonics | All: n = 1, 2, 3, 4 … | Odd only: n = 1, 3, 5 … |
| General form | fn = nv / 2L | fn = nv / 4L for odd n only |
For the same length L, the fundamental frequency of an open–open pipe is twice that of an open–closed pipe.
Pipe mode explorer
This animation is meant to help you see which patterns are allowed, what the end conditions force the wave to do, and why even harmonics fail in an open–closed pipe.
Phase relationships
Phase questions are easy to mishandle unless you focus on the stationary wave loops.
- All points between two adjacent nodes oscillate in phase.
- Points in neighbouring loops are in antiphase.
- The same principle works in both open–open and open–closed systems.
For example, in the second harmonic of an open–open pipe, the two sections of the wave are in antiphase. In the third harmonic of an open–closed pipe, the two moving sections on either side of the internal node are in antiphase.
Do not say that all points in the whole stationary wave are in phase. They are only in phase within the same loop between adjacent nodes.
End correction: when it matters
For many OCR A harmonic questions, you can ignore end correction unless it is mentioned explicitly. But in practical measurements with resonance tubes, the displacement antinode at an open end is slightly outside the physical end of the tube.
This means the effective length of the air column is a little longer than the measured tube length.
Include end correction only when dealing with measured resonance lengths or when the question explicitly hints at it. For standard harmonic sketching and most theoretical frequency questions, the simple formulas are usually enough.
Worked examples
Knowledge Check
- Open end = displacement antinode
- Odd harmonics only
- Open–open
- They are in antiphase / 180° out of phase
Exam-Style Questions
- The closed end must be a node
- The open end must be an antinode
- Only odd numbers of quarter wavelengths satisfy these boundary conditions
- Even harmonics would require an incorrect end condition at one end
(a) Calculate the fundamental frequency. [2 marks]
(b) State the frequency of the third harmonic. [1 mark]
(c) Determine the wavelength of the third harmonic. [2 marks]
- (a) f1 = v / 2L = 340 / (2 × 0.68) = 250 Hz
- (b) f3 = 3 × 250 = 750 Hz
- (c) λ3 = 2L/3 = 1.36 / 3 = 0.453 m
- Even harmonics are not allowed in an open–closed pipe
- The next allowed mode after the fundamental is the third harmonic
- This is because the pattern must keep a node at one end and antinode at the other
(a) Show that the tuning fork frequency is 500 Hz. [3 marks]
(b) State the next allowed resonant frequency for the same air column. [1 mark]
- (a) For first resonance, L = λ/4 so λ = 4L = 0.680 m
- f = v/λ = 340 / 0.680 = 500 Hz
- (b) Next allowed harmonic is the third, so frequency = 1500 Hz
- Within each loop, points oscillate in phase
- Adjacent loops are in antiphase
- The second harmonic of an open pipe has two loops
- The third harmonic of a closed pipe has three quarter-wave sections / two adjacent loop changes giving alternating phase regions
Describe two important differences in their harmonic series.
- The flute supports all harmonics, whereas the clarinet supports only odd harmonics
- The fundamental frequency of the flute is twice that of the clarinet for the same length
- The overtone pattern is therefore more closely spaced in the flute
- This contributes to different timbre / tone quality
Topic Summary
Open end
Displacement antinode.
Closed end
Displacement node.
Open–open pipe
All harmonics allowed; fn = nv / 2L.
Open–closed pipe
Odd harmonics only; f = v / 4L, 3v / 4L, 5v / 4L …