The shape of the lesson
By the end of the lesson, learners can
- Trap a wave between two ends: Step through the harmonics of a string fixed at both ends and watch nodes stay still while antinodes oscillate.
- Nodes and antinodes: The fixed geometry of a stationary wave.
- Strings, pipes and microwaves: Stationary waves form wherever waves reflect and overlap.
- Superposition: displacements add.
- Spacing: adjacent nodes λ/2 apart.
- Energy: stored, not transferred.
One picture every idea returns to
By the principle of superposition, when two waves meet, the resultant displacement is the sum of the individual displacements. Two waves of the same frequency travelling in opposite directions form a stationary wave: points of permanent zero amplitude (nodes) and points of maximum amplitude (antinodes). Adjacent nodes are half a wavelength apart, and a stationary wave stores energy rather than transferring it along its length.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, A wave that stays put.. Send two identical waves toward each other and they superpose into a pattern that does not travel: fixed points that never move and points that swing wildly. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Trap a wave between two ends | Step through the harmonics of a string fixed at both ends and watch nodes stay still while antinodes oscillate. Project tab 1 of the lab (Stationary waves) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Nodes and antinodes | The fixed geometry of a stationary wave. | Pairs |
| 21 to 30 min | Teach and check: Strings, pipes and microwaves | Stationary waves form wherever waves reflect and overlap. | Think-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 14 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Strings, pipes and microwaves, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Think-Pair-Share
Pose the question, give thirty seconds of silent thinking, then a minute to compare with a partner before a few pairs share with the class. The silent minute is the part that cannot be cut.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Adjacent nodes are half a wavelength apart, not a whole wavelength, so the wavelength is twice the node spacing. | A stationary wave does not travel and transfers no net energy along its length, unlike a progressive wave. At a node the two waves are always in antiphase, giving permanent cancellation, while at an antinode they reinforce. |
Stationary waves: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
State the principle of superposition.
Distinguish between a node and an antinode on a stationary wave.
On a stationary wave the adjacent antinodes are 0.25 m apart. Find the wavelength.
A string of length 1.2 m is fixed at both ends and vibrates in its fundamental mode. Find the wavelength of the wave on the string.
State two ways in which a stationary wave differs from a progressive wave.
Microwaves reflected from a metal sheet form a stationary wave with nodes 1.5 cm apart. Taking c = 3.0 × 10⁸ m s⁻¹, find the wavelength and frequency.
Answer key and marking notes
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements ✓
- A node is a point of permanently zero amplitude ✓
- An antinode is a point of maximum amplitude ✓
- Adjacent antinodes are λ/2 apart ✓
- λ = 2 × 0.25 = 0.50 m ✓
- The fundamental has a node at each end and one antinode, so the length is half a wavelength ✓
- λ = 2L = 2 × 1.2 = 2.4 m ✓
- A stationary wave does not transfer energy along its length, while a progressive wave does ✓
- A stationary wave has fixed nodes and antinodes, whereas every point on a progressive wave has the same amplitude ✓
- λ = 2 × node spacing = 2 × 0.015 = 0.030 m ✓
- f = c / λ = (3.0 × 10⁸) / 0.030 = 1.0 × 10¹⁰ Hz ✓
12 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A stationary wave forms when two waves of the same frequency travel:
- Ain opposite directions
- Bin the same direction
- Cat different speeds
- Dwith different frequencies
A point of zero amplitude on a stationary wave is a:
- Aantinode
- Bcrest
- Cnode
- Dcompression
Adjacent nodes are 0.40 m apart. The wavelength is:
- A0.20 m
- B0.80 m
- C0.40 m
- D1.6 m
A stationary wave, unlike a progressive wave, does not:
- Ahave a wavelength
- Bhave a frequency
- Chave nodes
- Dtransfer energy along its length
A string fixed at both ends vibrates in its first harmonic. The length equals:
- Ahalf a wavelength
- Bone wavelength
- Ca quarter wavelength
- Dtwo wavelengths
The principle of superposition states that the resultant displacement is the:
- Aproduct of the displacements
- Blarger displacement only
- Csum of the individual displacements
- Ddifference always
A tube closed at one end has its shortest resonance when its length is:
- Ahalf a wavelength
- Ba quarter of a wavelength
- Cone wavelength
- Dtwo wavelengths
A stationary wave on a string has 3 antinodes between its fixed ends and is 0.90 m long. The wavelength is:
- A0.30 m
- B0.90 m
- C1.8 m
- D0.60 m
At a node, the two component waves are always:
- Ain antiphase, cancelling
- Bin phase, reinforcing
- Cmoving the same way
- Dabsent
Microwave nodes are 2.0 cm apart. The wavelength is:
- A1.0 cm
- B2.0 cm
- C4.0 cm
- D8.0 cm
The distance between an adjacent node and antinode is:
- Ahalf a wavelength
- Ba quarter of a wavelength
- Cone wavelength
- Da whole number of wavelengths
At an antinode the amplitude is:
- Azero
- Bhalf the node value
- Calways negative
- Da maximum
Quiz key
Two opposite waves superpose into a stationary wave.
Nodes have permanent zero amplitude.
Adjacent nodes are λ/2 apart, so λ = 0.80 m.
A stationary wave stores energy rather than transferring it.
The fundamental fits half a wavelength between the ends.
Displacements add (vector sum) where waves meet.
Node at the closed end, antinode at the open end: length = λ/4.
Three antinodes means three half-wavelengths: 3(λ/2) = 0.90, so λ = 0.60 m.
Permanent cancellation at a node means the waves arrive in antiphase.
λ = 2 × node spacing = 4.0 cm.
Node to next antinode is λ/4.
Antinodes are points of maximum amplitude.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.