Theme C — Wave Behaviour
C.4.1 Standing Waves — IB Physics HL Revision Notes & Topical Questions
5 min read 8 parts Lesson 27 of 37 1 quiz
Curriculum Equation Key Wave Speed & Doppler Effect FormulasFast Exam Solution
v = f × λ | f_obs = f × (v ± v_o) / (v ∓ v_s) | T = 1 / f
Lesson summary
C.4.1 Standing Waves — IB PHYSICS SL notes: Explain how a standing wave forms from two identical waves travelling in opposite directions and contrast it with a travelling wave.Standing (stationary) wave — a wave pattern produced when two waves travelling in opposite directions al…
- Requirements
- Usual method
- Any wave type
- Appearance
- Non-example
- Increasing frequency
Part 1 of 6Formation
Formation
Standing (stationary) wave
- Standing (stationary) wave a wave pattern produced when two waves travelling in opposite directions along the same line superpose
Requirements
- Requirements the waves must have the same frequency and wavelength, and similar amplitude
Usual method
- Usual method a travelling wave reflects from a boundary and the reflected wave overlaps the incident wave
Any wave type
- Any wave type superposition applies to transverse and longitudinal waves
Examples
- Examples a vibrating guitar string, the air column in an organ pipe, microwaves reflected from a metal sheet and reflections in a narrow water tank
Appearance
- Appearance the pattern does not move along the medium
- Each point oscillates in place with its own fixed amplitude.
- Crests and troughs only move vertically.
Non-example
- Non-example if the half-cycles of the profile are not symmetrical, the incident and reflected waves never cancel at any point,
- so no standing wave forms
Part 2 of 6Resulting pattern
1
Nodes
- Nodes points where the two waves are always in antiphase, so the displacement is permanently zero
2
Antinodes
- Antinodes points where the waves are always in phase, so the amplitude is greatest
- It is up to twice the amplitude of either component wave.
3
Spacing
- Spacing adjacent nodes are λ/2 apart and a node and neighbouring antinode are λ/4 apart
4
Amplitude
- Amplitude it varies smoothly between a node and the next antinode
5
Phase
- Phase all points between two adjacent nodes oscillate in phase and are in antiphase with the neighbouring segment
6
Timing
- Timing when all displacements are zero the string is flat, and a quarter of a period later every antinode is at its peak
7
Frequency
- Frequency every point oscillates at the same frequency, with an amplitude fixed by its position
Part 3 of 6Comparison with a travelling wave
| Feature | Travelling wave | Standing wave |
|---|---|---|
| Energy | Transferred along the medium | Not transferred; stored in the oscillations |
| Amplitude | Same for all points | Zero at nodes to maximum at antinodes |
| Phase | Changes steadily along the wave | Same within a segment, opposite in neighbouring segments |
| Wavelength | Distance between points in phase | Twice the distance between adjacent nodes |
Part 4 of 6Energy in a standing wave
1Net flow
- Net flow it is zero past any point, because two equal waves carry energy in opposite directions
2Storage
- Storage energy is stored within each segment
- It is kinetic when the string passes through equilibrium and moves fastest.
- It is elastic potential when the string is at maximum displacement and momentarily at rest.
3Nodes
- Nodes no energy passes through a node
4Losses
- Losses absorption or radiation must be topped up by the source, otherwise the standing wave dies away
Part 5 of 6Demonstrations
String
- String fix one end and shake the other with a vibration generator
- At certain frequencies the string breaks into clear loops.
- The vibrator end moves only slightly, so it is treated as an approximate node.
Increasing frequency
- Increasing frequency the first clear pattern appears at the fundamental, then at twice that frequency, and so on
- Between these resonant frequencies the string moves only slightly.
Graph
- Graph plotting successive resonant frequencies against n gives a straight line through the origin with gradient f1
- The wave speed follows from v = 2L × f1.
Microwaves
- Microwaves reflection from a metal plate gives nodes that a small probe can locate
- The node spacing gives λ/2, and hence the speed or frequency.
Part 6 of 6More from the specification
More from the specification
Stationary wave amplitude scale
- Stationary wave amplitude scale superposition of opposing identical coherent waves creates spatial envelopes varying in displacement on a macroscopic scale.
Formation of standing waves by superposition
- Standing waves are produced when two identical waves travelling in opposite directions with equal frequency and amplitude superpose.
Nodes and antinodes
- A standing wave contains nodes where displacement is always zero and antinodes where amplitude of oscillation reaches a maximum.
Resonant modes in strings and pipes
- Standing wave harmonics in stretched strings and acoustic air columns depend on boundary conditions.
- (clamped ends or open/closed pipe terminations)
Natural resonant frequencies
- Natural resonant frequencies an oscillating system driven by an external periodic force resonates when driving frequency matches its natural harmonic frequencies.
Harmonic boundary conditions
- Harmonic boundary conditions pipes with one closed end produce only odd harmonics, whereas open pipes generate all integer harmonic multiples.
Damped wave systems
- Damped wave systems resistive damping forces remove mechanical energy from oscillating systems, decreasing resonance peak amplitude.
Heavy damping
- Damping greater than critical, giving a slow return without oscillation
Resistive force
- Friction and drag opposing motion.
Damping
- Energy loss to friction that reduces amplitude.
Harmonic
- Resonant mode of a stationary wave, with frequency a whole-number multiple of the fundamental where allowed.
Force
- A push or pull measured in newtons.
Natural frequency
- The frequency at which a system oscillates freely
Driving frequency
- The frequency of the periodic external force
Boundary condition
- Constraint at an edge fixing whether there is a node or antinode.
Forced oscillation
- Oscillation caused by a periodic driving force
Progressive wave
- Wave that transfers energy without transferring matter.
Key terms
| Term | Meaning |
|---|---|
| Standing wave | Pattern of fixed nodes and antinodes from two opposite travelling waves. |
| Node | Point of zero amplitude. |
| Antinode | Point of maximum amplitude. |
| Superposition | Resultant displacement is the sum of the individual ones. |
Avoid these mistakes
1
Taking the node spacing as one wavelength — it is half a wavelength.
2
Saying a standing wave transfers energy along the string.
3
Thinking all points on a standing wave have the same amplitude.
4
Believing nodes are where the string moves fastest — they are always at rest.
5
Assuming points on opposite sides of a node are in phase — they are in antiphase.
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