PHYC10003 Chap.8 Superposition and Standing Waves
Superposition and Standing Waves
Superposition adds signed displacements wherever waves overlap. Equal coherent waves travelling in opposite directions can create a standing pattern with fixed nodes and oscillating antinodes. This chapter connects those patterns to boundary conditions, fixed-string harmonics and resonance, then separates displacement nodes from pressure nodes and particle speed from phase speed.
Superposition is applied point by point and time by time. Constructive interference reinforces like-signed displacements, while destructive interference reduces the resultant; neither process erases the individual ideal waves. Displacement nodes remain fixed because cancellation occurs there at every time, while antinodes reach the largest oscillation amplitude.
Adjacent nodes are half a wavelength apart, and a node to its nearest antinode is one quarter wavelength. Fixed endpoints require nodes, so a string length contains an integer number of half-wavelength loops. This boundary rule determines allowed wavelengths and harmonic frequencies. Resonance occurs when a drive matches an allowed natural mode closely enough for a large response, with damping controlling its sharpness.
Particles away from nodes still move even though the amplitude envelope does not travel. In sound, pressure variation and particle displacement have different node positions, so an observed pattern must be tied to its physical variable before labels are assigned. A sketch of endpoints, loops, nodes, and antinodes checks the algebra before any harmonic formula is used.
What this chapter covers
- 01
Linear superposition
- 02
Constructive and destructive interference
- 03
Nodes and antinodes
- 04
Fixed-end boundary conditions
- 05
String harmonics
- 06
Resonance and particle speed
Worked example · free
Solve a fixed-string harmonic
- 1Write length as three half wavelengths.
- 1Solve wavelength as 0.80 m.
- 1Calculate frequency as 60 Hz.
- 1State adjacent node spacing as 0.40 m.
Key terms
- Superposition
- Superposition is the point-by-point algebraic addition of overlapping wave displacements.
- Node
- A displacement node is a fixed point of zero displacement in an ideal standing mode.
- Antinode
- An antinode is a location of maximum displacement amplitude in a standing wave.
- Harmonic
- A harmonic is an allowed mode whose frequency is an integer multiple of the fundamental for fixed endpoints.
- Resonance
- Resonance is a large steady response when driving frequency matches an allowed natural frequency closely.
Superposition and Standing Waves FAQ
Do waves disappear during destructive interference?
No. Their signed displacements cancel during overlap at particular places and times, but in an ideal linear medium each travelling wave continues afterward with its original form. Permanent standing-wave nodes require cancellation at the same positions throughout the motion, not at one instant.
How far apart are neighbouring nodes?
For a sinusoidal standing wave, adjacent displacement nodes are half a wavelength apart. A node to the nearest displacement antinode is one quarter wavelength along the medium.
Why must I identify pressure or displacement nodes?
In sound waves, pressure variation and particle displacement do not have nodes at the same positions. A visible or measured pattern must be tied to its physical quantity before its nodes are labelled.
Exam move
Begin superposition practice by adding signed displacements at selected positions and times. Use cases with reinforcement, exact cancellation, and partial cancellation, then explain why each travelling contribution continues after overlap in the ideal linear model. To study standing waves, sketch both extreme shapes and the equilibrium line.
Mark fixed nodes and antinodes, noting that neighbouring loops move in opposite phase while points inside one loop move together. Sketch the endpoints and count half-wavelength loops before using a harmonic formula. Translate loop count into mode number, wavelength, and frequency in that order. Verify that adjacent nodes are half a wavelength apart and that a node to its nearest antinode is one quarter wavelength.
Change length, speed, or mode number separately and predict the frequency trend before calculating. For resonance, treat the drive frequency as a compatibility test with allowed boundary patterns, not as permission to ignore the endpoints. Compare a matched drive with one between modes and describe how the stable node pattern changes.
For particle-speed questions, differentiate the local displacement in time and keep that result separate from travelling-wave phase speed. In sound demonstrations, write pressure or displacement beside every node label before interpreting the pattern. Use the physical boundary type to infer which quantity has a node or antinode there, since pressure and displacement patterns are spatially offset.
Conclude each mode calculation with a sketch-based check: the stated number of loops must fit the length, endpoints must satisfy their conditions, and the frequency must equal wave speed divided by the inferred wavelength.
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