Apply the wave model to the wave equation, superposition, standing waves, resonance and beats

A focused answer to the WACE Year 12 Physics Unit 4 dot point on wave motion and superposition. The wave equation, the principle of superposition, standing waves on strings and in pipes, resonance and beats.

Generated by Claude Opus 4.78 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

WACE wants you to use the wave model quantitatively, combine overlapping waves by superposition, and explain standing waves on strings and in air columns, plus the related ideas of resonance and beats. The unifying tool is that displacements add point by point, while the wave speed is fixed by the medium.

The wave equation

For any periodic wave the speed, frequency and wavelength are linked by

v=f\lambda,

where $f$ is in hertz and $\lambda$ in metres. The frequency is set by the source; the speed is set by the medium. So if a wave passes into a new medium and slows down, its frequency stays the same and its wavelength shrinks. The period is $T=\frac{1}{f}$ .

Superposition

When two or more waves meet, the resultant displacement at each point is the vector sum of the individual displacements. This is the principle of superposition. Where crests align with crests the waves reinforce (constructive interference); where a crest meets a trough they cancel (destructive interference). After overlapping, the waves pass through unchanged.

Standing waves

A standing wave forms when a wave reflects and superposes with itself in a bounded medium, producing fixed nodes (zero displacement) and antinodes (maximum displacement). Only certain wavelengths fit the boundaries, giving a discrete set of harmonics.

For a string fixed at both ends, length $L$ must hold a whole number of half-wavelengths,

L=\frac{n\lambda}{2}\quad\Rightarrow\quad f_n=\frac{nv}{2L},\quad n=1,2,3,\dots

The same applies to a pipe open at both ends. A pipe closed at one end must have a node at the closed end and an antinode at the open end, so it holds odd quarter-wavelengths and only odd harmonics are present,

f_n=\frac{nv}{4L},\quad n=1,3,5,\dots

Resonance

Every system has natural frequencies (its harmonics). When an external driving force oscillates at one of these natural frequencies, energy transfers efficiently and the amplitude grows large: this is resonance. It explains why a guitar body amplifies a plucked string and why air in a tube sings at particular pitches.

Beats

When two waves of slightly different frequencies $f_1$ and $f_2$ superpose, the amplitude rises and falls periodically. The number of these loudness pulses per second is the beat frequency,

f_{beat}=|f_1-f_2|.

Musicians use beats to tune instruments: the beats slow and vanish as two notes are brought into unison.

Choosing the right boundary model

Sketch the standing wave before calculating. Decide whether the medium has two fixed (or two open) ends, which support all harmonics, or one closed end, which supports only odd harmonics with a different formula. The diagram fixes how many half- or quarter-wavelengths fit the length.

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