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

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

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,

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,

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,

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

Constructive and destructive interference in detail

The principle of superposition is the engine behind every interference effect in the unit, so it pays to state it precisely. When two waves overlap, the resultant displacement at each point and instant is the algebraic sum of the individual displacements. If the waves are in phase (path difference a whole number of wavelengths), crest aligns with crest and the amplitudes add, giving constructive interference and maximum intensity. If they are exactly out of phase (path difference a half-integer number of wavelengths), crest aligns with trough and they cancel, giving destructive interference and, for equal amplitudes, zero intensity. A crucial point is that the waves emerge unchanged after overlapping; superposition is temporary. This single rule explains two-slit fringes, the nodes and antinodes of standing waves, and the loud-quiet cycle of beats, which is why it is the most important idea to get right in the wave strand.

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.