Describe the superposition of mechanical waves and explain constructive and destructive interference in terms of phase relationships

What this dot point is asking

QCAA wants you to state the principle of superposition for mechanical waves and apply it to constructive and destructive interference. The link between path-length difference and phase is the key idea: integer wavelengths of path difference (for sources in phase) give a maximum, half-integer wavelengths give a minimum.

The principle of superposition

When two or more waves meet at a point, the total displacement of the medium at that point is the sum of the displacements that each wave would produce on its own.

This is a linear principle: it applies to mechanical waves with small amplitudes (water, sound, strings) and to electromagnetic waves in vacuum.

After superposing, the waves continue past one another unchanged. They do not interact, only their displacements add at the moment of overlap.

Constructive interference

When two waves meet in phase (peaks line up with peaks, troughs with troughs), the resultant amplitude is the sum of the individual amplitudes. The waves reinforce.

For two identical waves of amplitude $A$, the resultant amplitude is $2A$ at points of full constructive interference. Wave energy depends on $A^2$, so the energy at a constructive maximum is $4$ times the energy of either wave alone (not $2$ times, because energy scales with the square of amplitude).

Destructive interference

When two waves meet exactly out of phase (peaks line up with troughs), the resultant amplitude is the difference of the individual amplitudes. For identical waves, the resultant is zero: total destructive interference.

Energy is not destroyed. It is redistributed to the constructive maxima elsewhere in the interference pattern.

Path-length difference rule

For two sources oscillating in phase at the same frequency, the type of interference at a point depends on the path-length difference $\Delta d$:

• Constructive: $\Delta d = n \lambda$, where $n = 0, 1, 2, 3, \ldots$
• Destructive: $\Delta d = (n + \tfrac{1}{2}) \lambda$, where $n = 0, 1, 2, 3, \ldots$

If the sources are out of phase by half a cycle, swap the two rules.

Where this leads next

Superposition is the foundation for standing waves (next dot point), for diffraction patterns (which you will meet in Year 12), and for double-slit interference of light (Unit 4 quantum context). The same path-difference rule applies in all three.

Examples in context

Example 1. A QPAC sound technician sets up two stage speakers $3.0 \text{ m}$ apart playing the same $680 \text{ Hz}$ test tone ($\lambda = 0.50 \text{ m}$). At a point $4.0 \text{ m}$ from one speaker and $4.25 \text{ m}$ from the other, path difference is $0.25 \text{ m} = \lambda/2$, so destructive interference produces a quiet patch. Moving $0.25 \text{ m}$ sideways shifts the path difference to $0.50 \text{ m} = \lambda$ for constructive (loud). QCAA EA Unit 2 thematic items use this two-source setup.

Example 2. A Sunshine Coast tidal study models wave interaction between an ocean swell and a tide-driven counter-flow at a river mouth. When the two surface-wave trains meet $180^\circ$ out of phase at $1.2 \text{ Hz}$, amplitudes near-cancel; in phase they double, creating standing crests visible to drone surveys. The same superposition principle underpinning QCAA Unit 2 IA1 data-test stems works at any wavelength.

Try this

Q1. State the principle of superposition. [2 marks]

• Cue. When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements.

Q2. Two speakers emit $1700 \text{ Hz}$ tones in phase. Calculate the wavelength using $v = 340 \text{ m s}^{-1}$ and the path-difference required for destructive interference. [3 marks]

• Cue. $\lambda = 0.20 \text{ m}$; destructive at $(n+\tfrac{1}{2})\lambda = 0.10, 0.30, 0.50 \text{ m}$.

Q3. A QPAC two-speaker setup at $680 \text{ Hz}$ ($v = 340 \text{ m s}^{-1}$) is placed $3.0 \text{ m}$ apart. (a) Calculate the wavelength. (b) Determine the path-difference for the first constructive maximum off-axis. (c) Explain why such interference is rarely heard outdoors and discuss one design consideration. [2+3+3 marks; ISMG: Analysis and interpretation, Evaluation]

• Cue. (a) $0.50 \text{ m}$; (b) $\Delta = \lambda = 0.50 \text{ m}$; (c) reflections off ground/walls scramble phase, design uses directional baffles.