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 IMATH_9

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

Worked example

Two speakers $1.0$ m apart emit $1.0$ kHz sound in phase. A listener stands $3.0$ m directly in front of one speaker. Find the path-length difference and predict the interference at the listener's position. Use $v = 343$ m s$^{-1}$.

Wavelength: $\lambda = 343 / 1000 = 0.343$ m.

Distance from far speaker: $\sqrt{3.0^2 + 1.0^2} = \sqrt{10} = 3.162$ m.

Path difference: $\Delta d = 3.162 - 3.0 = 0.162$ m.

Ratio: $0.162 / 0.343 = 0.472$.

This is close to a half wavelength, so the listener is near a destructive minimum but not exactly on one. Sound at this location is much quieter than at points where $\Delta d$ is an integer or half-integer multiple of $\lambda$.

Common traps

Treating interference as energy disappearing. Energy is conserved. Nodes (zero amplitude) exist alongside antinodes (large amplitude). Average over a full interference pattern recovers the input energy.

Confusing in-phase with same-amplitude. Constructive interference requires the waves to be in phase. Same-amplitude does not by itself guarantee constructive.

Forgetting that the rule flips for out-of-phase sources. If the sources oscillate $180°$ out of phase (one peak corresponds to the other's trough), integer path differences give minima.

Treating a small-amplitude resultant as no wave. Two waves of slightly different frequencies superpose to give beats (a slowly modulating amplitude). The destructive moments are real instants of zero displacement, not silence.

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.

In one sentence

Mechanical waves obey superposition (the total displacement is the sum of individual displacements at a point), and two sources in phase produce constructive interference at points where the path-length difference is an integer wavelength and destructive interference at points where the difference is a half-integer wavelength.