Superposition, coherence and interferenceEdexcel A-Level Physics: Revision notes
Section 1
Wavefronts and phase
A wavefront is a line or surface joining points on a wave that are in phase, for example all the crests of ripples. Wavefronts are perpendicular to the direction in which the wave travels. Circular ripples from a point source have circular wavefronts.
Phase describes how far through its cycle a point on a wave is. One full cycle is 2π radians (360°). The phase difference between two points or two waves is the fraction of a cycle by which one leads the other. Waves with a phase difference of 0 or a multiple of 2π are in phase, and with a phase difference of π (or an odd multiple) they are in antiphase.
Section 2
Superposition
The principle of superposition: when two or more waves meet at a point, the resultant displacement is the sum of the individual displacements at that point (taking direction into account).
After overlapping, the waves continue unchanged.
- Constructive interference: the displacements add to give a larger amplitude, when the waves are in phase.
- Destructive interference: the displacements partly or completely cancel, when the waves are in antiphase. If the amplitudes are equal the resultant is zero.
Section 3
Coherence
Two sources are coherent if they emit waves with the same frequency (and wavelength) and a constant phase difference. This does not mean they must be in phase, only that the phase difference does not change.
A stable interference pattern needs coherent sources. Two separate filament lamps are not coherent: atoms emit short random bursts of light, so the phase difference changes randomly many times a second and the pattern averages out to uniform brightness.
Coherent waves are obtained by using one source split into two (two aerials on one oscillator, two dippers on one motor, one light source through two slits) or by using a laser.
Coherent does not mean 'in phase'. Coherent sources have a constant phase difference, which may be any value.
Section 4
Path difference
The path difference is the difference in the distances travelled by two waves from their sources to the point where they meet. For two sources that are in phase:
- Path difference = nλ (n = 0, 1, 2, ...): waves in phase, constructive interference
- Path difference = (n + ½)λ: waves in antiphase, destructive interference
The path difference gives rise to a phase difference:
phase difference = 2π × (path difference) ÷ λ (in radians)
or 360° × (path difference) ÷ λ in degrees.
If the two sources are themselves in antiphase, the conditions for constructive and destructive interference are swapped.
A path difference of half a wavelength is a phase difference of π, and a path difference of one wavelength is a phase difference of 2π.
Section 5
Worked example
Two loudspeakers driven in phase by one generator emit 850 Hz in air (v = 340 m s⁻¹). A listener is 3.60 m from one and 4.20 m from the other.
λ = v/f = 340 ÷ 850 = 0.40 m
Path difference = 4.20 − 3.60 = 0.60 m = 1.5λ
Phase difference = 2π × 0.60 ÷ 0.40 = 3π rad, which is equivalent to π. The waves are in antiphase, so the listener hears a minimum.
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Superposition, coherence and interference
- Two identical loudspeakers are connected to the same signal generator, which produces a sine wave of frequency 850 Hz. They face a listener in a large room with sound-absorbing walls. The speed of sound in air is 340 m s⁻¹.Calculate the phase difference, in radians, between the two waves at a point where the path difference is 0.10 m.2 marks
- Two small dippers, fixed to the same motor, vibrate in a shallow tank of water at 12 Hz. Each dipper produces circular ripples that travel at 0.18 m s⁻¹. A point P on the water surface is 12.0 cm from dipper A and 15.0 cm from dipper B.Dipper B is replaced by one that vibrates exactly in antiphase with dipper A, at the same frequency. Explain what is now observed at P.2 marks
- A car radio receives a 100 MHz FM signal both directly from a transmitter and by reflection from a tall building. The speed of radio waves is 3.00 × 10⁸ m s⁻¹. At one position on the road the reflected signal has travelled 4.5 m further than the direct signal. Ignore any phase change on reflection.Calculate the wavelength of the signal and the phase difference between the direct and reflected signals at this position. State the effect on the received signal.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).