Walk slowly between two loudspeakers playing the same steady note and something eerie happens: loud, quiet, loud, quiet. You haven’t changed the volume — you’ve changed how far you are from each speaker. Superposition told us waves add. Interference tells us where they add and where they cancel, and the answer comes down to one number: the path difference.
📘 What you need to know
Interference is the effect observed when two or more waves superpose
Constructive: waves arrive in phase → displacements add → maximum
Destructive: waves arrive in antiphase → displacements cancel → minimum
Sources must be coherent: same frequency and a constant phase difference
Path difference = the difference in distance travelled by the two waves to a point
Constructive: path difference = nλ · Destructive: path difference = (n + ½)λ, with n = 0, 1, 2, 3…
What is interference?
Definition — interference
The effect observed due to the superposition of two or more waves
Superposition is the rule (add the displacements); interference is what you actually see when two waves overlap everywhere at once. Two special cases matter:
Constructive interference — the waves meet in phase, crest on crest or trough on trough. The displacements reinforce, giving a maximum. For two equal waves the resultant amplitude is 2A.
Destructive interference — the waves meet in antiphase, crest on trough. The displacements cancel, giving a minimum. For two equal waves the resultant is zero.
You get this pattern from two coherent sources — two loudspeakers fed the same signal, microwaves through a pair of slits, or a single laser beam split by a double slit.
Coherence
Not any two waves will do. For a stable, watchable interference pattern the sources must be coherent.
Coherent sourcesthe same frequencyand a constant phase difference
Both pairs have the same frequency, but only the left pair keeps a constant phase difference. On the right the second wave jumps phase, so the bright and dark regions would shuffle around faster than the eye can follow.
Two ordinary light bulbs never make an interference pattern, even side by side. Their atoms emit in tiny random bursts, so the phase difference between them changes billions of times a second and the fringes smear into a uniform glow. A laser works because its light is monochromatic (one frequency) and coherent. Same trick with the double slit: one source, split in two, so whatever the phase does it does to both.
Path difference
Definition — path difference
The difference in distance travelled by two waves, from their sources to the point where they meet
Both waves leave their sources in step. If one then travels exactly a whole number of wavelengths further than the other, it arrives back in step — crest lands on crest. If it travels half a wavelength further (or one and a half, or two and a half…), it arrives exactly out of step — crest lands on trough.
Count the wavelengths along each path. Six against six and a half: the extra half wavelength flips one wave upside down relative to the other, so they cancel at P. Both trains here are drawn to scale, at exactly 6.000λ and 6.500λ.
Where n = 0, 1, 2, 3… is any integer, and λ is the wavelength. Note that a path difference of zero counts as constructive (n = 0) — that’s why the point midway between two speakers is always loud.
Two coherent sources
measure the
Path difference
divide by λ
Whole → loud Half → quiet
Don’t confuse them:path difference is a distance (measured in metres, or in wavelengths). Phase difference is an angle (degrees or radians) describing how far through a cycle one wave is ahead of the other. A path difference of λ corresponds to a phase difference of 360°.
The interference pattern
Now let two coherent point sources ripple away together and look at the whole picture. Where a crest from one meets a crest from the other, the water leaps; where a crest meets a trough, it stays flat. Those points join up into lines of maxima and lines of minima fanning out across the tank.
Solid teal arcs are crests, dashed grey arcs are troughs. Blue and amber dots are constructive (crest+crest and trough+trough); green dots are destructive. Every point on the dotted centre line is the same distance from both sources, so its path difference is zero.
To decide what happens at any point on such a diagram, you don’t need a ruler. Just count the wavefronts from each source and subtract.
Path difference
Waves arrive
Phase difference
Result
0
In phase
0
Constructive (n = 0)
½λ
In antiphase
180°
Destructive (n = 0)
λ
In phase
360°
Constructive (n = 1)
1½λ
In antiphase
540°
Destructive (n = 1)
2λ
In phase
720°
Constructive (n = 2)
🔢 Deciding constructive or destructive
Find the two distances from the point back to each source (or count wavelengths on a wavefront diagram).
Subtract to get the path difference. Take the positive value.
Divide by λ. A whole number → constructive. A whole number plus a half → destructive.
Read off n. For 2λ, n = 2. For 2.5λ = (2 + ½)λ, so n = 2.
WE 1
On a wavefront diagram from two coherent sources, the number of wavelengths from each source to three points is measured. For each, determine the path difference, the value of n, and the type of interference. X: 4λ and 4λ · Y: 3.5λ and 5λ · Z: 6λ and 4λ
Step 1 — point X: subtract the distances4λ − 4λ = 0 = nλ with n = 0
→ constructive
Step 2 — point Y5λ − 3.5λ = 1.5λ = (n + ½)λ with n = 1
→ destructive
Step 3 — point Z6λ − 4λ = 2λ = nλ with n = 2
→ constructive
X: constructive · Y: destructive · Z: constructiveZero path difference is still constructive. Students often mark it as “nothing happening” — it’s the brightest point of all.
WE 2
Two loudspeakers, driven by the same amplifier, emit a 660 Hz note. A listener stands 4.00 m from one speaker and 5.50 m from the other. Taking the speed of sound as 330 m s⁻¹, determine whether the sound is loud or quiet at that spot.
Step 1 — find the wavelength of the soundλ = v / f = 330 / 660 = 0.50 mStep 2 — find the path difference5.50 − 4.00 = 1.50 mStep 3 — compare with λ1.50 / 0.50 = 3.0 = nλ with n = 3A whole number of wavelengths → constructive → loudSame amplifier means the sources are coherent. Step half a metre sideways and the path difference changes — which is exactly why you hear loud, quiet, loud as you walk.
WE 3
Laser light of wavelength 650 nm passes through a double slit. At a particular point on the screen the path difference between the two waves is 1.625 µm. Determine whether that point is bright or dark, and state the value of n.
Step 1 — convert both to the same unit
λ = 650 nm = 6.50 × 10⁻⁷ m, path difference = 1.625 × 10⁻⁶ m
Step 2 — divide the path difference by λ(1.625 × 10⁻⁶) / (6.50 × 10⁻⁷) = 2.5Step 3 — a whole number plus a half means destructive2.5λ = (2 + ½)λ, so n = 2Dark fringe, n = 2Careful with n: for 2.5λ the value of n is 2, not 3. Match the number in front of the bracket, not the rounded total.
💡 Top tips
Coherent means two things: same frequency and constant phase difference. Say both.
Zero path difference is constructive (n = 0), not “no interference”.
Divide the path difference by λ before deciding — a raw distance tells you nothing on its own.
Read n from the bracket: 1.5λ = (1 + ½)λ, so n = 1.
The conditions are in the data booklet — but you must know how to get the path difference.
Path difference is a distance; phase difference is an angle. One λ = 360°.
⚠ Common mistakes
Saying coherent just means “same frequency” — the constant phase difference is half the definition
Treating zero path difference as destructive
Reading n = 3 from a path difference of 2.5λ — it is n = 2
Mixing up path difference (a distance) with phase difference (an angle)
Thinking destructive interference destroys energy — the energy simply appears at the maxima instead
Forgetting to convert nm and µm to metres before dividing
Quick recap:Interference is what superposition looks like when two coherent sources (same frequency, constant phase difference) overlap. Whether a point is a maximum or a minimum depends on the path difference: nλ gives constructive, (n + ½)λ gives destructive. Count the wavelengths to a point from each source, subtract, and divide by λ.
You can now predict whether any point is bright or dark — but only if someone hands you the path difference. The next page does the geometry for you. Take two narrow slits a distance d apart and a screen a distance D away, and the bright fringes land in beautifully even steps you can measure with a ruler. That’s Young’s Double-Slit Experiment.
Path difference tripping you up?
Book a free meeting and we’ll work through interference conditions, coherence and past-paper questions together.
