The big idea: When two waves arrive at the same point, you add their displacements at every instant. This is called superposition.
If they arrive in step (crest on crest) the result is bigger — constructive interference.
If they arrive half a cycle out of step (crest on trough) and are equal, they cancel — destructive interference.
Two words to know: Amplitude = the height of a crest above the middle (how 'big' the wave is).
In phase = exactly in step (crests line up). Antiphase = half a cycle out of step (a crest lines up with a trough).
Just add the displacements: Crest + crest → a taller crest (constructive).
Equal crest + trough → zero (destructive).
Often two waves start in step from a source but travel different distances to reach a point. The extra distance one wave travels is the path difference. That single number decides whether they end up in step (constructive) or out of step (destructive).
New term — path difference: Path difference = how much further one wave travels than the other to reach the point, measured in metres.
What matters is how this compares with the wavelength λ.
- extra distance one wave travels to reach the point (m)
- a whole number: 0, 1, 2, 3, …
- wavelength of the waves (m)
Constructive (in step)
- Path difference = 0, λ, 2λ, 3λ, … (nλ)
- Waves arrive in phase
- Amplitudes add → bright / loud
Destructive (out of step)
- Path difference = ½λ, 1½λ, 2½λ, … ((n+½)λ)
- Waves arrive antiphase
- Equal amplitudes cancel → dark / quiet
Worked example — constructive or destructive?
Two loudspeakers play the same note (wavelength 0.80 m), in step. At a point P the sound from one speaker has travelled 1.6 m further than from the other. Is the sound at P loud (constructive) or quiet (destructive)?
Solution
- Start with the given constructive rule:
- Divide the path difference by the wavelength:
- That is a whole number (n = 2), so the rule is satisfied:
Final answer
Path difference = 2λ exactly, so the waves arrive in step → constructive: the sound at P is loud.
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How this is tested: Interference comes up with sound, light and microwaves.
- Paper 1A: add the amplitudes of two waves at a point, or pick constructive vs destructive from a path difference. - Paper 2: explain why two sources must be coherent, or outline why a point is quiet (destructive) using path difference.
Classic trap: for destructive cancellation to be complete the two amplitudes must be equal — otherwise only part cancels.
New word — coherent: Two sources are coherent if they keep a constant phase difference — the same wavelength and a fixed, unchanging step between them.
Only coherent sources give a steady interference pattern; if the phase difference kept changing, the bright and dark points would jump around and average out to nothing.
IB-style question — (a) why destructive
Two small loudspeakers are driven by the same signal (so they are coherent) and emit sound of wavelength 0.50 m. At a detector the sound from one speaker has travelled 0.25 m further than from the other. Outline why the detector picks up almost no sound.
Solution
- Compare the path difference with the wavelength:
- That matches the given destructive rule with n = 0:
- So one wave is half a cycle behind — it arrives antiphase.
- A crest from one speaker meets a trough from the other, and they are equal, so they cancel.
Final answer
The path difference is half a wavelength, so the waves arrive antiphase and (being equal) cancel — destructive interference, so almost no sound.
IB-style question — (b) why coherent
The same demonstration only gives a steady pattern of loud and quiet points if the two speakers are coherent. Explain what coherent means and why it is needed.
Solution
- Coherent means the two sources keep a constant phase difference (same wavelength, fixed step).
- Then the path difference at each point gives the same result every instant.
- So the loud and quiet points stay in fixed places — a steady pattern.
- If the phase difference drifted, those points would move around and average out to nothing.
Final answer
Coherent = constant phase difference. It is needed so the constructive and destructive points stay put, giving an observable, steady interference pattern.