Converting path difference to phase difference
For waves that leave their sources in phase, the phase difference on arrival is set entirely by the path difference: Δφ = (2π/λ) × path difference. One whole wavelength of extra path corresponds to 2π rad, and half a wavelength to π rad.
Δφ = (2π/λ) Δ · Δ = (Δφ/2π) λWhat this actually means
This is the translator between the two languages the syllabus uses. Questions about light usually speak in path difference; questions about oscillating sources usually speak in phase difference. Convert freely.
The proportionality is the useful bit: phase difference is proportional to path difference, with λ setting the exchange rate. Double the path difference, double the phase difference.
If the sources themselves have a phase difference φ₀, the total on arrival is Δφ = φ₀ + (2π/λ)Δ. Antiphase sources have φ₀ = π, which is exactly why their bright and dark positions swap over.
Once you have Δφ, reduce it modulo 2π before judging. A path difference of 2.5λ gives 5π, which reduces to π, so destructive.
In microwave and radio questions the wavelength is often centimetres or metres, so path differences of several wavelengths are common. Keep everything in the same units before dividing.
Forgetting to add the sources' own phase difference, so antiphase sources are analysed as if they were in phase.
Prove it — watch it be true
- Open the path-difference translator panel.
- Enter a path difference of 0.5λ and confirm the phase difference reads π rad.
- Enter 2.5λ and confirm it reads 5π rad, and that the reduced value shown is π rad.