Phase difference and path difference

Phase difference is how far one oscillation is ahead of or behind another, as a fraction of a cycle, measured in degrees or radians. Path difference is the difference in the distances two waves have travelled to reach a point, usually given in wavelengths. They are linked: a path difference of one whole wavelength is a phase difference of 360°, or 2π radians.

What is phase difference?

Every point on a wave goes through a repeating cycle of motion. Its phase says where in the cycle it is. The phase difference between two points, or two waves of the same frequency, says how far apart in the cycle they are. AQA's A-level specification says phase difference may be measured as angles, in radians or degrees, or as fractions of a cycle.

Points that are in phase move together: they reach their peaks at the same time. Points in antiphase do the opposite: one is at a peak when the other is at a trough.

Phase difference formula

For two points a distance x apart along a wave of wavelength λ:

For two oscillations with a time delay t between them, and period T, swap the distance for time: φ = 2πt/T.

Example 1: two points on a wave

A wave has wavelength 2.4 m. Find the phase difference between two points 0.60 m apart, and between two points 1.8 m apart.

• 0.60 m: φ = 2π × 0.60 ÷ 2.4 = π/2 rad (90°), a quarter of a cycle.
• 1.8 m: φ = 360 × 1.8 ÷ 2.4 = 270° (3π/2 rad), three quarters of a cycle.

Example 2: from a time delay

Two microphones pick up the same 250 Hz note, and one trace on an oscilloscope lags the other by 1.0 ms. The period is T = 1 ÷ 250 = 4.0 ms, so φ = 2π × 1.0 ÷ 4.0 = π/2 rad.

What is path difference?

When waves from two sources meet at a point, they have usually travelled different distances. The path difference is the difference between those distances:

If the two sources are coherent (same frequency, with a constant phase difference) and start in phase, the path difference alone decides what happens at the point.

Path difference and interference

Constructive interference happens where the waves meet crest to crest; destructive interference where they meet crest to trough. To go from one to the other, use phase difference = 2π × (path difference ÷ λ).

Example 3: two loudspeakers

Two loudspeakers connected to the same signal generator emit a 680 Hz note in phase. Sound travels at 340 m/s. A listener stands 3.00 m from one speaker and 3.75 m from the other. Loud or quiet?

1. Wavelength: λ = v ÷ f = 340 ÷ 680 = 0.50 m.
2. Path difference: 3.75 − 3.00 = 0.75 m.
3. In wavelengths: 0.75 ÷ 0.50 = 1.5λ, a whole number plus a half.
4. Half a cycle out of step, the two sounds cancel: quiet. At a point 3.00 m and 4.00 m away, the path difference would be 2λ, giving a loud spot.

Example 4: the double slit

In a double-slit experiment with light of wavelength 600 nm, the third bright fringe from the centre is where the path difference is 3λ = 3 × 600 nm = 1.8 µm. The second dark fringe is at a path difference of 1.5λ = 900 nm.

You can watch maxima and minima appear as two sources overlap in the wave interference simulator, and the diffraction grating guide uses the same path difference idea with many slits.

Common mistakes

• Mixing degrees and radians. Say which you are using, and set your calculator to match.
• Confusing the two ideas. Path difference is a distance (often in wavelengths); phase difference is an angle or fraction of a cycle.
• Forgetting the sources must be in phase. If the sources start in antiphase, a path difference of nλ gives destructive interference instead.
• Using total distance. Path difference is the difference between the two distances, not either distance on its own.
• Calling 360° a phase difference. Points a whole wavelength apart are in phase: the phase difference is 0 (or 2π).

To practise waves questions in a short timed paper, try today's A-level Physics: Year 1 paper.

Read next

Physics · Waves and lightWave simulator: ripple tank, interference and slitsFree wave simulator: a ripple tank with two sources, a double slit or a single slit, showing interference, diffraction, path difference and fringe spacing.