Phase difference: how far behind is that point?

By the end of this lesson

By the end you can explain what phase difference means, decide whether two points are in phase or in antiphase, and calculate the phase difference from their separation and the wavelength.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\lambda$	wavelength
$\phi$	phase difference

Runners on a circular track

Imagine two runners going round a circular track at the same speed, but one started later than the other.

They are always doing the same thing — running — but at any moment one is further round the lap. We could say they are out of step by a certain fraction of a lap.

Two points on a wave are like that. They oscillate with the same frequency, but one reaches its crest slightly after the other.

The fraction of a cycle between them is their phase difference.

Definition

Phase difference

The phase difference between two points on a wave is the fraction of a cycle by which one oscillation leads or lags the other.

It is measured in degrees or radians:

one complete cycle $= 360° = 2\pi$ radians;
half a cycle $= 180° = \pi$ radians.

Two points in phase have a phase difference of $0°$ (or $360°$ ): they reach their crests at the same time. Points one wavelength apart are in phase.

Two points in antiphase have a phase difference of $180°$ (or $\pi$ ): when one is at a crest, the other is at a trough. Points half a wavelength apart are in antiphase.

From distance to angle

A wave repeats every wavelength. So a separation of one wavelength corresponds to one full cycle, $360°$ .

If two points are a distance $\Delta x$ apart, the fraction of a cycle between them is

$\frac{\Delta x}{\lambda}$

Multiply that fraction by a full cycle to get the phase difference:

$\Delta\phi = \frac{\Delta x}{\lambda} \times 360°$

or, in radians,

$\Delta\phi = \frac{\Delta x}{\lambda} \times 2\pi$

The second form is the one used later in interference, where radians are standard.

Worked example: Worked example: two points a quarter-wavelength apart

Two points on a wave are separated by $0.15\ \text{m}$ . The wavelength is $0.60\ \text{m}$ .

Find the phase difference in degrees and in radians.

Have a go first?

Quick check

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

Multiple choiceQuestion 3 · Recall

Answer the 3 questions above to continue.