Physics 9702 · Paper 2

Path & Phase Difference

Two waves often start together, in step. But one may have to travel further to reach the same point.
That extra distance is the . That extra distance alone determines what happens when they meet. A whole wavelength further and they add up. Half a wavelength further and they cancel. All of interference builds on this.

Path difference

Path difference = how much further one wave travels than the other to reach the same point. Measure it in metres. Better still, measure it in wavelengths — how many wavelengths of extra distance.
One whole wavelength further: the wave lines up again, so the two are back in step (crest on crest, they add up).
Half a wavelength further: out of step (crest meets trough, they cancel).

Both waves have the same wavelength and reach the same point P. Wave 2 sets off further back, so it travels an extra length — the path difference. Here it is a whole 2 wavelengths, so the two arrive in step and add.

Phase and path difference

You met in the Waves chapter: how far out of step two waves are, given as an angle (in degrees).
In step = 0° (or a full turn: 360°, 720°…). Fully out of step = 180°, which we call .
Path difference (a distance) and phase difference (an angle) say the same thing — how far out of step — in two different ways. One whole wavelength of path is one full 360° of phase.

The two waves are out of step by a quarter wavelength: that gap is the path difference (a distance), and measured as an angle it is a 90° phase difference.

\text{phase difference} = \frac{\text{path difference}}{\lambda} \times 360^\circ

Symbols

$\lambda$ = wavelength (m)
$\text{path difference}$ = extra distance one wave travels (m)

Worked example

Turning a path difference into a phase difference

Two waves have wavelength 8 cm. One travels 2 cm further than the other before they meet. Are they in step? What is the phase difference?

Path difference in wavelengths: $\frac{2}{8} = \frac{1}{4}$ of a wavelength.
Phase difference: $\frac{1}{4} \times 360^\circ = 90^\circ$ .
90° is not 0° and not 180°, so the waves are partly out of step — they do not fully add, and do not fully cancel.

Answer

\text{phase difference} = 90^\circ

Path difference is a distance; phase difference is an angle. Swap between them with one fact: one whole wavelength of path is 360° of phase.

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