Coherent have the same frequency, wavelength, and velocity, and coherent waves have a constant phase relationship. A constant phase relationship means the phase difference is the same along all points of the wave at all times, i.e. if they have no phase difference this will be true at all points, if they have a phase difference of one wavelength this will be true at all points .

When two coherent waves meet, they combine to produce a new wave pattern in a process known as interference. Interference is evidence for the wave model of light.

Interference patterns can appear in the ways shown below.

Constructive interference occurs when two waves are in phase (the phase difference between them is a whole number of wavelengths). Destructive interference occurs when two waves are out of phase by half a wavelength (the phase difference between them is an odd number of half-wavelengths).

The above shows double-source interference. The waves emerge from the two slits in phase, but the slits cause them to be diffracted which results in a pattern of maxima (where constructive interference occurs) and minima (where destructive interference occurs). The path difference refers to the difference in the length of the path taken by the two waves to a certain point.

Maxima are produced when the path difference between waves is a whole number of wavelengths. Minima are produced when the path difference between waves is an odd number of half-wavelengths.

→ The central maximum is “of order $0$”, then out on either side the next one is “of order $1$”.

The following relationship links the path difference between waves, wavelength and order number of the maximum or minimum, and it can be used to solve problems involving these quantities.

$$ path \space difference =m\lambda \space or \space (m \space + \space \dfrac{1}{2})\lambda \space where \space m \in \N $$

A diffraction grating with more than two slits can also be used to produce an interference pattern, which models interference among many coherent sources.

The relationship linking grating spacing, wavelength, order number of the maximum and angle to the maximum is below, and can be used to solve problems involving these quantities.

$$ dsin\theta=m\lambda $$

→ To produce a pattern with more spaced out maxima, i.e. where $\theta$ is greater, the slit separation of the grating, $d$, should be decreased or a source with a greater wavelength should be used. The distance of the grating from the screen can also be increased to produce more spaced out maxima.

Where the number of slits on a diffraction grating is given, $N$, the following formula can be used to find the individual distance between two adjacent slits.

$$ d= \dfrac{1}{N} $$

→ $N$ must be the number of lines per metre, so if the number of lines per millimetre is quoted then this must be converted to the correct units.