Interference & coherence
Interference
Interference occurs when waves overlap and their resultant displacement is the sum of the displacement of
each wave.
Interference of two waves can either be:
•
•
In phase, causing constructive interference. The peaks and troughs line up on both waves. The
resultant wave has double the amplitude
In anti-phase, causing destructive interference. The peaks on one wave line up with the troughs of
the other. The resultant wave has no amplitude.
Conditions for constructive and destructive interference
In general, for waves emitted by two coherent sources very close together:
The condition for constructive interference is:
The condition for destructive interference is:
Where: λ = wavelength of the waves in metres (m)
n= 0, 1, 2, 3... (any other integer)
Coherence
Coherence means the waves have a constant phase relationship with each other and the same frequency.
This allows for a stable and predictable interference pattern to be observed.
Coherent sources - have the same frequency and constant phase difference.
Q. The diagram shows the interferences of coherent waves from two point sources.
Which row in the table correctly identifies the type of interference at points X, Y and Z.
Demonstrating two source interference
Two source interference can be demonstrated using
•
•
•
•
water waves in a ripple tank
sound
light and
microwaves
1. Using water waves
•
•
Two-source interference in can be demonstrated in water using ripple tanks.
A curved line represents each wavefront (peak or trough) The diagram below shows the wavefronts
from two-point sources. E.g. dropping two pebbles near to each other in a pond
The lines of maximum displacement occur when all the peaks and troughs line up with those on another
wave.
On a wavefront diagram, it is possible to count the number of wavelengths to determine whether constructive
or destructive interference occurs at a certain point.
2. Using sound waves
•
Two-source interference can be demonstrated with two speakers emitting a coherent sound.
Sound waves are longitudinal waves made up of compressions and rarefactions.
Constructive interference occurs when the compressions and rarefactions from each wave line up and
the sound appears louder.
Destructive interference occurs when a compression from one wave lines up with a rarefaction from the
other and vice versa. The two waves cancel each other out, so zero sound is heard.
This is the technology used in noise-cancelling headphones.
3. Using microwaves
•
Two-source interference for microwaves (and other electromagnetic waves) can be detected with a
moveable microwave detector.
The detector picks up a maximum amplitude or intensity in regions of constructive interference.
The detector picks up a minimum or zero amplitude, so no signal in regions of destructive interference.
4. Using light waves
•
Lasers are the ideal piece of equipment to analyze the diffraction and intensity patterns.
The diffraction pattern produced by a laser on a screen is made up of:
Areas of constructive interference - the bright strips or fringes
Areas of destructive interference - the dark fringes
Two source interference fringes
For two-source interference fringes to be observed, the sources of the wave must be:
•
•
Coherent (constant phase difference)
Monochromatic (single frequency)
A laser is an example of a coherent monochromatic light source. Other sources of light, such as a filament
bulb or a sodium lamp, are non-coherent, so they produce white light.
When two waves interfere, the resultant wave depends on the path difference between the two waves.
Remember the conditions for Constructive and destructive interference:
For constructive interference (or maxima), the difference in wavelengths will be an integer number of whole
wavelengths.
For destructive interference (or minima) it will be an integer number of whole wavelengths plus a half
wavelength.
Double slit interference pattern:
n = 0 is taken from the middle, n = 1 is one either side and so on.
Q. Two coherent sources of sound waves S and S are situated 65 cm apart in air as shown below.
The two sources vibrate in phase but have different amplitudes of vibration. A microphone M is situated 150
cm from S1 along the line normal to S1 and S2. The microphone detects maxima and minima of the intensity
of the sound. The wavelength of the sound from S1 to S2 is decreased by increasing the frequency. Determine
which orders of maxima are detected at M as the wavelength is increased from 3.5 cm to 12.5 cm.
Young's Double Slit Experiment
Young's double-slit experiment produces a diffraction and an interference pattern using either:
•
•
The interference of two coherent wave sources
A single wave source passing through a double slit
In this typical set-up for Young's double slit experiment:
The laser light source is placed behind the single slit. So, the light is diffracted, producing two light sources
at slits A and B
The light from the double slits is then diffracted, producing a diffraction pattern made up of bright and dark
fringes on a screen.
Calculations can be made for the double slit interference of light using the equation:
In this experiment:
•
D is much bigger than any other dimension, normally several meters long.
•
a is the separation between the two slits and is often the smallest dimension, normally in mm.
•
x is the distance between the fringes on the screen, often in cm. This can be obtained by measuring
the distance between the center of each consecutive bright spot.
The above equation shows that the separation between the fringes will increase if:
•
the wavelength λ of the incident light increases
•
the distance D between the screen and the slits increases
•
the separation x between the slits decreases