2D Wavefronts

According to the syllabus document, you need to know the following outcomes – Use this as a
checklist when studying this chapter:
Learners must be able to








Define a wavefront as an imaginary line that connects waves that are in phase (e.g. all at the crest
of their cycle).
State Huygen’s principle.
Define diffraction as the ability of a wave to spread out in wavefronts as they pass through a
small aperture or around a sharp edge.
Define interference as when two waves pass through the same region of space at the same time,
resulting in superposition of waves.
Explain the concepts of constructive and destructive interference.
Predict areas of constructive and destructive interference from a diagram / source material.
Apply Huygen’s principle to explain diffraction qualitatively. Light and dark areas can be
described in terms of constructive and destructive interference of secondary wavelets.
Sketch the diffraction pattern for a single slit.
EXTENTION WORK (Not for Exammination)



Use the approximation sin   m  for a slit of width a to calculate the position (angle from
the horizontal) of the dark bands inaa single slit diffraction pattern, where m=+1, +2, +3, ...
Investigate the interference of waves on the surface of water from two coherent sources,
vibrating in phase e.g. light waves through a double slit.
Draw an interference pattern marking nodal lines and noting positions of maximum interference
e.g. interference pattern for a double slit.
(Although the focus here is on light, it will be very useful to use water waves in a ripple tank to
demonstrate diffraction and interference.)
Page 1 of 7
Definition of a wave front
A wave front is an imaginary line that connects points on a wave that are in phase (e.g. all at the crest
of their cycle).
Huygen’s Principle: Every point on a wave-front may be considered
as a new source of circular wave-fronts.
Huygen’s principle explains why diffraction occurs as seen in the diagram below:
The diagram shows wave fronts approaching a gap and spreading out around the edges of the gap
Definition of diffraction: Diffraction is the bending of a wave-front around the
edges of an obstacle.
.
Diffraction can readily be observed in a ripple tank, for example,
The degree of diffraction will increase as:
1.
the wavelength increases (or frequency decreases)
2.
the aperture width decreases.
Page 2 of 7
Interference:
Principle of Superposition of Waves: If two waves arrive at the same point the resulting
disturbance is the vector sum of the disturbance due to
each wave.
If a crest of a wave meets a crest of another wave at the same point then the crests interfere
constructively and the resultant wave amplitude is greater.
If a crest of a wave meets a trough of another wave then they interfere destructively, and the overall
amplitude is decreased.
Consider two waves that are in phase, with amplitudes A1 and A2. Their troughs and peaks line up and
the resultant wave will have amplitude A = A1 + A2. This is known as constructive interference.
If the two waves are 180°, out of phase, then one wave's crests will coincide with another wave's
troughs and so will tend to cancel out. The resultant amplitude is A = |A1 − A2|. If A1 = A2, the
resultant amplitude will be zero. This is known as destructive interference.
When two sinusoidal waves superimpose, the resulting waveform depends on the frequency (or
wavelength), amplitude and relative phase of the two waves. If the two waves have the same
amplitude A and wavelength then the resultant waveform will have an amplitude between 0 and 2A
depending on whether the two waves are in phase or out of phase.
combined
waveform
wave 1
wave 2
Two waves in phase
Two waves 180° out
of phase
Constructive interference occurs when two waves arrive at the same point in phase.
Destructive interference occurs when two waves arrive at the same point out of phase.
Note: Where light interferes constructively it is brighter. Where light interferes destructively it is
dimmer – even black (no light). This effect can be seen in both Diffraction patterns when light
passes through a single slit and Interference patterns where light from the same source comes
through two slits.
Page 3 of 7
Interference effects of light with a single slit (Diffraction pattern)
After passing through a narrow slit, a homogeneous monochromatic beam of light is found to diverge
from the slit and displays a structure of alternating light and dark bands. This is as a result of
interference resulting from a number of wavelets starting within the slit (see P 6-3)
NOTE: This pattern consists of a bright central band with each band outside of this being
progressively dimmer. See page 6-2
Monochromatic light is light of the same frequency.
EXTENTION WORK
Interference effects
This next diagram shows what happens when water waves from two slits coincide with each other.
Areas of constructive and destructive interference result. The adding together of such waves is also
called superposition. The constructive interference is from a crest and a crest or a trough and a trough
meeting to produce a larger crest or deeper trough. The destructive interference occurs when a crest
from one wave cancels out the trough of another producing an area where there is no wave.
A line joining the points of zero disturbance is called a nodal line.
A line joining the points of maximum disturbance is called an antinodal line.
constructive
interference
destructive
interference
Nodal line
Page 4 of 7
A LIGHT interference pattern on a screen consists of bands of
light and dark (below). How far apart the bands are depends upon
the wavelength of the light and the distance between the slits.
They will be further apart if: 1) the wavelength is long and
2) the slits are close together.
Nodal lines are lines of zero
disturbance caused by destructive
interference. (crest & trough)
Anti nodal lines are lines of greater
disturbance caused by constructive
interference. (2 crests or 2 troughs)
(crest & trough)
Page 5 of 7
Questions:
1.
What is a nodal line?
2.
The diagram alongside shows the diffraction of one set of
wave fronts. Assume that they represent the crests of this
wave. Copy the diagram and
a)
b)
c)
d)
draw in the other set of wave fronts, assuming that they
too are crests.
between each set of wave fronts, use dotted lines to
draw in the troughs.
put in four nodal lines.
mark the lines of reinforcement with dotted lines.
The extent of the diffraction of light is proportional to λ, the wavelength of the light, and inversely
proportional to the width of the slit, a.
It can be shown that the angle Ө, at which the first dark line occurs, is given by
sin  
θ
m
a
λ
=
=
=
=
m
a
angle from the horizontal
1, 2, 3… (first fringe, second fringe, etc)
width of the slit
wavelength of light
Page 6 of 7
Example
The second dark fringe from a single slit diffraction pattern is found at an angle of 12° from the
horizontal. If the wavelength of the light is 4.8 x 10-7m,
a) calculate the width of the slit.
b) convert the measurement from (a) into mm.
Questions
1.
The wavelength of light that is incident on a single slit of width 0.004mm is 600nm. At what
angle, θ, will the
a) first dark fringe be found?
b) second dark fringe be found?
2.
What is the wavelength of the light used in a single slit diffraction experiment, if the first dark
fringe is found at angle of 10° from the horizontal and the width of the slit is 0.0028mm?
3.
In the diagram alongside, the distance L = 0.12m, a =
0.024mm and the first dark fringe is found at a
distance, x = 4.4mm. Calculate
a)
b)
a
the angle θ.
the wavelength of the light.
x
θ
L
Page 7 of 7