Wave superposition
•
•
If two waves are in the same place
at the same time they superpose.
This means that their amplitudes
add together vectorially
Positively when they are in phase
Wave superposition
• If two waves are in the same place at the same time they
superpose.
• This means that their amplitudes add together vectorially
Negatively when they are in antiphase
The conditions for two waves to
interfere with each other
• The waves must be
coherent.
This means there must
be a constant phase
difference between
them ( which also
implies that they have
to be of the same
frequency.)
Interference between
water waves from
coherent sources
Interference Is best understood with relatively long waves.
Both water waves or microwaves (3cm ) provide a reasonable model.
Coherence of the waves is ensured by obtaining the waves from a
single frequency microwave source ( a monochromatic source)
The waves are passed through two slits and are diffracted in the
process. (That is they begin to spread out as if the slit was at the centre
of a circular wave front)
On a wave front diagram the waves positively superpose where they
cross and negatively superpose in the centre of the gaps.
A microwave detector moved normal
to the source detects positive and
negative superposition called
interference fringes.
Each peak is produced by positive interference. Each trough
occurs because of negative interference
Central maximum phase difference 0
S1
X
S2
Here the path lengths from S1 and S2 to X
are the same
Next maximum phase difference 2π
S1
S2
Note that when the phase difference is 2π
the path length from one of the slits is
longer by a single wavelength
2π
Path difference 4π
Path difference 2π
Path difference 0
Measuring the difference in path length to the
first fringe
1st fringe
0.645m
0.670m
In this case λ= 0.670-0.645= 0.025m.
Zero or central
fringe
Notes
• If you measure over 2 fringes you would
have to divide your answer by 2 and so
on. (the fringes are equally spaces).
• You may choose to use maxima OR
minima. (they are equally spaced)
• Measurement of path difference is
impractical for interference involving
visible light! Why?
Interference with Light Sources
The geometry of the situation gives us
the relationship
ws

D
W is the distance
between adjacent
fringes
S is the slit
separation
D
laser
double slit
The laser is a coherent light source which is divided into two by the
fine double slit The screen is at a distance of 5-10m. The interference
pattern below is produced. The fringes are equally spaced.
In reality you would measure the total distance between the centre of several
visible fringes and divide by the number of dark intervals between them to
achieve a better value for w.
Diffraction from a Single Slit
Through a narrow single slit the wave front spreads out. If the slit is
wide the spreading is slight. If the slit is comparable in width with the
wavelegth of the wave the spreading is large.
Diffraction of water waves
from above
Diffracted laser light from a single
slit projected onto a screen
intensity
2nd order
1st order
Principle
maximum
Principle
maximum
Zero order
principle
maximum
1st order
2nd order
Principle
maximum
Principle
maximum
minimum
minimum
minimum
minimum
The diffraction grating
• A typical diffraction grating is an
arrangement of identically spaced
diffracting elements. Normally a large
number of parallel lines are ruled on
glass. The diffracting elements are the
gaps between the ruled lines. typically
there would be around 600 lines per mm.
Each line acts as a very
narrow slit
A
B
C
The light through each
slit is diffracted in all
directions.
Consider the light
through two slits
diffracted at angle θ to
the normal
If the light diffracted through
angle θ at A is in phase with the
light diffracted through θ at B it
must be in phase with the light
at this angle through every
other slit.
θ
θ
The path difference is the length AN
θ
A
X
N
d
Y
θ
Notice that:
to be in phase
the path difference (ie the
distance A to N)
has to be a multiple of the
wavelength λ
i.e (n λ )
The distance AN = d sinθ
So d sinθ = n λ
What you see
• When the waves interfere constructively
through each slit they are at an angle
given by the formula
• d sinθ = n λ
Question
• When a grating of 300 lines per millimetre
is illuminated with parallel beam of
monochromatic light normal to it a second
order principle maximum is observed at
18.90 to the straight through direction.
Calculate the wavelength of the light.