Lecture 22
Chapter 21
Physics II
08.06.2015
Interference
Course website:
http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII
Lecture Capture:
http://echo360.uml.edu/danylov201415/physics2spring.html
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ConcepTest 1 Standing Wave

What is the mode number of this
standing wave?
A) 4
B) 5
C) 6
D) Can’t say without
knowing what kind of
wave it is.
Number of antinodes = mode number
Interference
A standing wave is the interference pattern produced when two waves of equal
frequency travel in opposite directions.
In this section we will look at the interference of two waves
traveling in the same direction.
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Interference in One Dimension
The pattern resulting from the superposition of two waves is often called interference.
In this section we will look at the interference of two waves traveling
in the same direction.
The resulting amplitude is A  2a for
maximum constructive interference.
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The resulting amplitude is A  0 for
perfect destructive interference
Let’s describe 1D interference mathematically
Consider two traveling waves. They have:
1. The same direction, +x direction
2. The same amplitude, a
3. The same frequency, 
Let’s find a displacement at point P at time t:
,
P
,
+
0 Using a trig identity:
+
Δ
cos
sin
2
2
tells us what the source is
doing at t  0.
2
sin
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0 The phase constant 0
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Constructive/destructive interference
It is still a traveling wave
with an amplitude:
where   1 - 2 is the phase difference
between the two waves.
The amplitude:
 The amplitude has a maximum value A = 2a if
,
cos(/2)  1.
, , ,…
Conditions for constructive interference:
 Similarly, the amplitude is zero, A=0 if
cos(/2)  0.
Conditions for destructive interference
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,
, , ,…
Let’s look deeper in Δ
  2 - 1 is the phase difference between the two waves.
So, there are two contributions to the phase difference:
1.
2.
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– pathlength difference
-- inherent phase difference
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Inherent phase difference
These are identical sources:
These are not identical sources: out of phase
/
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Pathlength difference for constructive interference
Assume that the sources are identical
0. Let’s separate the sources with a pathlength ∆x
Conditions for
constructive interference:
Thus, for a constructive interference of two identical sources with
A = 2a, we need to separate them by an integer number of wavelength
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Pathlength difference for destructive interference
Assume that the sources are identical
0. Let’s separate the sources with a pathlength ∆x
Conditions for
destructive interference:
/2
Thus, for a constructive interference of two identical sources with A =0, we need to
separate them by a half integer number of wavelength
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Applications
Noise-cancelling headphones
It allows reducing unwanted sound by the addition
of a second sound specifically designed to cancel
the first (destructive interference).
 Thin transparent films, placed on
glass surfaces, such as lenses, can
control reflections from the glass.
 Antireflection coatings on the lenses
in cameras, microscopes, and other
optical equipment are examples of
thin-film coatings.
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ConcepTest 2 1D interference
Two loudspeakers emit waves
with
.
What, if anything, can be done to
cause constructive interference
between the two waves?


/ A) Move speaker 1 forward by 0.5 m
B) Move speaker 1 forward by 1.0 m
C) Move speaker 1 forward by 2.0 m
D) Do nothing
Interference
in two and three
dimensions
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A Circular or Spherical Wave
A linear (1D) wave can be written
,
A circular (2D) or spherical (3D) wave
can be written
,
where r is the distance measured
outward from the source.
 The amplitude a of a circular or
spherical wave diminishes as r
increases.
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Transition from 1D to 2D/3D interference
 The mathematical description of interference in two or three dimensions is
very similar to that of one-dimensional interference.
 The conditions for constructive and destructive interference are:
one-dimensional
Constructive:
two or three dimensions
Destructive:
where r is the path-length difference.
If the sources are identical (
Constructive if
Destructive if
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), the interference is
Example of 2D interference
 The figure shows two identical sources
that are in phase.
 The path-length difference r
determines whether the interference at a
particular point is constructive or
destructive.
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ConcepTest 3 2D Interference

Two in-phase sources emit sound
waves of equal wavelength and
intensity. At the position of the dot,
A) The interference is
constructive.
B) The interference is
destructive
C) The interference is somewhere
between constructive and destructive
D) There’s not enough information to
tell about the interference.
.
.
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ConcepTest 4 The Wave
At a football game, the “wave”
might circulate through the stands
and move around the stadium. In
this wave motion, people stand up
and sit down as the wave passes.
What type of wave would this be
characterized as?
A) polarized wave
B) longitudinal wave
C) lateral wave
D) transverse wave
E) soliton wave
The people are moving up and down, and the wave is
traveling around the stadium. Thus, the motion of the
wave is perpendicular to the oscillation direction of the
people, and so this is a transverse wave.
What you should read
Chapter 21 (Knight)
Sections
 21.5
 21.6
 21.7
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Thank you
See you tomorrow
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