General Physics II
Wave Optics
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Periodic Wave Characteristics
• Amplitude: Magnitude of
the maximum displacement
from zero line.
• Wavelength: Distance
between successive two
crests (or any two
successive identical points
on a wave).
• Frequency: Number of
crests passing a fixed point
per unit time.
• Period: Time taken between
two successive crests
passing a fixed point.
wavelength
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Electromagnetic Spectrum
Note that red light has the longest wavelengths
and violet the shortest in the visible spectrum.
speed of light in vacuum c = 2.99792458×108 m/s.
v = λ f . (speed = wavelength×frequency)
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Interference and Diffraction of Waves
• Interference and diffraction
are phenomena that are
exhibited by all waves.
• Interference occurs when two
or more waves overlap.
• Diffraction is associated with
the bending or spreading of
waves when they they
encounter an obstacle or go
through an opening.
• The concept that underlies
both diffraction and
interference is the principle of
superposition.
Diffraction of water waves
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Principle of Superposition
• Consider two waves on a
string or in water.
• The principle of
superposition states that
when two or more waves
overlap, the resultant wave
displacement at any point is
the sum of the
instantaneous values of the
individual wave
displacements at that point.
• The superposition principle
is true for all waves,
including electromagnetic
waves.
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Interference
• The nature of interference is
most easily understood if the
sources of the overlapping
waves are coherent.
• Two wave sources are
coherent if the waves have
the same frequency and have
a fixed phase relationship
relative to each other. The
latter means that the cycles of
vibration of the sources are
synchronized.
• In the figure to the right, the
sources are exactly in phase.
Thus, when source 1 is
emitting a crest (green
circles), so is source 2 (blue
circles), and so forth.
Outgoing waves are shown
as circles.
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Interference
• Consider waves from two
coherent sources, in which the
sources are in phase. The
waves have the same
amplitude.
• Along the line bisecting the two
sources (e.g., point a), the
waves travel the same distance
to each point on the line and so
arrive in phase. A crest arrives
simultaneously with a crest and
so by superposition, a crest with
twice the amplitude is obtained.
Thus, the waves reinforce each
other. This reinforcement of
overlapping waves is called
constructive interference.
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Constructive Interference
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Destructive Interference
• At some points in space (e.g.,
point c), the waves will arrive out
of phase by exactly one half
wavelength. This a crest will
arrive simultaneously with a
trough. Hence, the waves cancel
each other and the resultant
value of the wave displacement
is zero. Instances in which the
overlap results in total
cancellation are called
destructive interference.
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Destructive Interference
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Interference
• In general, interference can
result in a resultant wave
displacement anywhere
between total cancellation
and total reinforcement.
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General Conditions for Constructive and
Destructive Interference for Coherent Sources
that are in Phase
• If the path-length difference between the wave trains
is integral multiple of the wavelength, then there is
constructive interference:
Δr = r2 − r1 = mλ. (m = 0, 1, 2, 3,...)
• If the path-length difference between the wave trains
is a half-integral multiple of the wavelength, then there
is destructive interference:
⎞
1
Δr = r2 − r1 = m + ⎟⎟⎟ λ. (m = 0, 1, 2, 3,...)
2⎠
⎛
⎜
⎜⎜
⎝
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Two speakers are connected to a single sound source so
that they emit monochromatic sound waves in phase. A
person stands at point that is 4.0 m from one speaker and
3.25 m from the other. If the wavelength of the sound is 0.5
m,
1. the person hears a sound louder than either speaker by
itself.
2. the person hears a sound softer than either speaker by
itself.
3. the person hears a sound that is the same loudness as
either speaker by itself.
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Two-Source Interference of Light
Young’s Double-Slit Interference Experiment
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Two-Source Interference of Light
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Double-Slit Interference of Light
• Bright regions (fringes) are seen where there is
constructive interference and dark fringes where there is
destructive interference. The path-length difference
between the two wave trains is dsinθ. Thus, for two-slit
interference, we have:
d sinθ = mλ (m = 0, 1, 2,...)
(constructive interference)
⎛
⎞
⎜
1
d sinθ = ⎜ m + ⎟⎟ λ (m = 0, 1, 2,...)
⎜
2 ⎟⎠
⎝
(destructive interference)
• The position of the center of the mth fringe is given by
ym = L tanθm.
θm can be obtained from the condition for constructive or
destructive interference.
If θm is very small, i.e., ym L, then tanθ m ≈ sinθm. Hence,
ym = mλ , (bright fringes)
L d
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i.e.,
ym = mλ L. (bright fringes, small angles)
d
(1) Workbook: Chapter 17, Question 5
(2) Textbook: Chapter 17, Problem 9
Two-Source Sound Interference
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The Diffraction Grating
• A diffraction grating is a large
number of closely spaced parallel
slits, all having the same width and
center-to-center separation.
• If the slit separation is d, then the
path difference between waves from
adjacent slits is dsinθ. Thus, like twoslit interference, the condition for
constructive interference (bright
fringes) is
dsinθ = mλ (m = 0, 1, 2,..)
• Fringes for diffraction gratings are
usually called lines because they are
extremely sharp. Any small deviation
from the angles producing
constructive interference results in
almost complete destructive
interference.
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Diffraction Grating
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Workbook: Chapter 17, Questions 6, 7
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By passing red laser light through a diffraction grating,
you produce an interference pattern on a screen.
Changing to green laser light with the same diffraction
grating, you would produce a new pattern that is
1. the same as the red.
2. the same as the red, except that the
pattern is sharper.
3. similar to the red, but the pattern is
more spread out.
4. similar to the red, but the pattern is
more compressed.
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