6' Standing Waves, Beats, and Group Velocity

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6. Standing Waves, Beats, and Group Velocity
Superposition again Standing waves the sum of
two oppositely traveling waves
Beats the sum of two different frequencies
Group velocity the speed of information Going
faster than light...
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Superposition allows waves to pass through each
other.
Otherwise they'd get screwed up while overlapping
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Now well add waves with different complex
exponentials.

• It's easy to add waves with the same complex
  exponentials

where all initial phases are lumped into E1, E2,
and E3. But sometimes the complex exponentials
will be different!
Note the plus sign!
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Adding waves of the same frequency, but opposite
direction, yields a "standing wave."
Waves propagating in opposite directions
Since we must take the real part of the field,
this becomes
(taking E0 to be real)
Standing waves are important inside lasers, where
beams are constantly bouncing back and forth.
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A Standing Wave
The points where the amplitude is always zero are
called nodes. The points where the amplitude
oscillates maximally are called anti-nodes.
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A Standing WaveAgain
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A Standing Wave
Youve seen the previews. Now, see the movie!
Nodes
Anti-nodes
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A Standing Wave Experiment

• 3.9 GHz microwaves

Mirror
Input beam
The same effect occurs in lasers.
Note the node at the reflector at left (theres a
phase shift on reflection).
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Interfering spherical waves also yield a
standing wave

• Antinodes

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Two Point Sources
Different separations. Note the different node
patterns.
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Beats and Modulation
If you listen to two sounds with very different
frequencies, you hear two distinct tones.
But if the frequency difference is very small,
just one or two Hz, then you hear a single tone
whose intensity is modulated once or twice every
second. That is, the sound goes up and down in
volume, loud, soft, loud, soft, , making a
distinctive sound pattern called beats.
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Beats and Modulation
The periodically varying amplitude is called a
modulation of the wave.
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When two waves of different frequency interfere,
they produce beats.
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When two waves of different frequency interfere,
they produce beats.
Average angular frequency Modulation
frequency Average propagation number Modulation
propagation number
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When two waves of different frequency interfere,
they produce beats.
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When two waves of different frequency interfere,
they produce "beats."
Indiv- idual waves Sum Envel- ope Irrad- ian
ce
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When two light waves of different frequency
interfere, they also produce beats.
Take E0 to be real.
For a nice demo of beats, check out
http//www.olympusmicro.com/primer/java/interferen
ce/
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Group velocity

• a wave-group, with given whose
  amplitude is modulated so that it is limited to a
  restricted region of space at time t 0.

The energy associated with the wave is
concentrated in the region where its amplitude is
non-zero.
At a given time, the maximum value of the
wave-group envelope occurs at the point where all
component waves have the same phase..
This point travels at the group velocity it is
the velocity at which energy is transported by
the wave.
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Group velocity
If the maximum of the envelope corresponds to the
point at which the phases of the components are
equal, then???
For nondispersive medium, v is independent of k ,
so Vg v
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Group velocity
Light-wave beats (continued) Etot(x,t) 2E0
cos(kavexwavet) cos(DkxDwt) This is a rapidly
oscillating wave cos(kavexwavet) with a
slowly varying amplitude 2E0 cos(DkxDwt) The
phase velocity comes from the rapidly varying
part v wave / kave What about the other
velocitythe velocity of the amplitude? Define
the "group velocity" vg º Dw /Dk In
general, we define the group velocity as
carrier wave
irradiance
vg º dw /dk
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The group velocity is the velocity of a pulse of
light, that is, of its irradiance.
While we derived the group velocity using two
frequencies, think of it as occurring at a given
frequency, the center frequency of a pulsed wave.
Its the velocity of the pulse.
When vg vf, the pulse propagates at the same
velocity as the carrier wave (i.e., as the phase
fronts)
z
This rarely occurs, however.
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When the group and phase velocities are different
More generally, vg ? vf, and the carrier wave
(phase fronts) propagates at the phase velocity,
and the pulse (irradiance) propagates at the
group velocity (usually slower).
The carrier wave
The envelope (irradiance)
Now we must multiply together these two
quantities.
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Group velocity (vg) vs. phase velocity (vf)
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The group velocity is the velocity of the
envelope or irradiance the math.
The carrier wave propagates at the phase velocity.
And the envelope propagates at the group velocity
Or, equivalently, the irradiance propagates at
the group velocity
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Calculating the Group velocity
vg º dw /dk Now, w is the same in or out of
the medium, but k k0 n, where k0 is the
k-vector in vacuum, and n is what depends on the
medium. So it's easier to think of w as the
independent variable Using k w n(w) / c0,
calculate dk /dw ( n w dn/dw ) / c0
vg c0 / ( n w dn/dw) (c0 /n) / (1
w /n dn/dw ) Finally So the group velocity
equals the phase velocity when dn/dw 0, such as
in vacuum. Otherwise, since n increases with w,
dn/dw gt 0, and vg lt vf
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Calculating Group Velocity vs. Wavelength

• We more often think of the refractive index in
  terms of wavelength,so let's write the group
  velocity in terms of the vacuum wavelength l0.

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The group velocity is less than the phase
velocity in non-absorbing regions.

• vg c0 / (n w dn/dw)
• In regions of normal dispersion, dn/dw is
  positive. So vg lt c for these frequencies.

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The group velocity can exceed c0 whendispersion
is anomalous.

• vg c0 / (n w dn/dw)
• dn/dw is negative in regions of anomalous
  dispersion, that is, near a
• resonance. So vg can exceed c0 for these
  frequencies!

One problem is that absorption is strong in these
regions. Also, dn/dw is only steep when the
resonance is narrow, so only a narrow range of
frequencies has vg gt c0. Frequencies outside
this range have vg lt c0. Pulses of light (which
are broadband) therefore break up into a mess.