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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Standing waves: the
sum of two oppositely
Beats: the sum of two different frequencies
Group velocity: the speed of information
Going faster than light...
Otherwise they'd get screwed up while overlapping
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!
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.
The points where the amplitude is always zero are called “nodes.” The points where the amplitude oscillates maximally are called “anti-nodes.”
3.9 GHz microwaves
The same effect occurs in lasers.
Note the node at the reflector at left (there’s a phase shift on reflection).
Different separations. Note the different node patterns.
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.
The periodically varying amplitude is called a modulation of the wave.
Average angular frequency
Average propagation number
Modulation propagation number
Take E0 to be real.
For a nice demo of beats, check out: http://www.olympusmicro.com/primer/java/interference/
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.
If the maximum of the envelope corresponds to the point at which the phases of the components are equal, thenهلا
For nondispersive medium, v is independent of k , so
Vg = v
Light-wave beats (continued):
Etot(x,t) = 2E0 cos(kavex–wavet) cos(Dkx–Dwt)
This is a rapidly oscillating wave: [cos(kavex–wavet)]
with a slowly varying amplitude[2E0cos(Dkx–Dwt)]
The phase velocity comes from the rapidly varying part: v = wave / kave
What about the other velocity—the velocity of the amplitude?
Define the "group velocity:" vgºDw /Dk
In general, we define the group velocity as:
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. It’s 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):
This rarely occurs, however.
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.
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:
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 = wn(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0
vg = c0 / ( n + wdn/dw) = (c0 /n) / (1 + w /n dn/dw)
So the group velocity equals the phase velocity when dn/dw = 0,
such as in vacuum. Otherwise, since n increases with w, dn/dw > 0,
vg < vf
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.
vg = c0 / (n + wdn/dw)
In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies.
vg = c0 / (n + wdn/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 > c0. Frequencies outside this range have vg < c0.
Pulses of light (which are broadband) therefore break up into a mess.