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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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6 standing waves beats and group velocity
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...

superposition allows waves to pass through each other
Superposition allows waves to pass through each other.

Otherwise they'd get screwed up while overlapping

now we ll add waves with different complex exponentials
Now we’ll 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!

adding waves of the same frequency but opposite direction yields a standing wave
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.

a standing wave
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.”

a standing wave experiment
A Standing Wave: Experiment

Mirror

3.9 GHz microwaves

Input beam

The same effect occurs in lasers.

Note the node at the reflector at left (there’s a phase shift on reflection).

two point sources
Two Point Sources

Different separations. Note the different node patterns.

beats and modulation
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.

beats and modulation12
Beats and Modulation

The periodically varying amplitude is called a modulation of the wave.

when two waves of different frequency interfere they produce beats14
When two waves of different frequency interfere, they produce beats.

Average angular frequency

Modulation frequency

Average propagation number

Modulation propagation number

when two waves of different frequency interfere they produce beats16
When two waves of different frequency interfere, they produce "beats."

Indiv-

idual

waves

Sum

Envel-

ope

Irrad-

iance:

when two light waves of different frequency interfere they also produce beats
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/interference/

group velocity
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.

group velocity19
Group velocity

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

group velocity20
Group velocity

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:

“carrier wave”

irradiance

vgºdw /dk

the group velocity is the velocity of a pulse of light that is of its irradiance
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. 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):

z

This rarely occurs, however.

when the group and phase velocities are different
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.

the group velocity is the velocity of the envelope or irradiance the math
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:

calculating the group velocity
Calculating the Group velocity

vgºdw /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 = wn(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0

vg = c0 / ( n + wdn/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 > 0,

and:

vg < vf

calculating group velocity vs wavelength
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.

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

vg = c0 / (n + wdn/dw)

In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies.

the group velocity can exceed c 0 when dispersion is anomalous
The group velocity can exceed c0 whendispersion is anomalous.

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.

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