6. Standing Waves, Beats, and Group Velocity Group velocity: the speed of information Going faster than light... Superposition again Standing waves: the sum of two oppositely traveling waves Beats: the sum of two different frequencies

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. It's easy to add waves with the same complex exponentials: where all initial phases are lumped into E 1, E 2, and E 3. 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." Since we must take the real part of the field, this becomes: (taking E 0 to be real) Standing waves are important inside lasers, where beams are constantly bouncing back and forth. Waves propagating in opposite directions:

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 Again…

A Standing Wave You’ve seen the previews. Now, see the movie! Nodes Anti-nodes

A Standing Wave: Experiment 3.9 GHz microwaves Note the node at the reflector at left (there’s a phase shift on reflection). Mirror Input beam The same effect occurs in lasers.

Interfering spherical waves also yield a standing wave Antinodes

Two Point Sources Different separations. Note the different node patterns.

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

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 beats.

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. Take E 0 to be real. For a nice demo of beats, check out: primer/java/interference/

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 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 V g = v

Group velocity v g  d  /dk Light-wave beats (continued): E tot (x,t) = 2E 0 cos(k ave x–  ave t) cos(  kx–  t) This is a rapidly oscillating wave: [cos(k ave x–  ave t)] with a slowly varying amplitude [2E 0 cos(  kx–  t)] The phase velocity comes from the rapidly varying part: v =  ave / k ave What about the other velocity—the velocity of the amplitude? Define the "group velocity:" v g   /  k In general, we define the group velocity as: “carrier wave” irradiance

The group velocity is the velocity of a pulse of light, that is, of its irradiance. When v g = v , the pulse propagates at the same velocity as the carrier wave (i.e., as the phase fronts): 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. This rarely occurs, however. z

When the group and phase velocities are different… More generally, v g ≠ v , 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.

Group velocity ( v g ) vs. phase velocity ( v  )

The group velocity is the velocity of the envelope or irradiance: the math. And the envelope propagates at the group velocity: Or, equivalently, the irradiance propagates at the group velocity: The carrier wave propagates at the phase velocity.

v g  d  /dk Now,  is the same in or out of the medium, but k = k 0 n, where k 0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of  as the independent variable: Using k =  n(  ) / c 0, calculate: dk /d  = ( n +  dn/d  ) / c 0 v g  c 0  n  dn/d  ) = (c 0 /n) / (1 +  /n dn/d  ) Finally: So the group velocity equals the phase velocity when dn/d  = 0, such as in vacuum. Otherwise, since n increases with , dn/d  > 0, and: v g < v  Calculating the Group velocity

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 0.

The group velocity is less than the phase velocity in non-absorbing regions. v g = c 0 / (n +  dn/d  ) In regions of normal dispersion, dn/d  is positive. So v g < c for these frequencies.

The group velocity can exceed c 0 when dispersion is anomalous. v g = c 0 / (n +  dn/d  ) dn/d  is negative in regions of anomalous dispersion, that is, near a resonance. So v g can exceed c 0 for these frequencies! One problem is that absorption is strong in these regions. Also, dn/d  is only steep when the resonance is narrow, so only a narrow range of frequencies has v g > c 0. Frequencies outside this range have v g < c 0. Pulses of light (which are broadband) therefore break up into a mess.