T-Scale=

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Type Graphs Character Table Quantum Carpet

Time Behavior Pause at End Loop back to t=0 Continue

Time Start (% Period) =

Time End (% Period)=

Del-x Width (% L) =

Excitation (Max n) =

Left (% L) =

Right (% L)=

n-Mean (% Max n)=

Peak1 Mean (% L)=

OverAll Scale =

Peak2 Mean (% L)=

Peak2 Amp (% Peak1)=

Draw Ring

m/n Labels

m-Boxcar
Draw m-Bars	m-Bars Max =

Number of Frames =

Spatial Phasor Scale =

Fourier Amplitude Scale =

Phasor Amplitude Scale =

Mode-1 Wavenumber =

Mode-1 Amplitude =

Mode-2 Wavenumber =

Mode-2 Amplitude =

Self Coupling V(1,1) =

1st-Neighbor Coupling V(1,2) =

2nd-Neighbor Coupling V(1,3) =

Space-Asymmetry (Band gap) =

Time-Asymmetry (Gauge) =

Movie Time-Scale =

Number of x-Grid Points =

Number of Osillators C(n) =

Upper Brillouin Zone order =

Lower Brillouin Zone order =

Dispersion Dependence

Aspect Ratio {W/H} =

Red Level =

Green Level =

Blue Level =

Alpha Level =

Definition Level =

|ψ| Line Width
Re(ψ) Line Width
Im(ψ) Line Width
Phasor Line Width

Envelope	Real	Imaginary
Clock	Fill clock face
Phasor Hand	Location Point
Longitudnal Wave	Fourier IC	Spectral Colors

Chapter 1 n-Oscillator Wave Rings   (n = 2, 3, 4,...,12 ) and k-waves            Demonstrations of n coupled oscillators begin with a line of four (n=4) masses that form a ring. Motion of each mass is described by a phasor clock which plots the mass position versus its velocity. The gray clock or mass at the extreme right is a copy of the one at the extreme left and represents the completion of the ring. Each clock also is a plot of the real versus imaginary parts of a complex phasor exp(ikx-iwt) as explained in the text. The wave number k is the number of 'kinks' or wavelengths that fit on the ring. The first demo shows one (k=1) wave on the 4-oscillator ring, and the second demo shows two (k=2) waves. Finally, the same waves are shown for a twelve member ring (n=12). Things to notice<   • For n=4 and k=1 each clock is 2π/4 radians behind its neighbor to the left. For k=2 the setback is π. In general it is 2π(k/n). (See n=2 and n=3 examples, as well.)   • The wavevector k =k(2π/L) is defined to be k in units of (2π/L) radian per meter, where L is the length (circumference) of the ring. A clock at position x meters is setback by phase k x from the clock at x=0.   • Since the clock at x=L is the same as the original clock at x=0 the setback k x must be a integer multiple of 2π, or k x=k(2π/L)x=k(2π/L)L=2πk, where: k=0,1,2...   • If our unit of distance is the radius (r=L/2π) of the ring of oscillators then wavenumber k and wavevector k are the same value and it must be an integer k=0,1,2....
Chapter 2 Standing or Galloping Waves            Adding a wave with positive k (forward moving wave) to one with negative k (backward moving wave) gives a standing wave or, more generally, a galloping wave. If the amplitude of the +k wave is equal to that of its -k partner then it's a standing wave, otherwise it is a galloping wave.
Chapter 3 Group Waves            Adding two waves with k values of the same sign but slightly different magnitudes gives a group or beat pattern. The resulting wave group moves at velocity of           vg = (w1-w2)/(k1-k2) As k1 and k2 are close then vg is close to the derivative           vg = (dw/dk) of the dispersion function w=w(k).
C(n) Characters
Quantum Carpet
RelaWavity