T-Scale=

 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 Phasor Hand Location Point Longitudnal Wave Fourier IC Color
 Show Ring Molecule Show Dispersion Show k-Component waves
 Number of k-waves shown = Rank k-waves by Amp
 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