Time Behavior
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)=
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
Number of k-waves shown =
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