











































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









Chapter 1 nOscillator Wave Rings (n = 2, 3, 4,...,12 ) and kwaves 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(ikxiwt) 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 4oscillator ring, and the second demo shows two (k=2) waves. Finally, the same waves are shown for a twelve member ring (n=12).
• 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 = (w1w2)/(k1k2) 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 