Physics
Phonons
Nothing here knows what a wave is. Every mass obeys Hooke's law and pulls only on its four neighbours. Out of that comes a pulse that travels at a speed nobody wrote down, and a shortest possible wavelength that no continuous material would have.
Drag to orbit; each line is one spring, each crossing one mass, and a mass only ever feels its four neighbours
Sound speed a√(k/m)—
—
Excitation
48a, long2a, the limit
coarsefine
Numbers
Sound speed a√(k/m)—
Measured crest speed—
Frequency ω (lattice)—
ω if it were continuous—
Group velocity—
Highest ω the grid allows—
Followed mass: now—
Followed mass: net drift—
What to observe
- Tap the sheet and watch the highlighted mass: it bobs and settles back, itsnet drift near zero while the ring crosses the grid. The wave travels, the matter does not.
- Stiffen the springs and the measured crest speed rises to tracka√(k/m), though no rule here mentions a speed. Doubling k and m together changes nothing: only the ratio counts.
- A long plane wave sits on the dashed continuum line. Shorten it and the lattice curve bends below the line: short waves run slower than sound.
- At λ = 2a, the edge of the Brillouin zone, neighbours move opposite ways, the group velocity is zero and the wave only shudders in place. The same zone edge flattens the electron bands inBand Theory.
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