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 ever have.
48a, long2a, the limit
coarsefine
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—
Drag to orbit. Each line is one spring, each crossing one mass, and a mass only ever feels its four neighbours. The grid wraps around at the edges, so nothing reflects.
What to observe
- Tap it and watch the ring spread. Now ignore the ring and stare at the single highlighted mass: it bobs up, comes back down, and stops roughly where it began. Its net drift stays near zero while the pulse crosses the whole grid. The wave travels; the matter does not. A wave is a pattern moving through stuff, not stuff moving.
- Stiffen the springs and the pulse speeds up. Make the masses heavier and it slows down. The measured crest speed tracks a√(k/m) the whole way, and yet no rule in this simulation mentions a speed. Every spring only knows how hard to pull. Sound is what a lattice does, not something added to it.
- Double k and double m together: nothing changes. Only the ratiomatters, which is why stiff light materials carry sound fast (steel, about 5 km/s) and soft heavy ones carry it slowly (lead, about 1.2 km/s).
- Switch to a plane wave with a long wavelength. Its dot sits right on the dashed straight line, the continuum prediction, and it rides at exactly c. Now shorten the wavelength. The lattice curve bends away below the line, and the wave gets slower than sound. The smooth wave equation is only the long-wavelength approximation of something grainy.
- Push the wavelength slider to its limit, λ = 2a. The wave stops travelling altogether: group velocity zero, just a standing shudder at the highest frequency the grid can carry. There is a ceiling frequency, and it exists purely because matter comes in lumps. In a real crystal that ceiling is a few terahertz, and it is why solids stop conducting sound above it.
- Try to imagine a wave shorter than 2a. Neighbouring masses are already alternating up, down, up, down. There is nothing left to subdivide, so there is no such wave. Discreteness does not just approximate the continuum badly at short scales, it runs out of scales entirely.
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