Many Coupled Oscillators
Chain of masses — standing waves. Builds on Normal Modes.
You have a chain of 10 identical masses connected by identical springs, fixed at both ends (like beads on a string). Each mass can move transversely (up and down).
How many distinct normal modes does this system have?
Exactly 10 normal modes! A system with N masses has N degrees of freedom, and therefore exactly N normal modes. Each mode is a standing wave pattern: mode 1 has one half-wavelength across the chain, mode 2 has two, and so on up to mode 10. There is a maximum frequency (called the cutoff frequency) above which no modes exist — the discrete chain cannot support wavelengths shorter than twice the mass spacing.
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Watch each normal mode in isolation. Mode 1 has all masses moving together in a smooth arc. Mode 10 has adjacent masses moving in opposite directions. Try exciting a single mass and watch the wave patterns that emerge.
ω_n = 2√(k/m) · sin(nπ / 2(N+1))
At low n, frequencies are nearly equally spaced (like a continuous string). At high n, they crowd together near the cutoff. The sine function ensures a maximum frequency exists — the lattice has a natural speed limit.
Atoms in a solid form a 3D chain of masses and springs. Their normal modes are called phonons. The cutoff frequency determines the maximum phonon energy, which is directly related to the Debye temperature and heat capacity.
A guitar string is a chain of atoms so dense that it behaves as continuous. But at extremely high frequencies (terahertz range), the discrete atomic nature would impose a cutoff — a real string can’t vibrate at arbitrarily high frequencies.
A digital signal is a discrete chain of samples. The Nyquist frequency (half the sampling rate) is exactly analogous to the lattice cutoff frequency. You cannot represent frequencies higher than Nyquist — they ‘alias’ to lower frequencies.
Einstein and Debye used the normal modes of atomic chains to explain why the heat capacity of solids drops at low temperatures. The discrete mode spectrum means fewer modes are excited as temperature decreases.
“A chain of N masses is the bridge between single oscillators and continuous waves. It has N modes, a cutoff frequency, and a dispersion relation. Make N very large, and waves emerge from the discrete world.”
Chain of Oscillators Lab
Build a chain of masses and springs with adjustable N (from 2 to 50). Excite individual normal modes or create arbitrary initial conditions. Watch the dispersion relation emerge as you increase N.
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Start with N=3 masses and view all 3 normal modes. Notice that mode 1 is a smooth wave, mode 2 has a node in the middle, and mode 3 has adjacent masses moving opposite. Now increase to N=10 and watch the mode spectrum: low modes have long wavelengths and nearly equally spaced frequencies. High modes crowd together near the cutoff. Try N=50 — the low modes look exactly like standing waves on a continuous string! Pluck a single mass and watch how the wave propagates and reflects from the boundaries.
The dispersion relation ω(k) of a discrete chain is NOT linear — it curves and saturates at the cutoff frequency. This means high-frequency waves travel slower than low-frequency waves. Only in the long-wavelength limit does the chain behave like a continuous, non-dispersive string. Every real material has this discrete structure hiding at the atomic scale.