OBSERVATION CHAMBER / ACTIVE
Phononic Band Gap Observatory
Change alternating masses to open a frequency gap. Select a Bloch wave and observe its acoustic or optical motion in a periodic chain.
89 / LATTICE / BAND GAP
Two masses. Two branches. One forbidden interval.
Canvas 2D is unavailable. Use the controls and numerical readings below.
Cyan: acoustic. Amber: optical. The shaded interval has no real-k Bloch mode. Click the plot to select k; 6–12 cells below show longitudinal displacement, amplified for visibility.
BLOCH MODE / NUMERICAL READOUT
Read the selected mode
- Angular frequency ω
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- Gap width Δω
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- Band edges ω− / ω+ at π/a
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- Displacement u / v (real gauge)
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- Mass-weighted qA / qB
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- B phase relative to gauge
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- Model time
- 0
- Complex matrix residual
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- Numerical checks
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Numerical check details
Generated report
OBSERVATION GUIDE
Follow a mode through the lattice
- Set M / m = 1. The two branches meet at the zone edge. Increase the ratio and watch the gap open.
- Select k = 0 and compare branches. Acoustic motion has equal displacements; optical motion satisfies m u + M v = 0. The zero-frequency acoustic mode is a static translation.
- Choose a nonzero k and press PLAY. Adjacent cells gain phase ka. Change K to rescale frequency while keeping the mode shape.
This is an infinite, lossless, one-dimensional periodic harmonic chain. The visible ends are only the edge of the drawing. No finite-chain driving, transmission, defect localization, edge state or topological invariant is calculated.
FORMULA / CONVENTION
Diatomic Bloch eigenproblem
ω±² = K(1/m + 1/M) ± K √[(1/m + 1/M)² − 4 sin²(ka/2)/(mM)]
D(k) = [[2K/m, −K(1+e⁻ⁱᵏᵃ)/√(mM)], [−K(1+eⁱᵏᵃ)/√(mM), 2K/M]]
a = 1 is the unit-cell length; adjacent particles are a/2 apart. m = 1 fixes the mass unit. K uses the reference spring unit and ω is angular frequency in the resulting time unit.
The normalized mass-weighted eigenvector is (x, y exp(ika/2)); physical displacements are u = x/√m and v = y/√M. Cell n uses Re[(u, v exp(ika/2)) exp(i nka − iωt)]. The plot scales both displacements by one common factor.
At equal masses and the zone edge, both frequencies coincide. The displayed vectors use the one-sided branch limit: (1, 1)/√2 for acoustic and (1, −1)/√2 for optical in the real gauge. This is a basis choice in a degenerate space.
OBSERVATION POLICY
The chain model, calculations, report and PNG are produced in your browser. No model data is uploaded. This chamber does not access a camera or microphone or measure a material.
Route
This chamber belongs to Structures & Symmetries.
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