M070 · Discrete mechanics

Mechanical Lattice & Dispersion

Build a monoatomic or alternating-mass chain, launch an acoustic or optical wave packet, and compare crest, envelope and energy motion. Insert a mass defect, excite a finite-chain eigenmode and measure its localization. Rail clicks place the packet; the finite modes and infinite-lattice curves remain separate measurements.

Acquired normalized time $0$

Total mechanical energy $0$

Kinetic energy $0$

Bond potential energy $0$

Signed energy defect $0$

Chain momentum $0$

Independent net wall impulse $0$

Momentum minus initial and wall impulse $0$

Half-step displacement difference · first ten time units $0$

Actual normalized integration step $0$

Maximum integration energy defect · full record $0$

Record / model domain $0$

Infinite-chain carrier group velocity $0$

Infinite-chain carrier phase velocity $0$

Infinite-chain band-gap width $0$

Finite-chain energy centroid $0$

Acquired energy-centroid displacement per time $0$

Selected finite eigenfrequency $0$

Selected mode participation count $0$

Selected finite-mode harmonic energy share $0$

Maximum eigenpair residual $0$

RELEASE / ACQUIRE / AUDIT

How does a packet travel when its crests and envelope move differently?

Moving sites

N

40.00\,

Alternating mass ratio

m_B/m_A

1.00\,

Normalized bond stiffness

\kappa

1.00\,

Middle-site mass multiplier

m_d/m_{host}

1.00\,

Carrier wave number / reduced-zone edge

2q/\pi

0.22\,

Packet envelope width in sites

\sigma

5.00\,

Packet center along the rail

s_0/L

28.00\,

Normalized release amplitude

A

0.12\,

Packet branch · acoustic zero, optical one

b

0.00\,

Release · packet zero, eigenmode one

j

0.00\,

Finite eigenmode selector

s

20.00\,

Normalized observation window

\tau_{end}

50.00\,

Maximum normalized step

h_{max}

0.03\,

Inspect acquired record

t/t_{end}

0.00\,\%

Normalized quantities; use the rail/profile and keyboard ranges. The record stops at a model boundary or step budget; no clipping is applied.

All quantities are normalized by the reference mass, nearest-neighbour spacing and reference stiffness. Motion is longitudinal; the lower profile uses a vertical plotting axis. Walls are fixed. The infinite diatomic reference uses equal springs and a two-site cell; a single mass defect does not change that bulk reference. A Gaussian packet has finite bandwidth, truncation and reflections, so finite-chain energy-centroid drift need not equal the carrier group velocity. The optical branch can have negative group velocity. Finite eigenmodes use a symmetric mass-weighted solve with a displayed residual. Harmonic energy is exact for this linear model.

Start with the essentials

Focus question: Which velocity carries the packet’s energy?
One-sentence intuition: Compare the carrier’s group derivative with finite-chain energy-centroid motion, then check a mass defect through its eigenvector and participation count.

Core mathematical model

Longitudinal force balance

m_i\ddot u_i=\kappa(u_{i+1}-2u_i+u_{i-1})

Two fixed endpoints supply external momentum impulse but do no work.

Equal-spring diatomic dispersion

\omega_\pm^2=\kappa\left(\frac1{m_A}+\frac1{m_B}\right)\pm\kappa\sqrt{\left(\frac1{m_A}+\frac1{m_B}\right)^2-\frac{4\sin^2q}{m_Am_B}}

The nearest-neighbour spacing is one; the two-site cell has length two.

Phase and group velocities

v_p=\omega/q,\qquad v_g=\partial\omega/\partial q

A derivative at the carrier is an infinite-chain, narrow-band reference.

Finite mass-weighted eigenproblem

M^{-1/2}KM^{-1/2}e_j=\omega_j^2e_j,\qquad e_j^{\mathsf T}e_j=1

The displayed residual tests the matrix equation for every computed mode.

Localization and energy motion

P_j=\left(\sum_i e_{ji}^4\right)^{-1},\qquad \bar s_E=\frac{\sum_i s_iE_i}{\sum_iE_i}

Participation measures the number of mass-weighted sites; the centroid follows local energy.

Common difficulties

An optical mode is mechanical

Typical misconceptionThe optical branch is a simulation of light.

Better mental modelIts name describes opposite sublattice motion; here both branches are spring-chain vibrations.

A bulk gap is not every finite eigenvalue

Typical misconceptionNo finite mode can occur outside the infinite bands.

Better mental modelA localized defect mode need not lie in a bulk band; inspect the finite eigenvector.

Energy drift includes boundaries

Typical misconceptionThe centroid speed always equals group velocity.

Better mental modelFinite bandwidth, envelope truncation and reflected waves change the finite-chain measurement.

Run the experiment

01
Track the acoustic packet
Acquire the uniform chain and scrub its space-time record.
What to observe: The envelope, crests and energy centroid are different observables.
02
Open a band gap
Select alternating masses and vary their ratio.
What to observe: The two reduced-zone branches separate; equal masses close the gap.
03
Reverse the optical envelope
Acquire the optical packet and compare the sign of group velocity.
What to observe: Its envelope can move opposite the phase crests.
04
Measure a defect mode
Select the light-defect release, inspect the eigenvector, then change the selected mode.
What to observe: A high-frequency localized mode has a small participation count.