Mechanical Lattice and Dispersion

How does a wave packet travel when its crests and its envelope move at different speeds? For a narrow-band packet far from boundaries, crests move at the phase velocity and the envelope carries energy at the group velocity, the slope of the dispersion curve. A mass defect can support a localized vibration that stays near the defect.

Launch a wave packet in a mass–spring chain. Compare dispersion, reflection and a localized defect mode.

Carrier group velocity $0$

Carrier phase velocity $0$

Bulk band-gap width $0$

Finite-chain energy centroid $0$

Normalized time $0$

Inspected site $0$

Site velocity $0$

Left spring force $0$

Right spring force $0$

Total mechanical energy $0$

Signed energy defect $0$

Maximum record energy defect $0$

Total momentum $0$

Net wall impulse $0$

Momentum balance defect $0$

Displayed displacement gain $0$

Selected finite-mode angular frequency $0$

Mass-weighted participation $0$

Selected mode energy share $0$

Largest eigenpair residual $0$

Numerical integration step $0$

Energy-centroid displacement per time $0$

Follow the wave packet

Alternating mass ratio

m_B/m_A

1.00\,

Carrier / reduced-zone edge

2q/\pi

0.75\,

Defect mass multiplier

m_d/m_{\rm host}

1.00\,

Packet branch

b

0.00\,

Bond stiffness

\kappa

1.00\,

Moving sites

N

18.00\,

Packet width

\sigma

2.70\,

Packet release centre

s_0/L

28.00\,\%

Release amplitude

A

0.20\,

Packet or finite mode

j

0.00\,

Finite-mode quantile

r_{\rm mode}

20.00\,\%

Observation window

\tau_{\rm end}

32.00\,

Maximum integration step

h_{\rm max}

0.03\,

Probe position

r_{\rm probe}

50.00\,\%

Defect position

r_d

50.00\,\%

The fixed-end chain has alternating masses and equal linear springs. A finite Gaussian release is recorded with velocity Verlet, including independent wall-force quadrature. Infinite periodic-chain dispersion is a separate reference; finite bandwidth, truncation and reflections affect energy-centroid motion. The defect does not change the bulk reference. A defect-mode release uses the finite-chain mass-weighted eigenproblem. Participation is mass weighted. Displacements, mass sizes, guide hardware and playback are schematic; no friction, impacts, spring mass or quantum phonons are computed.

Focus question: When do the packet and its crests move in different directions?
One-sentence intuition: The slope of the dispersion curve sets carrier group velocity. On the optical branch the slope can be negative even when the phase velocity is positive.

Core mathematical model

Each site feels two springs

\begin{gathered}F_i^L=-\kappa(u_i-u_{i-1})\\F_i^R=\kappa(u_{i+1}-u_i)\\m_i\ddot u_i=F_i^L+F_i^R\end{gathered}

Both wall displacements are zero. Local energy assigns half of each internal bond energy to each neighbour, and an end bond entirely to its end site.

Two branches of a periodic reference

\begin{gathered}\mu=m_B/m_A\\S=\kappa(1+1/\mu)\\D^2(q)=S^2-\frac{4\kappa^2}{\mu}\sin^2q\\D(q)\ge0\\\omega_\pm^2=S\pm D(q)\end{gathered}

The reduced zone runs from zero to half pi because the cell length is two. The lower branch is acoustic; the upper is optical.

Phase and group velocities

v_{\rm ph}=\frac{\omega}{q},\qquad v_g=\frac{d\omega}{dq}

The group-velocity prediction belongs to a narrow-band carrier packet in an infinite lattice. The finite energy-centroid drift can differ after dispersion, truncation and reflection.

The bulk gap for a heavy second mass

\begin{gathered}\omega_{\rm ac}^{\max}=\sqrt{2\kappa/\mu}\\\omega_{\rm op}^{\min}=\sqrt{2\kappa},\quad\mu\ge1\end{gathered}

Equal masses close this folded-zone gap. One local defect changes the finite modes but leaves the periodic bulk reference unchanged.

The finite-chain eigenproblem

\begin{gathered}H=M^{-1/2}KM^{-1/2}\\He_p=\omega_p^2e_p\\u_i\propto e_{pi}/\sqrt{m_i}\end{gathered}

The displayed participation uses the squared mass-weighted eigenvector. It is an effective participation measure and can be noninteger.

An independent light-defect limit

\begin{gathered}\omega_d^2=\frac{4\kappa}{\mu_d(2-\mu_d)}\\r=-\frac{\mu_d}{2-\mu_d},\quad0<\mu_d<1\end{gathered}

For an otherwise uniform infinite chain, successive displacement amplitudes away from the defect have the displayed ratio. Finite walls introduce a small correction.

Common difficulties

What is conserved?

Typical misconceptionA quiet bead means the vibration energy has vanished.

Better mental modelVelocity, displacement and local elastic energy change through the cycle. The full undamped chain conserves total energy; fixed walls can exchange momentum.

Run the experiment

01
Open the gap
Compare Acoustic packet and Open the band gap. Change the alternating mass ratio.
What to observe: Two branches separate in frequency; the finite spectrum can also contain a defect mode outside the bulk bands.
02
Follow a reverse envelope
Choose Envelope runs backward and inspect the space-time record.
What to observe: The optical carrier has positive phase velocity and negative group velocity; boundary reflections are visible separately.
03
Move a light defect
Choose Trap a defect mode, then drag the gold handle along the chain.
What to observe: The localized mode follows the defect. Near a wall, its finite frequency can shift while the bulk reference stays fixed.