M070 · Waves on a chain of masses

Mechanical Lattice & Dispersion

How does a wave packet travel when its crests and its envelope move at different speeds? The crests slide through the packet at the phase velocity, while the envelope, and the energy with it, moves at the group velocity. The group velocity is the slope of the curve of frequency against wavenumber. A mass defect in the chain can hold a vibration in place around it instead of letting it travel.

Build a chain of equal or alternating masses, launch an acoustic or optical wave packet, and compare how the crests, the envelope and the energy move. Add a mass defect, excite one of the chain's normal modes and measure how localized it is. In the 3D view, pick a site to see the two spring forces on it, and review how energy and wall impulse change over the recorded run. Click the rail in the data row to place the packet. The finite-chain modes and the infinite-lattice curves are computed separately.

Inspected site number $0$

Inspected longitudinal displacement $0$

Inspected longitudinal velocity $0$

Force from left bond $0$

Force from right bond $0$

Net longitudinal force $0$

Inspected longitudinal acceleration $0$

Drawing displacement / equilibrium spacing scale $0$

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 the chain and record its motion

How does a wave packet travel when its crests and its envelope move at different speeds?

Inspected bond forcesTeal shows the left bond's force, gold the right bond's and violet their sum. All three share one force scale at each instant.Inspected velocityBlue arrow with its own scale, separate from force. Rightward is positive.Prepared release outlinesWire outlines mark the starting positions, which stay fixed.Inspect site along the chain

s/L

50.00\,\%

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\,\%

All quantities are normalized. You can work the rail and the displacement profile directly, or set the sliders from the keyboard. The record stops when the motion leaves the model's valid range or the step limit is reached; the motion itself is never clipped.

The 3D apparatus is schematic. Displacements along the chain are drawn at a fixed scale for each record, separate from the spacing between masses, and the body sizes and coils do not model collisions. Colour shows each site's current energy relative to the current peak. All quantities are in units of the reference mass, the nearest-neighbour spacing and the reference stiffness. The masses move only along the chain; the profile below plots displacement on a vertical axis. The walls are fixed. The infinite two-mass reference uses equal springs and a two-site cell, and a single mass defect does not change it. A Gaussian packet has a finite bandwidth, is cut off at the chain ends and reflects from the walls, so the energy centroid in the finite chain need not drift at the carrier group velocity. The optical branch can have a negative group velocity. The finite-chain modes come from a symmetric mass-weighted matrix, and the residual is shown. For this linear model the harmonic energy is exact.

Physics tutorial

Waves carry an envelope through a discrete chain

BackgroundWhen masses are joined to their neighbours by springs, the whole chain vibrates in collective modes. If the masses alternate between two values, the frequencies of an infinite chain split into two bands, the acoustic and optical branches. In a finite chain, a single mass that differs from the rest can instead hold a mode that stays localized around it.

Why it mattersMIT lecture notes on lattice vibrations derive the acoustic and optical branches for an infinite chain. Here you compare that prediction with the motion of a finite chain whose ends are fixed.

Start with the essentials

Focus question: Which velocity carries the packet’s energy?
One-sentence intuition: Compare the group velocity at the carrier, the slope of the dispersion curve, with how fast the energy centroid of the finite chain actually moves. Then check a mass defect by its mode shape and by how many sites take part.

Core mathematical model

Newton’s second law for each site

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

The two fixed ends push on the chain and change its momentum, but they do no work on it.

Dispersion of a two-mass chain with equal springs

\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}}

Neighbouring sites are one unit apart, so the repeating two-site cell is two units long.

Phase and group velocities

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

The derivative taken at the carrier wave number is a reference for a narrow-band packet on an infinite chain.

Normal modes of the finite chain

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

The residual shown checks how well every computed mode satisfies this matrix equation.

How localized a mode is, and where the energy sits

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

The participation count measures how many sites share the mass-weighted mode shape. The energy centroid follows where the local energy is.

Common difficulties

The optical branch is still a mechanical vibration

Typical misconceptionThe optical branch must be simulating light.

Better mental modelThe name comes from the two sublattices moving against each other. Here both branches are vibrations of a chain of springs.

A finite chain can have modes outside the bulk bands

Typical misconceptionA finite chain cannot have a mode whose frequency lies outside the bands of the infinite chain.

Better mental modelA mode localized at a defect does not have to fall inside a bulk band. Look at the eigenvector of the finite chain to see it.

Walls and packet width affect the energy drift

Typical misconceptionThe energy centroid always moves at the group velocity.

Better mental modelA real packet has a finite bandwidth, its envelope is cut off at the chain ends, and waves reflect from the walls. Each of these changes what the finite chain measures.

Run the experiment

01
Check the forces on one site in 3D
Pause, choose a site to inspect and use the fine motion step. Compare the teal and gold bond forces with their violet sum and with the acceleration readout.
What to observe: The masses move only along the chain. The displacement profile below is just a graph, and the body size and drawing magnification are schematic.
02
Follow energy and wall impulse
Open Energy & wall impulse, scrub through the acquired record, then return to Motion & modes.
What to observe: The full energy breakdown and the signed numerical error follow the same cursor. The fixed walls exchange momentum with the chain but do no work.
03
Track the acoustic packet
Acquire a record for the uniform chain and scrub through its space-time plot.
What to observe: The envelope, the crests and the energy centroid are three different things to measure.
04
Open a band gap
Select alternating masses and vary their ratio.
What to observe: The two branches in the reduced zone pull apart; with equal masses the gap closes.
05
Watch the optical envelope run backward
Acquire the optical packet and check the sign of its group velocity.
What to observe: The envelope can travel in the opposite direction to the crests.
06
Measure a defect mode
Select the light-defect release, inspect the eigenvector, then change the selected mode.
What to observe: The high-frequency mode stays localized, so its participation count is small.

Core mathematical model

Newton’s second law for each site

Dispersion of a two-mass chain with equal springs

Phase and group velocities

Normal modes of the finite chain

How localized a mode is, and where the energy sits

Common difficulties

The optical branch is still a mechanical vibration

A finite chain can have modes outside the bulk bands

Walls and packet width affect the energy drift

Run the experiment

Check the forces on one site in 3D

Follow energy and wall impulse

Track the acoustic packet

Open a band gap

Watch the optical envelope run backward

Measure a defect mode