Why does changing κ alter the wave speed?

κ plays the role of a nearest-neighbor spring constant in angle space; larger κ stiffens the chain against curvature in the discrete profile, increasing band dispersion and apparent pulse speeds in the linear regime.

Are the ends fixed in angle?

The discrete Laplacian uses θ = 0 ghost neighbors beyond the ends, approximating pinned ends for the coupling term.

Coupled pendulum chain

A row of identical pendula with weak springs coupling neighboring angular displacements behaves like a discrete elastic medium: impulses travel as waves and reflect from free or fixed ends. The equations θ̈ᵢ = −(g/L) sin θᵢ + κ(θᵢ₋₁ − 2θᵢ + θᵢ₊₁) are integrated in time with light damping so patterns remain readable. Normal-mode language from linear algebra applies after linearization for small angles; here nonlinear sine terms keep large-amplitude kicks interesting.

Who it's for: Waves on a lattice, coupled oscillators, and Newtons-cradle analog discussions.

Key terms

Coupled pendula
Discrete wave equation
Normal modes
Group velocity
Reflection

How it works

Identical pendula hung in a row with weak springs between neighbors: θ̈ᵢ = −(g/L) sin θᵢ + κ(θᵢ₋₁ − 2θᵢ + θᵢ₊₁). Try the center impulse and watch the wave travel and reflect — a discrete chain analog of Newton’s cradle.

Frequently asked questions

Why does changing κ alter the wave speed?: κ plays the role of a nearest-neighbor spring constant in angle space; larger κ stiffens the chain against curvature in the discrete profile, increasing band dispersion and apparent pulse speeds in the linear regime.
Are the ends fixed in angle?: The discrete Laplacian uses θ = 0 ghost neighbors beyond the ends, approximating pinned ends for the coupling term.