Coupled Wave Machine (3D)

The machine

A row of N = 64 bars (pendulums) hangs from one long torsion wire. Each bar can swing about the wire axis by an angle θ_i. Neighbouring bars feel each other only through the twist of the wire between them, so a disturbance has to travel along the wire as a wave. This is the classic Shive / Bell Labs wave machine, here with an optional pendulum restoring torque (gravity on the bobs) and a motor that twists the left end.

Equation of motion

μi θ̈i = c² (θi+1 − 2θi + θi−1) − μi ω0² sin θi − μi γi θ̇i

Here c² = κ/I₁ (wire torsion constant over the inertia of a normal bar, in bars²/s²), μ_i = I_i/I₁ is the relative inertia of the bob, ω₀² = m g ℓ/I is the pendulum frequency (a heavier bob has both more inertia and more gravity torque, so ω₀ does not depend on μ) and γ is a friction rate. For small angles sinθ ≈ θ and the equation is linear (the Klein–Gordon lattice); the pendulum check box switches on the full sinθ, which is the discrete sine-Gordon equation.

Dispersion and the band gap

A plane wave θ_i = A cos(k i − ω t) satisfies ω² = ω0² + (4c²/μ) sin²(k/2). For long waves (k small) this is ω² ≈ ω₀² + (c k)²/μ: with ω₀ = 0 the wave speed is c/√μ for every wavelength (no dispersion), but with gravity on, long waves are faster than short ones in phase and the group speed vg = (c²/μ) sin k / ω drops to zero at the cutoff ω₀ and at the band top ω_max = √(ω₀² + 4c²/μ). A motor below ω₀ (or above ω_max) cannot launch a travelling wave: the disturbance dies out exponentially (evanescent) within a few bars, exactly like light below the plasma frequency.

Reflection at a change of load

If the bobs become μ₂ times heavier on the right, the wire stays the same but the characteristic impedance changes, Z ∝ √(κ I) ∝ √μ. Continuity of angle and torque at the step gives the amplitude coefficients R = (√μ1 − √μ2)/(√μ1 + √μ2) and T = 1 + R (valid when the pulse is long compared with the bar spacing). Going into a heavier medium R < 0: the reflected pulse is inverted, as at a fixed end; going into a lighter one R > 0, as at a free end. The foam absorber is a sponge layer whose damping rises smoothly (as s²) over the last 16 bars, so almost nothing returns.

Kinks

The sine-Gordon equation has the exact travelling solution θ = 4 arctan exp[γ_L(x − v t)/d] with d = c/ω₀ and γ_L = 1/√(1 − v²/c²): a 360° twist that runs along the chain like a particle and is Lorentz-contracted as v approaches c. The page launches one at 0.4 c; check 4 compares the simulated position with x₀ + v t after 12 s.

Numerics and honest simplifications

The page integrates the 64 equations with the velocity-Verlet (kick-drift-kick) scheme at a fixed step of 5 ms, which is symplectic: energy only oscillates slightly and does not drift. The four in-page checks run separate copies of the model: lattice dispersion for a clamped standing mode, the reflection and transmission coefficients, energy conservation at 126° amplitudes, and the kink speed. Simplifications: the wire has no mass of its own (its inertia is lumped into the bars), friction is linear, the motor imposes the angle exactly, the foam is a numerical sponge, not a material model, and the lattice is a good approximation of a continuous string only for wavelengths much longer than the bar spacing (the dispersion formula above is exact for the lattice).