Space Elevator Tether Wave Propagation

Tether Material

Safety factor 1.5×

Counterweight altitude 1.5× GEO

Playback speed 6×

Readouts

Transverse wave speed c = √(σ_eff/ρ)

—

One-way travel time (anchor → counterweight)

—

Simulated elapsed time

—

How this 2D model differs from the 3D view

The 3D simulator plugs the material into the closed-form taper equation and renders the resulting static shape. This 2D simulator instead treats the tether as a real vibrating string and numerically integrates the transverse wave equation along it, so the picture is a live physics solve, not a lookup:

∂²u/∂t² = c(r)² · ∂²u/∂s²        u(s,t) = sideways displacement

c(r)² = T(r) / μ(r) = [σ_eff·A(r)] / [ρ·A(r)] = σ_eff / ρ

A(r) cancels exactly — because the taper is *designed* to keep stress
σ = T(r)/A(r) constant at every altitude, the wave speed c is the same
number everywhere along the cable, from the ground station to the
counterweight, however wildly A(r) itself varies.

An explicit finite-difference scheme steps u(s,t) forward: the anchor end is fixed (u=0, bolted to the ground station), the counterweight end carries real inertia (its own equation of motion, driven by the tension of the last segment), and everywhere in between the same uniform c drives the update — verified against a standalone numerical check that the simulated pulse crosses a known distance in the predicted time.

Material — sets ρ and σ, hence c. Carbon nanotube's huge specific strength gives it a transverse wave speed around 8 km/s; steel is only ~650 m/s.
Safety factor — divides the working stress σ_eff = σ/SF, so c shrinks as √(1/SF).
Counterweight altitude — sets the tether length, hence the one-way and round-trip travel time (the speed itself is unaffected).
Pluck the tether — injects a displacement pulse at a random point and lets it propagate and reflect off both ends, exactly like plucking a real string — except this "string" is up to a hundred thousand kilometres long.

Real-world relevance: this is the same mechanism (transverse traveling waves) engineers must damp out of a real tether — climbers, wind loads at the anchor, and orbital tugs from the Moon would all excite it, and a fixed wave speed everywhere makes the resonant modes predictable.

Static taper profile (reference) Live transverse wave u(s,t) GEO altitude

Click / tap the tether to pluck it there