Silk Ribbon Dance — 2D Standing Waves and Resonance Spectrum

Waves on a ribbon

For small transverse motion the ribbon obeys the wave equation with damping, ∂²y/∂t² = c² ∂²y/∂x² - γ ∂y/∂t, where c = √(T/μ). With both ends fixed the allowed standing waves have fn = n c / (2L); with one end free they are fn = (2n-1) c / (4L). Driving the hand end at one of these frequencies pumps energy in each cycle and the amplitude grows until damping balances it.

What this view adds

A live time trace of the ribbon at 1 m and a Hann-windowed frequency spectrum of that trace, with the predicted harmonics drawn over it. Move the hand frequency across a harmonic and the peak jumps up.

Simplifications

A real silk ribbon has bending stiffness, air drag that depends on speed, and moves in 3D with twist; this 1D string model has none of that, so the harmonics stay exactly evenly spaced. The explicit finite-difference scheme runs at half the stability limit (Courant number 0.5) on 240 nodes. Large amplitudes are not physical for the small-angle wave equation, and the picture is vertically exaggerated.