A wind chime tube is not a string and not an air column — it is a free-free vibrating bar (both ends unsupported, hanging from a node point). Its fundamental bending frequency comes from Euler-Bernoulli beam theory:
- f1 = (β12 / 2πL²)·√(EI/ρA), with β1L = 4.730 for the first free-free mode.
- Collecting the material/cross-section terms into one constant C gives the compact form used here: f1 = C / L² — frequency drops with the square of length, not linearly like a plucked string (f ∝ 1/L) or a wind instrument's air column.
- C ≈ 115 Hz·m² for a thin-walled 19 mm aluminum tube, matching real wind-chime tube lengths of ~0.15–0.8 m ringing from roughly 180 Hz to 5 kHz.
Drag the length slider: halving the tube's length quadruples its frequency (two octaves up), not one — the signature of bending-mode resonance. Raise the wind gust strength to swing the central clapper into the tubes at random; each strike rings that tube's own sine tone with a short exponential decay, exactly like a real hanging chime.
This is the flat 2D companion to the 3D wind-chime simulation: the same free-free bar acoustics, viewed as a side-on rig so the length→pitch relationship is directly readable.