Spring Keyboard

What the far bead heard after the last strike: time across, frequency up, a short-window spectrogram from the engine. Dotted: when the model says each frequency arrives, after one length of wire, three, five… Below F_C (dashed) every echo is a rising chirp, the low notes first; above it, faint falling ones.

Left: frequency against wavenumber for the two wave families of Wittrick’s helix (bright where the beads drive them), and the spring’s modes (dots, k = nπ/L). Right: the round trip’s group delay 2L/c_g. It starts at the echo time T_D, climbs without limit at F_C, where the waves stop, and falls again above.

The struck spring from the side, its twist exaggerated and slowed so a round trip takes two seconds; colour: how fast the wire turns. Below: that twisting speed along the wire. The pulse leaves the struck bead (left) and spreads as it goes, its quick ripples trailing.

Low down the modes sit near the note's harmonics, since the echoes come back about once a period; towards F_C the waves slow and the modes crowd together, and above it the sound changes character.

The modes pile up just below F_C (blue tick), where the waves crawl and the bead mounts hardly damp them: that band rings longest.