Vibrating String

Differential Equations · Axiom Academy

Vibrating String: The Wave Equation Discover how guitars, violins, and pianos turn a differential equation into music A stretched string of length L , fixed at both ends, obeys u_ tt = c^2 u_ xx — one equation drives the tension, the pitch, and the tone of everything from a guitar to a piano. Set the tension and pluck the string — watch it vibrate at its fundamental mode. The tighter the string, the faster the wave travels: . Pick a harmonic — read off its frequency Every mode is a standing wave with n-1 silent nodes in between. Its frequency is always an exact whole-number multiple of the fundamental: f_n = n f_1 — the harmonic series that makes a string sound musical. Where you pluck changes the tone The pluck position sets how much of each harmonic goes into the mix. Pluck the center and the even harmonics vanish — a mellow, pure tone. Pluck near the end and they roar back — a bright, twangy tone. Same string, same tension, different sound. Every guitar note, violin tone, and piano chord is a live solution to u_ tt =c^2u_ xx : tension sets the wave speed, the harmonic series f_n=nf_1 sets the pitch family, and the pluck point sets the timbre. The same equation — with different boundary shapes — governs bridge and building vibration modes , signal transmission through cables , and even the Schrödinger equation's wavefunctions.

This is the written version of the interactive lesson above. See the full Differential Equations course.