Musical Instruments

PDEs · Axiom Academy

REAL WORLD Musical Instruments How PDEs explain the mathematics of sound and music Every musical instrument you've ever heard—from guitars and violins to drums and pianos—produces sound through vibrations governed by partial differential equations. The wave equation isn't just abstract mathematics; it's the reason a guitar sounds different from a violin, and why a drum can produce so many different tones. When you pluck a guitar string or strike a drum, you're setting up a physical system that solves a PDE in real-time. The mathematics determines everything: the pitch, the harmonics, and the unique timbre that makes each instrument recognizable. String instruments like guitars, violins, and pianos all work on the same principle: a string stretched between two fixed points vibrates when disturbed. The vibration of the string is governed by the 1D wave equation: where u(x,t) is the displacement of the string at position x and time t The frequency (pitch) of a vibrating string depends on three key physical properties: Try adjusting the sliders to see how each parameter affects the pitch. Notice how shortening the string (like pressing a fret on a guitar) raises the pitch, while increasing tension also makes the note higher. The Fundamental Frequency Formula The fundamental frequency of a vibrating string is given by the elegant formula: L = length of string (m) T = tension force (N) ρ = linear mass density (kg/m)

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