String Vibration: How a Single Pluck Becomes a Chord of Harmonics

The equation behind every guitar string

A stretched string — a guitar string, a violin string, a rope shaken at one end — obeys the one-dimensional wave equation, one of the oldest and most-studied partial differential equations in physics, first derived by Jean le Rond d'Alembert in 1746:

∂²y/∂t²  =  c² · ∂²y/∂x²,     c = √(T / μ)

y(x,t) = transverse displacement of the string at position x, time t
T      = string tension
μ      = linear mass density (mass per unit length)
c      = wave speed along the string

Tightening the string (raising T) or using a lighter string (lowering μ) both increase the wave speed c, which is exactly why tuning a guitar string tighter raises its pitch, and why the thick, heavy bass strings on a guitar sound lower than the thin treble strings at the same tension and length.

Fixed ends turn travelling waves into standing waves

A string clamped at both ends cannot support just any wave — only waves whose displacement is forced to be exactly zero at both endpoints for all time. That constraint only permits a discrete family of shapes, the normal modes, each a whole number of half-wavelengths fitting exactly between the two fixed ends:

f_n = n · (c / 2L) = n · f_1,    n = 1, 2, 3, ...

L   = string length
f_1 = fundamental frequency (n=1)
f_n = the n-th harmonic (overtone), a whole-number multiple of f_1

The n=1 mode is the fundamental: the whole string moves up and down together as one long arc, with no interior point standing still. The n=2 mode splits the string into two arcs, moving in opposite directions, with one motionless point exactly in the middle. Every mode has nodes — points that never move — and antinodes — points of maximum displacement — and the n-th harmonic always has exactly n-1 interior nodes.

Why plucking a string produces every harmonic at once

A real pluck does not excite a single clean mode — it excites a whole superposition of harmonics simultaneously, each with its own amplitude determined by exactly where and how the string was disturbed. the Fourier theorem guarantees this decomposition is always possible: any shape the string can take, however irregular, can be written as a weighted sum of the pure sine-shaped normal modes. Plucking near the middle emphasises the fundamental and odd harmonics; plucking near one end brings in far more high harmonics, which is why a note played close to the bridge on a guitar sounds brighter and thinner — richer in high-frequency content — than the same note played over the sound hole.

Why real strings decay, and why that decay shapes timbre

An idealised wave equation with no damping term predicts a string that vibrates forever once plucked. Real strings lose energy continuously — to air resistance, to internal friction as the string flexes, and by radiating sound through whatever the string is attached to — and each harmonic decays at its own rate, generally faster for higher harmonics. This uneven decay is a large part of why a plucked note's timbre changes noticeably over its lifetime: bright and complex immediately after the pluck, warmer and simpler as the higher harmonics die away first, leaving mostly the fundamental ringing at the end.

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