The Vibrating String: Standing Waves, Harmonics and Timbre

The one-dimensional wave equation

An ideal stretched string's transverse displacement y(x,t) obeys the 1D wave equation, with a wave speed set entirely by the string's own physical properties — its tension T and its mass per unit length μ:

∂²y/∂t² = c² * ∂²y/∂x²
c = sqrt(T / μ)

Fixed ends force a discrete set of notes

A real string is clamped at both ends, y(0,t) = y(L,t) = 0, and that boundary condition is the whole reason a string produces musical notes rather than an arbitrary continuum of tones: only wavelengths that fit an exact number of half-waves between the two fixed ends can exist as a steady standing-wave pattern at all, so the allowed frequencies form a discrete harmonic series:

λ_n = 2L / n              (n = 1, 2, 3, ...)
f_n = n * c / (2L) = n * f_1     (f_1 = the fundamental)

Nodes, antinodes, and everything happening at once

The nth harmonic has n antinodes (points of maximum motion) and n+1 nodes (motionless points, including the two fixed ends). A plucked string does not vibrate in only one of these modes at a time — the general motion of any real pluck is a superposition of the fundamental and every harmonic above it simultaneously, exactly as Fourier's theorem guarantees for any periodic shape, with the relative strength of each harmonic determined entirely by the initial pluck shape.

Why tension and thickness change the pitch

Since c = √(T/μ) and f_1 = c / (2L), the fundamental frequency rises with the square root of tension and falls with the square root of linear density — which is exactly the physical basis of tuning a guitar (turning the tuning peg changes T) and of why a bass string sounds lower than a treble string of the same length and tension (a thicker, heavier-wound string has a larger μ, so it needs a lower frequency to satisfy the same wave speed relationship).

The overtone spectrum of a real pluck

Decomposing a triangular pluck shape (displaced at one point x₀ and released) into its Fourier sine series shows the amplitude of the nth harmonic scales as roughly (1/n²) · sin(nπx₀/L):

A_n ∝ (1 / n²) * sin(n * π * x₀ / L)

Plucking near one end (x₀ close to 0 or L) pushes energy into higher harmonics, since sin(nπx₀/L) stays large for more values of n, giving a brighter, twangier tone. Plucking exactly at the midpoint (x₀ = L/2) sends sin(nπ/2) to zero for every even n, silencing every even harmonic completely and leaving only the odd ones — a purer, more open sound that any guitarist plucking over the middle of the string is producing whether they know the Fourier series behind it or not.

Frequently asked questions

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