A vibrating guitar string supports standing waves at discrete harmonic frequencies f_n = n·v/(2L), where n is the harmonic number, L is the string length, and v = √(T/μ) is the wave speed determined by tension T and linear mass density μ. The fundamental (n = 1) gives the pitch of the note; higher harmonics (n = 2, 3, 4…) are integer multiples of the fundamental and give the guitar its characteristic timbre. A standard A₄ string tuned to 440 Hz with L = 0.65 m requires a wave speed of v = 2 × 0.65 × 440 ≈ 572 m/s. The guitar body acts as an acoustic resonator, coupling to the string via the saddle and radiating sound through the soundhole and top plate.
Adjust string tension, length, and linear density to change the wave speed and frequencies. Select harmonic modes 1–8 to see the animated standing wave with nodes (fixed points) and antinodes (maximum displacement) labelled. The harmonic overview canvas shows all eight modes simultaneously. Click "Fourier Synthesis" to combine all harmonics up to the selected mode, revealing how the pluck waveform is built from a superposition of sinusoids.
What is a standing wave on a guitar string?
A standing wave forms when a wave travelling along the string reflects from the fixed endpoints (nut and bridge) and interferes with the incoming wave. At specific frequencies, the reflected and incident waves combine to create a pattern with stationary nodes (zero displacement, always) and antinodes (maximum displacement, oscillating). The string length must equal an integer multiple of half-wavelengths: L = n·λ/2, giving λ_n = 2L/n and f_n = v/λ_n = n·v/(2L).
How does string tension affect pitch?
Wave speed v = √(T/μ), so pitch f₁ = v/(2L) = √(T/μ)/(2L). Doubling tension multiplies wave speed by √2 ≈ 1.414, raising pitch by a musical fifth (approximately). Guitar tuning machines adjust tension: tightening raises pitch, loosening lowers it. A standard light-gauge guitar string (μ ≈ 3 g/m, L ≈ 0.65 m) tuned to E₂ (82 Hz) requires a tension of about T = (2Lf₁)² × μ = (2 × 0.65 × 82)² × 0.003 ≈ 66 N.
What are nodes and antinodes?
Nodes are points on the string that never move — displacement is always zero there. For mode n, there are n+1 nodes (including the two fixed endpoints). Antinodes are midpoints between adjacent nodes where displacement is maximum. For mode n there are n antinodes. Lightly touching a guitar string at the 12th fret (midpoint, L/2) forces a node there, suppressing odd harmonics and producing a higher-pitched, flute-like "harmonic" — the note one octave above the open string.
A pure sine wave sounds thin and electronic. When you pluck a guitar string, the initial triangular displacement shape excites many harmonics simultaneously. Each harmonic decays at a different rate (higher harmonics decay faster), giving the attack and sustain phases of the note distinct timbral characters. The relative strengths of harmonics — the harmonic spectrum — is what distinguishes a guitar from a piano or violin playing the same fundamental frequency.
Fourier's theorem states that any periodic waveform can be expressed as a sum of sinusoids at integer multiples of the fundamental frequency, each with its own amplitude and phase. For a plucked string, the Fourier series is y(x,t) = Σ Aₙ sin(nπx/L) cos(2πfₙt + φₙ). The "Fourier Synthesis" panel in this simulator superimposes harmonics n=1 up to the selected mode with decreasing amplitude Aₙ = 1/n, approximating the triangular pluck shape.
Pitch f₁ ∝ 1/L, so halving the string length doubles the frequency (raises pitch by one octave). On a guitar, pressing a string against a fret shortens the vibrating length. The 12th fret is at exactly half the string length, producing the octave. Fret positions follow 2^(1/12) ≈ 1.0595 — each fret shortens the string by a factor of the 12th root of 2, raising pitch by one semitone in equal temperament.
Linear mass density μ (kg/m) is the mass per unit length of the string. For the same tension and length, a heavier string (higher μ) has lower wave speed v = √(T/μ) and lower pitch. Guitar strings use different diameters — low E strings are much heavier than high e strings — to produce a range of pitches while keeping tensions in a comfortable range for playability. Bass strings are often wound with metal wire around a core to increase μ without excessive stiffness.
The string itself radiates very little sound because it is too thin to push air efficiently. The bridge saddle transfers string vibration to the top plate (soundboard), which vibrates as a much larger radiating surface. The air cavity inside the body resonates at the Helmholtz frequency f_H = (c/2π)√(A/VL_neck), where A is the soundhole area, V is body volume, and L_neck is the soundhole neck length. For a standard guitar, f_H ≈ 100 Hz, reinforcing bass response. The Chladni patterns of the top plate reveal which vibrational modes the soundboard supports.
Both guitar and piano strings produce overtone series, but their initial excitation differs. A piano hammer strikes the string near one end (typically at 1/7 of string length), suppressing the 7th harmonic — a deliberate design choice to avoid an out-of-tune note. A plucked guitar string's spectrum depends on where you pluck: near the soundhole gives a full, rich tone (many harmonics); near the bridge gives a bright, nasal tone (higher harmonics stronger). The different decay rates and body resonances then shape the final timbre.
The transverse motion of an ideal string under tension T with linear density μ satisfies the wave equation ∂²y/∂t² = (T/μ)·∂²y/∂x², where y(x,t) is the transverse displacement. This is a second-order PDE whose solutions are sinusoidal waves travelling at speed v = √(T/μ). With fixed boundary conditions y(0,t) = y(L,t) = 0, the allowed solutions are standing waves y_n = sin(nπx/L)·cos(2πf_n t), where f_n = nv/(2L). Real strings also have bending stiffness, which slightly raises higher harmonic frequencies — the effect called inharmonicity, audible as a slight out-of-tune quality in thick piano bass strings.