Standing Waves & Resonance: Modes, Chladni Patterns & Q-Factor

Nodes and antinodes, not travelling energy

A standing wave forms when two counterpropagating waves of equal frequency and amplitude overlap — instead of carrying energy along, the superposition freezes into fixed nodes (zero displacement) and antinodes (maximum displacement). On a string of length L fixed at both ends, only sine-shaped modes that vanish at x=0 and x=L survive, giving a fundamental f₁ = c/(2L) and harmonics that are exact integer multiples: f₂=2f₁, f₃=3f₁, and so on — the harmonic series that underlies Western musical scales.

u(x,t) = Σₙ Aₙ · sin(nπx/L) · cos(ωₙt + φₙ)      ωₙ = nπc/L,  c = √(T/μ)
Guitar A-string (L=0.65m, T=81N, μ=3.8×10⁻⁴ kg/m): c≈462 m/s, f₁≈355 Hz ≈ A4

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Resonance and the Q-factor

Drive a lightly damped oscillator at its natural frequency ω₀ and the steady-state amplitude diverges toward F₀/(mγω₀). The quality factor Q = ω₀/γ measures how sharply that peak is defined — it also equals ω₀ times stored energy over power dissipated. A plucked guitar string in air has Q of a few hundred to a couple of thousand; a quartz crystal reaches 10⁴-10⁶; an optical Fabry-Pérot cavity can exceed 10⁹. High Q means a narrow, precisely tuned resonance that rings for a long time — and, notoriously, it's also why the Tacoma Narrows bridge (Q ≈ 40) could still be driven to destructive amplitude by wind.

Chladni figures and inharmonic drums

Extend standing waves to two dimensions and a rectangular membrane's modes become u_mn(x,y,t) = A·sin(mπx/a)·sin(nπy/b)·cos(ω_mnt) — sand sprinkled on a vibrating plate collects along the nodal lines, producing the Chladni figures Ernst Chladni demonstrated in 1787. A circular drum is different: its modes are governed by Bessel functions, whose zeros are irrational rather than evenly spaced, so a timpani's overtones are inharmonic — not integer multiples of the fundamental — which is exactly why a drum sounds less "pure" than a plucked string.

Frequently asked questions

Why does a plucked string only ring out at specific frequencies?

Fixing both ends of a string forces the displacement to zero there, which only a discrete set of sine-shaped standing-wave modes can satisfy. The fundamental is f1 = c/(2L), and every allowed higher mode is an exact integer multiple of it — the harmonic series that underlies Western musical intervals.

What does the Q-factor of a resonance actually tell you?

Q = omega0 divided by the damping rate compares stored energy to energy dissipated per cycle. A high-Q system, like an optical cavity with Q up to 10^9, rings for a very long time at a very sharply defined frequency; a low-Q system, like a plucked string in air, damps out quickly and resonates over a wider frequency band.

Why do circular drums sound less pure than strings?

A string's overtones are integer multiples of the fundamental because the boundary condition produces evenly spaced sine-mode frequencies. A circular membrane's modes are set by the zeros of Bessel functions, which are irrational multiples of each other, so a timpani's overtones are inharmonic rather than a clean harmonic series.

Try it live

Everything above runs in your browser — open Standing Waves in Pipes and step through harmonic modes, tension, and a plucked-string blend of overtones to hear and see resonance directly. Nothing is installed, nothing is uploaded.

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