Статичні хвилі та резонанс: Моди, візерунки Чла́нджі та коефіцієнт добротності

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

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.

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