A standing wave is a stationary spatial pattern formed by the superposition of two counterpropagating waves of equal frequency and amplitude. Unlike a travelling wave, which carries energy through space, a standing wave distributes it into fixed nodes (zero displacement) and antinodes (maximum displacement). From a vibrating guitar string to a drum membrane to a Fabry–Pérot optical cavity, resonance and normal modes govern the frequencies at which physical systems prefer to oscillate.
1. Standing Waves on a String
The 1-D wave equation for transverse displacement u(x,t) on a string of tension T and linear density μ is:
With fixed-end Dirichlet boundary conditions u(0,t)=u(L,t)=0, the general solution is a Fourier series of standing wave modes:
where ωₙ = nπc/L. The fundamental (n=1) frequency is f₁ = c/(2L). Harmonics are integer multiples: f₂=2f₁, f₃=3f₁, …
| Mode n | Frequency fₙ | Nodes (incl. ends) | Antinodes |
|---|---|---|---|
| 1 (fundamental) | f₁ = c/2L | 2 | 1 |
| 2 (1st overtone) | 2f₁ | 3 | 2 |
| 3 (2nd overtone) | 3f₁ | 4 | 3 |
| n | nf₁ | n+1 | n |
For a guitar A-string (L=0.65 m, T=81 N, μ=3.8×10⁻⁴ kg/m), c≈462 m/s, f₁≈355 Hz ≈ A4. The frequency ratio between successive harmonics is always 1:2:3:…, giving the harmonic series that underpins Western musical intervals.
2. Resonance and the Q-Factor
When a lightly damped oscillator is driven by an external periodic force F₀·cos(ωt), the steady-state amplitude is:
where ω₀ = √(k/m) is the natural frequency and γ is the damping coefficient. At resonance ω = ω₀ the amplitude diverges to A_res = F₀/(mγω₀). The quality factor Q = ω₀/γ characterises how sharply the resonance peak is defined:
| System | Typical Q | Bandwidth Δf / f₀ |
|---|---|---|
| Plucked guitar string (in air) | 200–2 000 | 0.0005–0.005 |
| Quartz crystal oscillator | 10⁴–10⁶ | 10⁻⁶–10⁻⁴ |
| Optical Fabry–Pérot cavity | 10⁶–10⁹ | 10⁻⁹–10⁻⁶ |
| Room acoustic mode (RT₆₀~0.5 s, f=100 Hz) | ≈ 100 | ≈ 0.01 |
| Tacoma Narrows bridge (pre-collapse) | ≈ 40 | ≈ 0.025 |
3. 2-D Membrane: Normal Modes
For a rectangular membrane (drum) with fixed edges (u=0 on all four sides), the 2-D wave equation yields modal solutions:
For a square membrane (a=b) the eigenvalue λₘₙ = m² + n² can be degenerate: modes (m,n) and (n,m) share the same frequency but different spatial patterns. Any linear combination of degenerate modes is also a valid mode — the physical shape depends on how the membrane is excited.
4. Chladni Figures
In 1787, Ernst Chladni demonstrated that sand sprinkled on a metal plate organises itself along the nodal lines of excited modes when the plate is stroked with a violin bow. These Chladni figures are a direct visual map of where displacement is zero.
The nodal pattern of the (m,n) mode of a square plate is the union of lines where:
For degenerate modes the linear combination A·sin(mπx/L)sin(nπy/L) + B·sin(nπx/L)sin(mπy/L) produces richer curved patterns (avoiding lines). Changing the ratio A/B continuously morphs between all Chladni figures at that frequency.
Historical context
Chladni's patterns fascinated Napoleon Bonaparte, who funded further research. Sophie Germain later developed the plate vibration equation governing their mathematical origin. The same mathematics appears in quantum mechanics: Chladni patterns are visually identical to probability density plots of 2-D particle-in-a-box wave functions.
5. Circular Membrane: Bessel Functions
A circular drum membrane (radius R, fixed at r=R) has modes described by Bessel functions:
where α_{mn} is the n-th positive zero of the Bessel function Jₘ. Unlike the rectangular case, these zeros are irrational — so circular drum overtones are inharmonic (not integer multiples of f₁), which is why timpani do not play a perfectly clear pitch. The fundamental is α₀₁ ≈ 2.405, giving f₁ = 2.405c/(2πR).
6. Fourier Analysis and the FFT
Any complicated wave shape on a finite string can be decomposed into normal modes by the Fourier series. The discrete Fourier transform (DFT) of N samples at spacing Δt:
reveals which frequencies are present in a signal with resolution Δf = 1/(NΔt) and Nyquist bandwidth fₘₐₓ = 1/(2Δt). The Cooley–Tukey FFT computes this in O(N log N) instead of O(N²), making real-time audio analysis feasible.
Physical resonance appears as a sharp peak in the power spectral density |X[k]|². The FWHM of the peak equals Δω = ω₀/Q. For a room mode at 80 Hz with Q≈100, the resonance bandwidth is ≈0.8 Hz — a very narrow peak that causes audible "booming" in small rooms.
7. Applications Across Physics
| Domain | Standing wave | Key quantity |
|---|---|---|
| Musical strings | Transverse string modes | f = n·c / 2L |
| Organ pipes | Longitudinal air column (open/closed) | f = n·v / 2L or (2n−1)v/4L |
| Microwave oven | EM cavity modes | 2.45 GHz TE/TM modes, 12 cm λ |
| Laser cavity | Fabry–Pérot longitudinal modes | Δν = c / 2L |
| Quantum well (semiconductor) | Electron wave function (particle-in-box) | Eₙ = n²π²ℏ²/2mL² |
| Room acoustics | Axial/tangential/oblique room modes | f_{pqr} = c/2 · √[(p/Lₓ)²+(q/Ly)²+(r/Lz)²] |
| Seismology | Earth's free oscillations (Normal modes) | ₀S₂ fundamental ≈ 0.3 mHz |
8. Interactive: Chladni Patterns & Wave Modes
Adjust the mode numbers m and n and the degenerate-mode mixing angle θ to generate the corresponding Chladni figure on a square plate. The nodal lines (zero displacement) are shown in white; sand would accumulate there. The right panel shows the 1-D string mode for mode number n.
When m = n the two degenerate modes are the same pattern. For m ≠ n, mixing angle θ ∈ [0°,90°] interpolates between mode (m,n) and (n,m). The sand concentrates at nodal lines: their count equals m + n − 2 internal lines plus the fixed edges.