Standing Waves

How standing waves form

Two identical waves traveling in opposite directions:

y₁ = A·sin(kx − ωt)  (rightward)
y₂ = A·sin(kx + ωt)  (leftward)

Sum (using trig identity):

y = y₁ + y₂ = 2A·sin(kx)·cos(ωt)

Result — a pattern that DOESN'T translate. Every point oscillates in place at frequency ω, with amplitude depending on x. At nodes (sin(kx) = 0, i.e., kx = nπ), no motion ever. At antinodes (sin(kx) = ±1), max motion.

Common standing-wave systems

System	Fixed ends?	Wavelength of nth mode	Frequency of nth mode
String, fixed both ends (guitar)	Both nodes	λ_n = 2L/n	f_n = n·v/(2L)
String, free both ends	Both antinodes	λ_n = 2L/n	Same
String, fixed one end (whip)	Node + antinode	λ_n = 4L/(2n-1)	f_n = (2n-1)·v/(4L)
Open pipe (flute)	Both antinodes	λ_n = 2L/n	f_n = n·v/(2L)
Closed pipe (clarinet)	One node, one antinode	λ_n = 4L/(2n-1)	f_n = (2n-1)·v/(4L)

Harmonic structure

For string fixed both ends — fundamental is f₁ = v/(2L), harmonics are integer multiples: f₂ = 2f₁, f₃ = 3f₁, etc.

For closed-end pipe — fundamental is f₁ = v/(4L), harmonics are ODD multiples: f₃ = 3f₁, f₅ = 5f₁, ... (no even harmonics).

This gives different instruments different timbres. Clarinet has prominent odd harmonics → distinctive sound vs flute (all harmonics).

Real-world examples

System	Standing wave behavior
Guitar string (E2 = 82 Hz, low)	Fixed both ends; integer harmonics 82, 164, 246 Hz, etc.
Violin string	Same physics; bowing excites multiple harmonics
Flute	Open pipe; integer harmonics
Clarinet	Closed pipe; odd harmonics only
Organ pipes	Mixed types — open pipes (some), closed pipes (others)
Singing in shower	Bathroom acts as resonant cavity; certain frequencies "ring"
Microwave oven	Standing waves of EM energy → "hot spots" — why food is on a turntable
Atom (quantum standing waves)	Electron orbitals are 3D standing waves; energies quantized
Laser cavity	Light waves bounce between mirrors, forming standing waves at allowed frequencies

JavaScript — standing wave calculations

// Frequencies on a fixed-end string
function stringHarmonics(L, v, num_harmonics = 5) {
  const f1 = v / (2 * L);
  return Array.from({length: num_harmonics}, (_, i) => (i + 1) * f1);
}

// Guitar low E: 0.65 m string, v = 165 m/s
console.log(stringHarmonics(0.65, 165, 5));
// [127, 254, 381, 508, 635] Hz

// String tuning: tension required for desired frequency
function tensionForFrequency(L, mass_per_length, freq, harmonic = 1) {
  // f = n·v/(2L), v = √(T/μ)
  // T = μ·v² = μ·(2L·f/n)²
  const v = 2 * L * freq / harmonic;
  return mass_per_length * v * v;
}

console.log(`A4 (440 Hz) on 0.65m string, μ=0.0005: ${tensionForFrequency(0.65, 0.0005, 440).toFixed(0)} N`);
// ~163 N

// Open pipe (flute): same as string fixed both ends
function openPipeHarmonics(L, v, num = 5) {
  const f1 = v / (2 * L);
  return Array.from({length: num}, (_, i) => (i + 1) * f1);
}

// Closed pipe (clarinet): only odd harmonics
function closedPipeHarmonics(L, v, num = 5) {
  const f1 = v / (4 * L);
  return Array.from({length: num}, (_, i) => (2 * (i + 1) - 1) * f1);
}

console.log('Flute (0.5m, 343 m/s):', openPipeHarmonics(0.5, 343, 4));
// [343, 686, 1029, 1372] Hz — all integers
console.log('Clarinet (0.5m, 343):', closedPipeHarmonics(0.5, 343, 4));
// [171.5, 514.5, 857.5, 1200.5] — only odd

// Compute nth mode displacement at position x
function standingWaveDisplacement(amplitude, n, L, x, t, omega) {
  const k = n * Math.PI / L;
  return 2 * amplitude * Math.sin(k * x) * Math.cos(omega * t);
}

// Find nodes for nth mode
function nodes(n, L) {
  // sin(nπx/L) = 0 → x = mL/n for m = 0, 1, ..., n
  return Array.from({length: n + 1}, (_, m) => m * L / n);
}

console.log(`3rd mode of 1m string nodes: ${nodes(3, 1)}`); // [0, 0.333, 0.667, 1.000]

// Hydrogen atom rough quantum standing wave (Bohr radius)
function bohrRadius(n) {
  return 0.529e-10 * n * n;  // n=1, 2, 3...
}

console.log(`H atom n=1: r = ${(bohrRadius(1) * 1e10).toFixed(3)} Å`);  // 0.529
console.log(`H atom n=4: r = ${(bohrRadius(4) * 1e10).toFixed(3)} Å`);  // 8.464

Where standing waves matter

• Music. All instruments — strings, winds, percussion, voice — produce standing waves. Pitch determined by fundamental frequency.
• Acoustics. Concert halls, recording studios — designed to control standing waves and avoid unwanted resonances.
• Microwave ovens. Standing waves create hotspots; turntable distributes the food through them.
• Lasers. Optical cavity supports standing waves at allowed wavelengths; only those amplify by stimulated emission.
• Quantum mechanics. Atomic orbitals are 3D standing waves; energy quantization arises from boundary conditions on Schrödinger's equation.
• RF and antennas. Standing waves on poorly-matched transmission lines (VSWR) reduce power transfer.
• Architecture. Bridge resonance (Tacoma Narrows famous failure) — natural frequencies of structures must avoid environmental excitation.

Common mistakes

• Confusing standing with traveling waves. Traveling waves carry energy; standing waves oscillate in place (energy bounces back and forth).
• Misidentifying nodes/antinodes. Nodes are stationary (zero displacement). Antinodes have max displacement. Don't swap them.
• Wrong harmonic series for closed pipes. Closed (one end) — only ODD harmonics (3f₁, 5f₁, ...). Open or both fixed — all integer harmonics. Different timbres result.
• Forgetting boundary conditions matter. Free vs fixed end determine where nodes/antinodes are. Different boundaries → different frequencies.
• Treating fundamental as the only frequency. Real instruments produce many harmonics simultaneously. Their relative amplitudes determine timbre. Pure fundamental sounds artificial (sine wave).
• Assuming all 1D systems behave identically. Strings, pipes, drum heads (2D), boxes (3D) — different geometries give different mode patterns.