Resonance

The phenomenon

Every oscillator has a natural frequency ω₀ at which it likes to oscillate (= √(k/m) for spring-mass; √(g/L) for pendulum; etc.).

If you apply a periodic driving force F(t) = F₀·cos(ωt), the system oscillates at the DRIVING frequency ω, not the natural one. But the steady-state amplitude depends on how close ω is to ω₀:

Driving frequency vs ω₀	Steady-state behavior
ω << ω₀	Oscillation in phase with force; small amplitude
ω = ω₀ (resonance)	90° phase lag; MAXIMUM amplitude (limited by damping)
ω >> ω₀	Oscillation 180° out of phase; small amplitude

The amplitude vs frequency curve has a peak at resonance, with width inversely related to damping.

Quality factor Q

Q measures how "sharp" a resonance is:

Q = ω₀ / Δω = m·ω₀ / b

where b is damping coefficient and Δω is the bandwidth (width where amplitude = ½ peak).

System	Approximate Q
Car suspension	~5
Tuning fork	~1000
Pendulum (vacuum)	~100,000
Quartz crystal oscillator	~10⁴-10⁶
Atomic clock (Cs)	~10⁹
LIGO (gravitational wave detector)	~10⁶
Helium superfluid	~10¹¹

Higher Q = narrower resonance, more selective, but slower energy decay.

Real-world resonance

System	Resonance role
Wine glass shatters at right pitch	Singer matches glass's natural frequency
Tacoma Narrows Bridge collapse	Wind vortex shedding matched torsional natural frequency
Millennium Bridge swaying	Pedestrian footstep frequency synced with lateral mode
Mexico City earthquake (1985)	Soft-soil shock wave frequency matched buildings' natural frequencies → many failed
Tuning forks	Designed for specific frequencies; small Q
Musical instruments	Resonance amplifies certain harmonics → distinctive timbre
Microwave ovens	Driver at 2.45 GHz matches water's molecular resonance (loosely)
Atomic spectroscopy	Atoms absorb only at specific frequencies (resonance with electronic transitions)
LIGO interferometer	Mirror suspensions tuned to high Q for sensitivity

JavaScript — resonance simulation

// Driven damped harmonic oscillator
function drivenOscillator(omega_0, gamma, omega_drive, F_0, m, t_max = 50, dt = 0.001) {
  let x = 0, v = 0;
  const trajectory = [];
  for (let t = 0; t < t_max; t += dt) {
    const F = F_0 * Math.cos(omega_drive * t);
    const a = (F - 2 * gamma * m * v - omega_0 * omega_0 * m * x) / m;
    v += a * dt;
    x += v * dt;
    if (Math.round(t * 100) === t * 100) trajectory.push({ t, x, v });
  }
  return trajectory;
}

// Steady-state amplitude as function of driving frequency
function steadyStateAmplitude(omega_0, gamma, omega_drive, F_0, m) {
  // A = F_0 / (m · √((ω₀² - ω²)² + (2γω)²))
  const denominator = m * Math.sqrt(
    Math.pow(omega_0*omega_0 - omega_drive*omega_drive, 2) +
    Math.pow(2 * gamma * omega_drive, 2)
  );
  return F_0 / denominator;
}

// Q factor
function qualityFactor(omega_0, gamma) {
  return omega_0 / (2 * gamma);
}

// Sweep driving frequency, find peak
const omega_0 = 10;     // natural freq
const gamma = 0.5;      // damping
const F_0 = 1, m = 1;

const responseCurve = [];
for (let omega = 1; omega <= 20; omega += 0.1) {
  responseCurve.push({
    omega,
    A: steadyStateAmplitude(omega_0, gamma, omega, F_0, m)
  });
}

const peak = responseCurve.reduce((a, b) => a.A > b.A ? a : b);
console.log(`Peak at ω = ${peak.omega.toFixed(1)} (vs ω₀ = ${omega_0})`);
console.log(`Q ≈ ${qualityFactor(omega_0, gamma).toFixed(1)}`);

// Tuned circuit: resonant frequency f = 1/(2π√(LC))
function lcResonantFrequency(L, C) {
  return 1 / (2 * Math.PI * Math.sqrt(L * C));
}

console.log(`L = 100 µH, C = 300 pF: f = ${(lcResonantFrequency(100e-6, 300e-12) / 1e6).toFixed(2)} MHz`);
// ~919 kHz — typical AM radio range

// Wine glass resonance — natural frequency
function wineGlassFrequency(thickness, radius, height, density = 2500, E = 70e9) {
  // Approximate; depends on mode
  // Lowest mode ~ √(E/ρ) · t / (R²)
  const v_sound = Math.sqrt(E / density);
  return v_sound * thickness / (4 * Math.PI * radius * height);
}

console.log(`Wine glass (1mm, 4cm radius, 12cm tall): ${wineGlassFrequency(0.001, 0.04, 0.12).toFixed(0)} Hz`);
// ~340 Hz (~ E4)

// Pendulum resonance: try driving at various frequencies
function pendulumResonance(L, drive_freq, drive_amplitude_angle, gamma = 0.05, t_max = 30, dt = 0.01) {
  const g = 9.81;
  const omega_0 = Math.sqrt(g / L);
  const omega_drive = 2 * Math.PI * drive_freq;
  let theta = 0, omega = 0;
  let max_amplitude = 0;
  for (let t = 0; t < t_max; t += dt) {
    const driving = drive_amplitude_angle * Math.cos(omega_drive * t);
    const alpha = -(g/L) * Math.sin(theta) - 2 * gamma * omega + driving;
    omega += alpha * dt;
    theta += omega * dt;
    if (Math.abs(theta) > max_amplitude) max_amplitude = Math.abs(theta);
  }
  return { omega_0, omega_drive, max_amplitude };
}

// Pendulum natural f = √(g/L)/(2π); for L = 1m, f ≈ 0.498 Hz
console.log(pendulumResonance(1, 0.498, 0.01));  // Resonance — large amplitude
console.log(pendulumResonance(1, 1.0, 0.01));     // Off resonance — small amplitude

Where resonance matters

• Engineering — avoiding catastrophic resonance. Buildings, bridges, airplanes, ships designed to avoid resonance with environmental forces.
• Engineering — exploiting resonance. MRI, radio, microwaves, atomic clocks all rely on tuned resonance.
• Music. Instruments designed for specific resonant frequencies. Singers, pipes, strings, drums all use natural frequencies.
• Electronics. LC resonant circuits for tuning, filtering, frequency synthesis.
• Atomic physics. Atomic clocks lock to electron transition frequencies; cesium clock at 9.192 GHz defines the second.
• Astronomy. Stellar oscillations (helioseismology) reveal interior structure; resonant orbits of planets/moons.
• Medical imaging. Magnetic Resonance Imaging — proton resonance in magnetic field.

Common mistakes

• Confusing natural and driving frequency. Natural is system property. Driving is external. Resonance is when they match.
• Forgetting damping always present. Real systems have damping — amplitude doesn't grow infinite even at exact resonance. Q factor measures sharpness; bandwidth is finite.
• Treating resonance as inevitable failure. Many systems intentionally USE resonance (radios, MRI). Failure occurs when resonance is unintended and amplitude exceeds material limits.
• Ignoring multiple modes. Real structures have many natural frequencies (modes). Need to consider all relevant ones — primary mode, secondary, etc.
• Conflating resonance with sympathetic vibration. Resonance — driving force directly drives the oscillator. Sympathetic vibration — energy transfers between coupled oscillators (similar effect, related but different).
• Forgetting transient vs steady-state. Initial transient response is different from steady-state. At resonance, system takes time to build up amplitude (~Q cycles).