Wave Speed

Wave speed formulas

Speeds in common media

JavaScript — wave speed calculations

// String wave speed
function stringWaveSpeed(tension, linearDensity) {
  return Math.sqrt(tension / linearDensity);
}

// Guitar string: T = 100 N, μ = 0.005 kg/m
console.log(`Guitar string speed: ${stringWaveSpeed(100, 0.005).toFixed(0)} m/s`); // ~141

// Sound speed in gas
function soundInGas(T_K, gamma, M_kg_per_mol) {
  const R = 8.314;
  return Math.sqrt(gamma * R * T_K / M_kg_per_mol);
}

console.log(`Sound in air at 20°C: ${soundInGas(293, 1.4, 0.029).toFixed(0)} m/s`); // ~343
console.log(`Sound in helium: ${soundInGas(293, 1.667, 0.004).toFixed(0)} m/s`);     // ~1007
console.log(`Sound in xenon: ${soundInGas(293, 1.667, 0.131).toFixed(0)} m/s`);      // ~178 (low!)

// Sound in liquid
function soundInLiquid(bulkModulus, density) {
  return Math.sqrt(bulkModulus / density);
}

// Water at 20°C: B ≈ 2.2 GPa, ρ = 1000 kg/m³
console.log(`Sound in water: ${soundInLiquid(2.2e9, 1000).toFixed(0)} m/s`); // ~1483

// Sound in solid (longitudinal)
function soundInSolid(youngsModulus, density) {
  return Math.sqrt(youngsModulus / density);
}

// Steel: E = 200 GPa, ρ = 7850 kg/m³
console.log(`Sound in steel: ${soundInSolid(200e9, 7850).toFixed(0)} m/s`); // ~5050

// Light in medium
const c = 3e8;
function lightSpeed(refractiveIndex) {
  return c / refractiveIndex;
}

console.log(`Light in glass (n=1.5): ${(lightSpeed(1.5) / 1e8).toFixed(2)}e8 m/s`); // 2.00e8

// Mach number
function machNumber(speed, soundSpeed = 343) {
  return speed / soundSpeed;
}

console.log(`Concorde at 2179 km/h: Mach ${machNumber(2179/3.6).toFixed(2)}`); // ~1.76

// Distance from lightning by sound vs light
function lightningDistance(timeBetweenFlashAndThunder) {
  // Light travels essentially instantly. Sound at 343 m/s.
  return timeBetweenFlashAndThunder * 343;
}

console.log(`5 seconds delay → ${lightningDistance(5)} m = ${(lightningDistance(5)/1000).toFixed(2)} km`);
// Note: rule of thumb is ~3 sec/km

// Tuning a guitar string
function guitarTension(L, mass, frequency) {
  // For string of length L, mass m, fundamental f:
  // λ = 2L → v = 2Lf → T = μv² where μ = m/L
  const mu = mass / L;
  const v = 2 * L * frequency;
  return mu * v * v;
}

// 0.65 m string, 5 g mass, A4 = 440 Hz
console.log(`Guitar tension: ${guitarTension(0.65, 0.005, 440).toFixed(0)} N`); // ~647 N

Where wave speed matters

• Acoustics. Sonar (water sound speed), audio engineering (room dimensions vs sound speed), aircraft sonic booms.
• Music. Tuning instruments — wave speed in strings depends on tension. Length of organ pipes for desired pitch.
• Optics. Refraction (speed change at boundary), prisms (dispersion), fiber optics (signal speed).
• Telecommunications. Signal propagation delays (~1 ms per 300 km in fiber). Limits high-frequency trading, VR latency.
• Seismology. Earthquake wave timing reveals Earth's interior structure. P-S delay gives distance to epicenter.
• Astronomy. Light-travel-time → "look-back time"; we see distant objects as they were in the past.
• Aerospace. Mach number determines aerodynamic regime (subsonic, supersonic, hypersonic).

Common mistakes

• Thinking source speed adds to wave speed. A moving source DOESN'T speed up the wave; it changes the frequency observed (Doppler effect). Wave speed is medium property.
• Confusing wave speed with particle speed. A water wave moves at ~5 m/s, but water molecules move only a few cm. Wave speed is energy propagation; particle speed is local oscillation.
• Forgetting medium dependence. Sound speed varies with T, pressure, humidity, and medium. Use the right formula for the right medium.
• Treating refractive index as fixed. n varies with wavelength (dispersion). Different colors have different n in glass — that's how prisms work.
• Mixing speed and frequency. Pitch (frequency) is set by source. Speed is medium. Wavelength = speed/frequency, so it changes when sound enters water from air.
• Light slower than c "violates relativity". Light in matter (glass, water) IS slower than c. The "speed limit" is c in vacuum. In matter, light bounces off atoms (interactions), effectively slowing.