Doppler Effect

Formulas

For sound (or any classical wave), with source speed v_s and observer speed v_o relative to medium (sign convention — positive when toward each other):

f' = f · (v + v_o) / (v − v_s)

where v is wave speed in medium.

Common cases:

Setup	Frequency observed
Source moving toward stationary observer	f' = f · v / (v − v_s) (higher)
Source moving away from stationary observer	f' = f · v / (v + v_s) (lower)
Observer moving toward stationary source	f' = f · (v + v_o) / v (higher)
Observer moving away from stationary source	f' = f · (v − v_o) / v (lower)
Both moving toward each other	f' = f · (v + v_o) / (v − v_s)
Both moving same direction (source ahead, observer chasing)	f' = f · (v − v_o) / (v − v_s)

Numerical examples

Scenario	Doppler shift
Ambulance siren (1 kHz, v_s = 30 m/s, approaching)	f' = 1000 × 343/(343 − 30) ≈ 1096 Hz (+96 Hz)
Ambulance receding	f' ≈ 919 Hz (−81 Hz)
Pitch drop (passing): 1096 → 919 Hz, ratio 1.19	Music interval ~3 semitones
Police radar (24 GHz, target at 30 m/s)	Δf ≈ 4.8 kHz, easily measured
Distant galaxy (z = 0.1) recession speed	~30,000 km/s
CMB photons we observe	Originally peaked in visible/UV; now microwave (z ≈ 1100)

Relativistic Doppler (for light)

For relative speed v (along line of sight), frequency observed:

f' = f · √((1 − β) / (1 + β))   (receding)
f' = f · √((1 + β) / (1 − β))   (approaching)

where β = v/c. As v → c, redshift z = (λ' − λ)/λ → ∞.

For non-radial motion (transverse), there's a transverse Doppler shift even with no radial velocity (purely relativistic effect, no classical analog).

JavaScript — Doppler calculations

// Classical Doppler (sound)
function classicalDoppler(f_source, v_wave, v_source, v_observer, approach = true) {
  // Convention: speeds are magnitudes; approach=true if motion brings them together
  if (approach) {
    return f_source * (v_wave + v_observer) / (v_wave - v_source);
  } else {
    return f_source * (v_wave - v_observer) / (v_wave + v_source);
  }
}

// Approaching ambulance, 1000 Hz siren, 30 m/s
console.log(`Ambulance approach: ${classicalDoppler(1000, 343, 30, 0, true).toFixed(0)} Hz`); // 1096
console.log(`Ambulance recede: ${classicalDoppler(1000, 343, 30, 0, false).toFixed(0)} Hz`); // 919

// Speed from observed Doppler shift (radar)
function speedFromRadarShift(deltaF, f_source) {
  // Reflected wave: factor 2
  // Δf/f = 2v/c → v = c·Δf/(2f)
  const c = 3e8;
  return c * deltaF / (2 * f_source);
}

console.log(`Δf = 4 kHz at 24 GHz: speed = ${speedFromRadarShift(4000, 24e9).toFixed(2)} m/s`); // 25
console.log(`= ${(speedFromRadarShift(4000, 24e9) * 3.6).toFixed(0)} km/h`); // 90 km/h

// Relativistic Doppler
function relativisticDoppler(f_source, v_relative, c = 3e8) {
  const beta = v_relative / c;
  // Negative beta = receding; positive = approaching
  if (beta < 0) {
    return f_source * Math.sqrt((1 + beta) / (1 - beta));
  } else {
    return f_source * Math.sqrt((1 + beta) / (1 - beta));
  }
}

// Galaxy at 30,000 km/s (z = 0.1)
console.log(`Galaxy 30,000 km/s: f' = ${(relativisticDoppler(1, -3e7).toFixed(4))}`);  // ~0.905

// Cosmological redshift
function redshift(z) {
  // z = (λ_obs - λ_emit) / λ_emit; v ≈ z·c for small z
  return z * 3e8;  // approximate recession speed
}

console.log(`z=0.5: v ≈ ${(redshift(0.5)/1000).toFixed(0)} km/s`);  // 150,000

// Rectifying for very high z
function highZRedshiftSpeed(z) {
  // Special relativity: 1+z = √((1+β)/(1−β))
  // Solve for β
  const sq = (1 + z) * (1 + z);
  return (sq - 1) / (sq + 1);  // β = v/c
}

console.log(`z=1: β = ${highZRedshiftSpeed(1).toFixed(3)}`); // 0.6 (60% of c)
console.log(`z=10: β = ${highZRedshiftSpeed(10).toFixed(3)}`); // 0.984

// Doppler effect on sound from passing car
function passingCarPitch(f, v_car, v_sound = 343) {
  return {
    approaching: f * v_sound / (v_sound - v_car),
    passing: f,  // when momentarily perpendicular
    receding: f * v_sound / (v_sound + v_car)
  };
}

console.log(passingCarPitch(440, 30));  // approach 482, recede 405 Hz

Where Doppler effect shows up

• Radar. Police speed measurement, military tracking, weather radar (storm motion).
• Astronomy. Stellar radial velocities, exoplanet detection, galaxy recession (cosmological redshift), 21-cm line in radio astronomy.
• Medicine. Doppler ultrasound — measure blood flow velocity, fetal heart monitoring, vascular imaging.
• Aviation. Air traffic control, military targeting, weather avoidance.
• Sports. Baseball pitch radars, tennis serves, formula 1 telemetry.
• Cosmology. Hubble's law (recession velocity ∝ distance), CMB studies, dark energy.
• Music. Pitch-shifting effects in sirens, train whistles — distinctive "doppler" feel as object passes.

Common mistakes

• Wrong sign convention. Source approaching observer raises pitch (smaller denominator). Don't flip the sign accidentally.
• Using classical formula for high speeds. Classical Doppler (with v/v_wave) breaks at v_wave. For light at relativistic speeds, must use special-relativistic formula.
• Ignoring difference between source and observer motion. Source motion gives one shift; observer motion gives slightly different. For small v/v_wave, they're nearly equal; otherwise, they differ.
• Using radial component only. Doppler depends on velocity COMPONENT along line of sight. Transverse motion doesn't shift frequency (in classical) or has only a tiny relativistic shift.
• Confusing redshift z and recession velocity. z = Δλ/λ. For small z, v ≈ zc. For large z (e.g., z > 1), need relativistic formula. For very large z (early universe), space itself expanded, not just motion.
• Forgetting medium dependence for sound. Doppler frequency depends on the medium's wave speed. Same source-observer geometry in air vs water gives different shifts because c_sound differs.