Sound Waves

Basics

Sound waves are longitudinal pressure variations. As a sound wave passes, molecules oscillate back and forth along the direction of wave propagation — alternately compressing and rarefying the medium.

For sinusoidal sound:

p(x, t) = p_max · sin(kx − ωt)

where p is pressure variation from atmospheric (small — sound at threshold of hearing is ~10⁻⁵ Pa, vs 10⁵ Pa atmospheric).

Speed in different media

Medium	Speed
Air (0°C)	331 m/s
Air (20°C)	343 m/s
Air (100°C)	387 m/s
Helium (20°C)	1,007 m/s
Water (fresh, 20°C)	1,482 m/s
Seawater (20°C)	1,531 m/s
Iron	5,120 m/s
Aluminum	6,420 m/s
Diamond	12,000 m/s

Speed in air increases ~0.6 m/s per °C. Cold winter air → slightly slower sound (and more sound bending due to thermal layers).

Frequency ranges

Range	Frequencies	Examples
Infrasound	< 20 Hz	Volcanic activity, large explosions, earthquake aftershocks
Audible to humans	20 Hz - 20 kHz	Music, speech, ambient sounds
Speech (humans)	~85-255 Hz fundamental, harmonics to ~3 kHz	—
Bass guitar low E	~41 Hz	—
A4 (concert pitch)	440 Hz	Tuning standard
Soprano top note	~1300 Hz	—
Ultrasound (cleaning)	20-40 kHz	Lab instruments, jewelry cleaners
Medical ultrasound	1-20 MHz	Imaging
Bat echolocation	9-150 kHz	Hunt insects

Intensity and decibels

Sound intensity I (W/m²) measures power per unit area. The decibel scale:

L = 10 · log₁₀(I / I_ref)

where I_ref = 10⁻¹² W/m² (threshold of hearing at 1 kHz).

Sound	Approximate dB
Threshold of hearing	0
Whisper at 1 m	30
Normal conversation	60
City traffic	80
Lawnmower	90
Chainsaw	110
Threshold of pain	120
Jet engine at 30 m	140
Stun grenade	170
Krakatoa eruption (1883)	~310 dB at source

JavaScript — sound calculations

// Sound speed in air (Celsius)
function soundSpeedInAir(T_C) {
  return 331.4 + 0.6 * T_C;  // approximate
}

console.log(`Sound at 20°C: ${soundSpeedInAir(20).toFixed(0)} m/s`); // 343
console.log(`Sound at 0°C: ${soundSpeedInAir(0).toFixed(0)} m/s`); // 331
console.log(`Sound at 100°C: ${soundSpeedInAir(100).toFixed(0)} m/s`); // 391

// Wavelength of sound at given frequency in air
function soundWavelength(frequency, soundSpeed = 343) {
  return soundSpeed / frequency;
}

console.log(`A4 (440 Hz): λ = ${soundWavelength(440).toFixed(2)} m`); // 0.78
console.log(`Bass (100 Hz): λ = ${soundWavelength(100).toFixed(2)} m`); // 3.43
console.log(`20 kHz: λ = ${(soundWavelength(20000) * 1000).toFixed(0)} mm`); // 17

// dB calculations
function intensityToDecibels(intensity) {
  return 10 * Math.log10(intensity / 1e-12);
}

function decibelsToIntensity(dB) {
  return 1e-12 * Math.pow(10, dB / 10);
}

console.log(`Conversation 60 dB: ${decibelsToIntensity(60).toExponential(2)} W/m²`);  // 10⁻⁶
console.log(`Two 60 dB sources: ${intensityToDecibels(2 * 1e-6).toFixed(2)} dB`);  // 63 dB

// Adding sounds in dB (intensities add)
function addDB(dB1, dB2) {
  return 10 * Math.log10(Math.pow(10, dB1/10) + Math.pow(10, dB2/10));
}

console.log(`Two 70 dB sources: ${addDB(70, 70).toFixed(2)} dB`);  // 73 dB
console.log(`70 dB + 80 dB: ${addDB(70, 80).toFixed(2)} dB`);  // 80.4 dB

// Sound pressure to dB (atmospheric pressure → effective)
function pressureToDB(pressure_Pa) {
  const p_ref = 20e-6;  // 20 µPa = threshold
  return 20 * Math.log10(pressure_Pa / p_ref);
}

console.log(`Whisper 20 mPa = ${pressureToDB(0.02).toFixed(0)} dB`);
console.log(`Conversation 0.02 Pa = ${pressureToDB(0.02).toFixed(0)} dB`);
console.log(`Jet 200 Pa = ${pressureToDB(200).toFixed(0)} dB`);

// Distance estimation from echo timing
function distanceFromEcho(timeDelay, soundSpeed = 343) {
  // Echo time = 2× distance / speed
  return timeDelay * soundSpeed / 2;
}

console.log(`0.1 s echo: ${distanceFromEcho(0.1).toFixed(1)} m`);
console.log(`1 s echo: ${distanceFromEcho(1).toFixed(0)} m`);

// Sonar depth from echo
function sonarDepth(timeDelay) {
  // Water sound speed
  return distanceFromEcho(timeDelay, 1500);
}

console.log(`0.5 s sonar: ${sonarDepth(0.5)} m`);  // 375 m

// Frequency from string vibration
function stringFrequency(L, tension, mass_per_length, harmonic = 1) {
  const v = Math.sqrt(tension / mass_per_length);
  return harmonic * v / (2 * L);
}

// Guitar A string: 0.65m, ~80 N tension, ~0.005 kg/m
console.log(`Guitar A: ${stringFrequency(0.65, 80, 0.005).toFixed(1)} Hz`); // ~97 Hz

Where sound matters

• Communication. Speech, music, telephony, broadcast.
• Acoustic engineering. Concert halls, recording studios, noise control, hearing aids.
• Sonar. Marine navigation, fish finding, submarine detection.
• Medical imaging. Ultrasound for prenatal, cardiac, abdominal imaging — uses MHz-frequency sound.
• Industrial NDT. Ultrasonic detection of cracks/voids in materials.
• Music. Instrument design, audio engineering, sound synthesis.
• Animal communication. Bat echolocation, dolphin sonar, bird song.

Common mistakes

• Treating dB as linear. dB is logarithmic. 60 dB is 10⁶× more intense than 0 dB. Doubling intensity adds ~3 dB, not 60.
• Forgetting Doppler effect for moving sources. Moving source/observer → frequency shift. Don't ignore for ambulances, jets, sports physics.
• Ignoring medium dependence. Sound speed varies dramatically: 343 m/s in air, 5000+ in steel. Pitch from a string in water vs air differs.
• Confusing sound and ultrasound. Both are pressure waves; just different frequencies. Audible 20-20k Hz; ultrasonic above 20 kHz.
• Conflating loudness with intensity. Loudness is psychoacoustic (perception), influenced by frequency. Intensity is physical (W/m²). The ear is most sensitive at 1-4 kHz.
• Confusing dB SPL with dB other. dB scales depend on reference. dB SPL (sound pressure) uses 20 µPa. dB SIL (sound intensity) uses 10⁻¹² W/m². dB(A) is A-weighted (mimics ear response).