# Sound Waves

> Sound is a longitudinal pressure wave: frequency sets pitch and wavelength, amplitude sets loudness. The physics behind 3D audio, delay and Doppler.

- Category: Physics › Mechanics
- Level: core
- Tags: waves, acoustics, audio, longitudinal, doppler
- Updated: 2026-11-01
- URL: https://gdcs.coody.app/physics/mechanics/sound-waves

## What it is

A loudspeaker cone pushes and pulls the air in front of it. Each air parcel only jiggles back and forth around its home position, but the pattern of **compressions** (crowded, high pressure) and **rarefactions** (spread out, low pressure) travels away at the speed of sound, about 343 m/s in room-temperature air. Because the jiggle is along the travel direction, sound is a **longitudinal** wave.

The **frequency** `f` is how many compressions pass per second (pitch); the **wavelength** `λ = v/f` is the distance between them. A 440 Hz A4 is 0.78 m long in air; 100 Hz bass is 3.4 m, which is why it bends around walls. **Amplitude** sets the pressure swing; intensity grows with its square, and perceived loudness follows the logarithm of intensity (decibels).

For game audio that physics shows up as distance attenuation (`1/r²` intensity, `−6 dB` per doubling of distance), propagation delay (thunder after lightning, far gunshots), Doppler shift for moving sources, and low-pass filtering for occlusion. Web Audio's `PannerNode` and Three.js's `PositionalAudio` implement the attenuation; delay and Doppler are often up to you.

## Key formulas

- **Particle displacement:** $\xi(x,t) = A \sin(kx - \omega t),\quad k = \dfrac{2\pi}{\lambda},\ \omega = 2\pi f$
- **Pressure (90° ahead of displacement):** $p(x,t) = p_0 - B\,\dfrac{\partial \xi}{\partial x} = p_0 - B k A\cos(kx - \omega t)$
- **Wave speed:** $v = f\lambda$
- **Intensity and level:** $I \propto A^2,\qquad L = 10\log_{10}\dfrac{I}{I_0}\ \text{dB}$
- **Doppler (moving source):** $f' = f\,\dfrac{v}{v - v_s}$ — v_s is the source speed toward the listener.

## Rules of thumb

- Speed of sound ≈ 331 + 0.6·T(°C) m/s in air; about 1480 m/s in water, so underwater audio should arrive almost instantly.
- Human hearing: 20 Hz–20 kHz, wavelengths 17 m down to 1.7 cm.
- Point source in open air: intensity ∝ 1/r², so the level drops 6 dB per doubling of distance. Games often use a gentler rolloff for playability.
- Delay a far sound by `distance / 343` seconds; 100 m is a noticeable 0.29 s.

## Implementation

Positional sound with physical delay and Doppler (Three.js + Web Audio):

```ts
import * as THREE from "three"

const SPEED = 343
const prev = new THREE.Vector3()
const toListener = new THREE.Vector3()

export function updateSound(sound: THREE.PositionalAudio, source: THREE.Object3D, listener: THREE.Object3D, dt: number) {
  sound.setRefDistance(2) // full volume within 2 m
  sound.setRolloffFactor(1) // inverse-distance falloff
  toListener.subVectors(listener.position, source.position)
  const dist = toListener.length()
  // Radial speed of the source toward the listener (m/s).
  const vs = prev.sub(source.position).multiplyScalar(-1 / dt).dot(toListener.normalize())
  prev.copy(source.position)
  const rate = SPEED / (SPEED - THREE.MathUtils.clamp(vs, -150, 150))
  sound.setPlaybackRate(rate) // Doppler pitch
  return dist / SPEED // seconds of delay to apply when (re)starting a one-shot
}
```

## Common mistakes

- Drawing sound as a transverse wiggle: air moves along the direction of travel, not sideways.
- Confusing amplitude with frequency: louder does not mean higher pitch.
- Scaling volume linearly with distance; perceived loudness is logarithmic and intensity falls off as 1/r².
- Applying Doppler from the frame-to-frame position delta without clamping, so a teleport produces a squeal.

## Interactive visual

Rows of air parcels shuffle left and right as the grey speaker pushes them (200× slow motion). Orange parcels are compressed, cyan ones rarefied, and the yellow curve above is the pressure. Raise the frequency to pack compressions closer (shorter λ); raise the amplitude to make the parcels swing further.

## Prompt

```text
Write a TypeScript Three.js helper for physically-flavoured positional audio. Inputs: a THREE.PositionalAudio, the source Object3D, the AudioListener's Object3D, dt and speed of sound (default 343 m/s). Each frame: compute distance, apply inverse-distance rolloff with refDistance 2 m, compute the source's radial velocity toward the listener and set playbackRate = c/(c − v_s) clamped to [0.5, 2], and expose delayFor(distance) for one-shots. Ignore velocity on frames where the source moved more than 10 m (teleport). Unit tests: a stationary source gives rate 1; approaching at 34.3 m/s gives rate ≈ 1.111; delay for 343 m is 1 s.
```

## Related

- [Spring-Mass Oscillator](https://gdcs.coody.app/physics/mechanics/spring-mass-oscillator.md) — Hooke's law, damping and a periodic push: the forced oscillator behind springy cameras, wobbly UI and suspension, and why driving at ω₀ explodes.
- [EM Wave Propagation](https://gdcs.coody.app/physics/electromagnetism/em-wave-propagation.md) — Light is E and B regenerating each other. A 1D Yee grid (FDTD) leapfrogs them in time, slows them in glass and blows up past the Courant limit.
- [Delta Time](https://gdcs.coody.app/game-feel/timing/delta-time.md) — Delta time is the seconds since the last frame. Multiply speeds and rates by it so a game plays the same at 30, 60 or 144 fps.

## Further reading

- [OpenStax University Physics — Sound Waves](https://openstax.org/books/university-physics-volume-1/pages/17-1-sound-waves)
- [MDN — Web audio spatialization basics](https://developer.mozilla.org/en-US/docs/Web/API/Web_Audio_API/Web_audio_spatialization_basics)

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Source: Game Development Cheat Sheet (https://gdcs.coody.app)