Explore the rigorous physics of sound — its mechanical nature, propagation through matter, interaction with boundaries, and extreme phenomena—from the sub-audible rumble of tectonic plates to the quantum vibrations of phonons in crystal lattices. Build the physical vocabulary and mathematical intuition that every subsequent unit in this module depends upon.
Understanding sound as a quantifiable physical phenomenon provides the foundational mechanics required for precise audiovisual synchronization, spatial audio design, and the translation of auditory data into visual forms. Every visual parameter you will later map—position, scale, brightness, colour, motion—must be grounded in a physical acoustic quantity. This unit builds that dictionary.
This unit treats sound as an objective physical phenomenon: waves in a medium, governed by the laws of mechanics and thermodynamics. What you will carry forward from this unit:
Sound is Acoustic vibrations—elastic, mechanical waves of varying pressure propagating through a material medium (gas, liquid, or solid). Three properties distinguish sound from other wave phenomena:
Frequency ($f$): The number of complete oscillation cycles per second, measured in Hertz (Hz). It is the primary physical determinant of perceived pitch. Period ($T$): The duration of one complete cycle. $T = 1/f$. Wavelength ($\lambda$): The spatial distance the wave travels during one complete cycle—the physical length of one compression–rarefaction pair. $$\lambda = \frac$$ where $v$ is the speed of sound in the given medium. Examples in air at 20 °C ($v ≈ 343$ m/s):
| Frequency | Wavelength | Physical intuition |
|---|---|---|
| 16 Hz (threshold of hearing) | ~21.4 m | Longer than a tennis court |
| 100 Hz (bass) | ~3.4 m | About the height of a room |
| 1 kHz (mid-range reference) | ~34.3 cm | Roughly the length of a forearm |
| 10 kHz (high treble) | ~3.4 cm | About the width of two fingers |
| 20 kHz (upper threshold) | ~1.7 cm | About the width of a thumbnail |
A sound wave involves two coupled oscillating quantities that are often conflated:
The speed of sound is not a universal constant. It is dictated by the medium's elasticity (resistance to compression) and density (inertia): $$v = \sqrt{\frac{\rho}}$$ where $K$ is the bulk modulus (or, for solids, the appropriate elastic modulus) and $\rho$ is the density. In ideal gases, this reduces to: $$v = \sqrt{\frac{M}}$$ where $\gamma$ is the adiabatic index, $R$ is the universal gas constant, $T$ is absolute temperature, and $M$ is molar mass. This explains why sound speed in air depends primarily on temperature and molecular composition, not on pressure. In air (~343 m/s at 20 °C; ~331 m/s at 0 °C): The speed increases by approximately 0.6 m/s for every 1 °C rise in temperature. This temperature dependence is the physical basis for acoustic refraction in the atmosphere (see §III.5).
Historical milestone: In the first half of the 17th century, Francis Bacon suggested in Novum Organum (1620) measuring the speed of sound by timing the delay between a visible flash and the arrival of its sound—the same principle as counting seconds between lightning and thunder. The first reasonably accurate measurements were made by the French Academy of Sciences in 1738, yielding ~332 m/s. The voice-altering effect (clarified): Breathing helium (less dense, lower molar mass than air) increases the speed of sound in the vocal tract to ~927 m/s. This shifts the resonant formant frequencies of the vocal tract upward, producing the "Donald Duck" effect. Crucially, the fundamental frequency of the vocal folds does not change—the folds still vibrate at the same rate. What changes is the filtering: the vocal tract's resonances shift, altering the spectral envelope (timbre), not the pitch. Conversely, sulfur hexafluoride (SF₆, molar mass 146 g/mol vs. air's ~29 g/mol) slows sound to ~134 m/s, lowering formants and creating a "Darth Vader" timbre. This distinction—source frequency vs. filter resonance—is foundational and will recur in Unit 3.3 (Timbre). In water (~1 480 m/s in fresh water at 20 °C; ~1 530 m/s in seawater): Sound travels over four times faster in water than in air because water's bulk modulus (~2.2 GPa) is roughly 15 000 times greater than air's, while its density is only ~800 times greater. The net effect favours speed. Historical experiment: The first quantitative measurement of underwater sound speed was conducted in 1826 on Lake Geneva by Charles-François Sturm and Jean-Daniel Colladon. They struck a submerged bell at one end of a 14 km stretch while simultaneously igniting a gunpowder flash visible through a telescope. A hydrophone (a simple listening tube) at the far end detected the sound. The measured speed: ~1 435 m/s—remarkably close to the modern value. The SOFAR channel (clarified mechanism): At ocean depths of roughly 600–1 200 m, a unique acoustic waveguide forms. The mechanism involves two competing effects on sound speed:
Note on diamond: Diamond is elastically anisotropic. The speed of sound varies with crystallographic direction and wave polarisation. The ~12 000 m/s figure applies to longitudinal waves along the stiffest axis [100]. Along other axes, it can be 10–15 % lower. This anisotropy is exploited in diamond-based acoustic sensors and quantum phononics research.
A point source radiating sound uniformly into free space produces a spherical wavefront. The total acoustic power is distributed over the surface of an ever-expanding sphere ($A = 4\pi r^2$), so intensity decreases with the square of the distance: $$I(r) = \frac$$ Consequence: every doubling of distance reduces sound pressure level by approximately 6 dB. A sound measured at 100 dB at 1 m will be ~94 dB at 2 m, ~88 dB at 4 m, and so on (in an ideal free field with no reflections or absorption).
Visual mapping note: This law governs how visual intensity, scale, or opacity should attenuate with simulated distance in a spatial audio–visual environment. Any 3D visual music system must implement inverse-square attenuation (or an artistic variant of it) to maintain perceptual realism.
When a sound source moves relative to a listener (or vice versa), the perceived frequency shifts: $$f' = f \cdot \frac{v \pm v{\text{obs}}}{v \mp v}}$$ where $v$ is the speed of sound, $v{\text{obs}}$ is the observer's velocity (positive if moving toward the source), and $v}$ is the source's velocity (positive if moving toward the observer).
Practical note: At the speeds encountered in everyday life (cars, aircraft), the Doppler shift is modest. At 30 m/s (~108 km/h), a 1 kHz source shifts by roughly ±9 %. At supersonic speeds ($v_{\text{src}} > v$), the wavefronts pile up into a Mach cone, producing a sonic boom—a shockwave, not a periodic sound.
As sound waves propagate, they interact with their environment in six classical ways. Each has direct implications for acoustic design, spatial audio, and visual representation.
Higher frequencies are absorbed more readily than lower ones. The mechanisms are molecular viscosity and thermal conduction at the wave's compression–rarefaction boundaries, which convert acoustic energy irreversibly into heat. The absorption coefficient increases roughly with the square of frequency in air.
Phenomenon: Distant thunder sounds like a low rumble because the high-frequency components of the original crack have been absorbed over kilometres of atmosphere, while the low frequencies survive. Similarly, music heard through a wall is predominantly bass.
Sound waves encountering irregularities, rough surfaces, or particles comparable to or smaller than their wavelength are scattered—energy is dispersed in multiple directions. Scattering is frequency-dependent: high frequencies (short wavelengths) scatter from small objects; low frequencies (long wavelengths) pass around them.
Example: Sound diffusing through a dense forest is scattered by trunks, branches, and leaves, creating a diffuse, ambient quality distinct from the direct sound in an open field.
Sound bounces back when it encounters a surface or boundary significantly larger than its wavelength. The angle of incidence equals the angle of reflection (specular reflection). Reflection creates echoes (discrete, delayed repetitions) and reverberation (dense, overlapping reflections that decay over time). The fraction of energy reflected vs. absorbed depends on the acoustic impedance mismatch at the boundary (see §IV).
Waves bend and spread as they pass around obstacles or through openings comparable in size to their wavelength.
Phenomenon: You can hear the low-frequency bass from a distant nightclub around a building corner, but not the high-frequency vocals. Bass wavelengths (~3–10 m) diffract easily around structures; treble wavelengths (~3–10 cm) are blocked, creating an acoustic "shadow." Diffraction limits the directivity of loudspeakers: a small speaker radiates high frequencies in a narrow beam but low frequencies nearly omnidirectionally. This is why subwoofers can be placed almost anywhere in a room, while tweeters must be aimed at the listener.
The trajectory of a sound wave bends when it passes through a medium with a spatial gradient in sound speed (e.g., a temperature or pressure gradient).
Phenomenon—Acoustic mirages and nighttime propagation: During the day, the ground is warmer than the air above it. Sound speed is higher near the ground, causing wavefronts to bend upward, creating an acoustic shadow at distance. At night, a temperature inversion forms (cool ground, warmer air above), bending wavefronts downward toward the ground. Sound travels dramatically farther at night. This is why distant trains, highways, or church bells seem louder after dark.
When two or more waves occupy the same space, their pressures add algebraically (the principle of superposition, valid for linear acoustics).
Acoustic impedance is the single most important concept for understanding how sound energy transfers between media, why resonators work, and why the ear needs an ossicular lever system (Unit 3.2). Characteristic (specific) acoustic impedance of a medium: $$Z = \rho \cdot v$$ where $\rho$ is density and $v$ is sound speed. Units: Pa·s/m (also called the rayl).
| Medium | $Z$ (rayl) |
|---|---|
| Air (20 °C) | ~415 |
| Water (20 °C) | ~1.48 × 10⁶ |
| Steel | ~4.7 × 10⁷ |
| Human soft tissue | ~1.6 × 10⁶ |
| Why it matters: | |
| When a sound wave encounters a boundary between two media with different impedances, part of the energy is transmitted and part is reflected. The reflection coefficient depends on the impedance ratio: | |
| $$R = \left(\frac\right)^2$$ |
Amplitude represents the physical magnitude of the pressure deviation from ambient atmospheric pressure. Because human hearing spans an enormous dynamic range—from the threshold of hearing (~20 µPa) to the threshold of pain (~20 Pa), a ratio of 1 000 000:1 in pressure—a linear scale is impractical. The decibel (dB), named after Alexander Graham Bell, uses a base-10 logarithmic scale, aligning with the Weber–Fechner law of human perception. Sound Pressure Level (SPL): $$Lp = 20 \log!\left(\frac{p_{\text{rms}}}\right) \text$$ where $p_0 = 20$ µPa (the reference pressure, approximately the threshold of hearing at 1 kHz). The logarithmic arithmetic:
The theoretical maximum for an undistorted, periodic sound wave in Earth's atmosphere at 1 atm is 194 dB SPL. At this level, the rarefaction (low-pressure) phase of the wave reaches a perfect vacuum (0 Pa absolute). The wave can no longer be sinusoidal; it becomes asymmetric. Critical clarification: This limit applies to continuous periodic waves. Transient phenomena—explosions, lightning, sonic booms, volcanic eruptions—produce shockwaves, not periodic sound waves. Shockwaves are non-linear pressure discontinuities that can exceed 194 dB because they are not constrained by the sinusoidal rarefaction limit. The 1883 eruption of Krakatoa is estimated to have produced a peak overpressure corresponding to ~310 dB at the source. This was a shockwave, not a sound wave in the strict acoustic sense. It circled the Earth four times and was heard clearly 3 000 miles (4 800 km) away. The distinction matters: 310 dB is not "louder sound"; it is a fundamentally different physical phenomenon.
Frequency determines the perceived highness or lowness of a sound (pitch). The standard human hearing range is ~20 Hz to ~20 000 Hz, though the upper limit declines with age (presbycusis, explored in Unit 3.2). Human hearing sensitivity is not flat across this range. The ear canal naturally resonates around 3–4 kHz, making us most sensitive in this region—a sensitivity tuned by evolution for detecting speech consonants and the spectral content of infant cries. The equal-loudness contours (Fletcher–Munson / ISO 226) quantify this non-uniformity and are explored in Unit 3.2. Pitch perception is logarithmic: the interval from 100 Hz to 200 Hz (an octave) is perceived as the same musical "distance" as 1 000 Hz to 2 000 Hz. This logarithmic relationship is the reason musical scales use multiplicative ratios rather than additive frequency steps, and it will be central to the tuning systems in Unit 3.5.
The physical world of vibration extends far beyond human perception. Each band exhibits unique physical behaviours and offers distinct possibilities for visual music.
Characteristics: Massive wavelengths (tens to hundreds of metres). Very little atmospheric absorption. Penetrates solid structures readily. Sources: Seismic waves, volcanic eruptions, severe weather, ocean surf, wind turbines, large machinery. Biological use: Elephants, whales, and rhinoceroses use infrasound for long-distance communication (up to 10 km through ground or air). Pigeons may use infrasonic cues for navigation.
Curious phenomenon—the "Ghost Frequency": Infrasound around 18.9 Hz is close to the resonant frequency of the human eyeball. Exposure to high-amplitude 19 Hz sound can cause the eyeballs to vibrate slightly, leading to blurred or distorted peripheral vision and feelings of dread, unease, or "presence." Researcher Vic Tandy (1998) demonstrated that a 19 Hz standing wave in his laboratory explained persistent reports of a "grey apparition" and feelings of being watched. The phenomenon was not supernatural; it was acoustic resonance of the eye. Tandy's work, published in the Journal of the Society for Psychical Research, remains a elegant example of physics dissolving mystery.
The narrow band of mechanical vibrations that the human auditory system evolved to decode. This is the domain of music, speech, and environmental sound—and the primary focus of this course.
Characteristics: Wavelengths become so short (millimetres to micrometres) that ultrasound begins to behave more like a directional beam of light than a spreading wave. Diffraction is minimal; reflection and refraction dominate. Applications:
Curious phenomenon—Acoustic levitation: By creating a precise standing wave of ultrasound, pressure nodes (points of zero acoustic pressure variation) can be formed. Small objects—water droplets, polystyrene beads, even small insects—can be trapped at these nodes and levitated against gravity without any physical contact. Modern arrays of ultrasonic transducers can manipulate multiple particles in 3D, creating "acoustic holograms" of floating matter—a direct bridge to visual music's interest in sculpting space with sound. Biological use—Echolocation: Bats emit ultrasonic pulses (up to ~200 kHz) and analyse returning echoes to navigate and hunt in complete darkness. Dolphins produce high-frequency clicks (up to ~150 dB) using phonic lips in the nasal passages, focus them through the fatty melon structure in the forehead, and receive returning echoes through the acoustically transparent lower jaw, which conducts sound to the middle ear.
Characteristics: In gases, the wavelength of hypersound becomes shorter than the mean free path of the molecules themselves. The continuum assumption of fluid mechanics breaks down, and the wave can no longer propagate as a traditional pressure wave. Propagation in condensed matter: In liquids and solids, hypersound is generated and detected using Brillouin scattering (interaction of light with acoustic phonons) or picosecond ultrasonics (ultrafast laser pulses generating and detecting GHz–THz acoustic pulses). Quantum regime: At these frequencies, sound is quantised into phonons—the quantum-mechanical analogues of photons for mechanical vibration. In solid-state physics, thermal energy in a crystal is modelled as a gas of thermal phonons (the Debye model). Hypersound is thus the frontier where classical acoustics meets quantum mechanics and materials science.
At low amplitudes, the wave equation is linear: waves superpose without interacting, and the speed of sound is independent of amplitude. This is the regime of everyday listening and of virtually all musical sound. At very high amplitudes (typically >140–150 dB SPL), the linear approximation breaks down:
Boundary note: Non-linear acoustics is not explored further in this module. It is mentioned here for completeness and to explain why the "louder sound" model breaks down at extreme amplitudes.
Definition: Resonance is the dramatic amplification of oscillation that occurs when an external driving force matches (or closely approaches) the natural frequency of a system. Energy input is maximally efficient at resonance because each cycle of the driving force adds energy in phase with the system's existing motion.
A Helmholtz resonator is a cavity of air with a narrow neck (the classic example: a bottle). The air in the neck acts as an oscillating mass; the air in the cavity acts as a compliant spring. Together they form a simple harmonic oscillator with a single, well-defined resonant frequency: $$fH = \frac \sqrt{\frac{V \cdot L}}}$$ where $v$ = speed of sound, $A$ = neck cross-sectional area, $V$ = cavity volume, and $L_{\text{eff}}$ = effective neck length (including end corrections).
Demonstration: Blow across the top of a bottle. The pitch depends on the volume of air inside. Add water (reducing $V$): the pitch rises. The bottle is a tunable Helmholtz resonator. Applications:
In an enclosed rectangular space, sound waves reflect between parallel walls and create standing waves at frequencies determined by the room's dimensions: $$f_ = \frac\sqrt{\left(\frac\right)^2 + \left(\frac\right)^2 + \left(\frac\right)^2}$$ where $n_x, n_y, n_z$ are non-negative integers and $L_x, L_y, L_z$ are the room dimensions. At these frequencies, sound pressure is dramatically amplified at some locations (antinodes) and cancelled at others (nodes), creating "boomy" bass or "dead spots." Calculating and treating room modes is a critical step in studio design, concert hall acoustics, and spatial audio installation.
Modern physics has developed engineered structures with acoustic properties not found in nature:
The physics in this unit is not merely theoretical. Several accessible tools allow you to see, measure, and analyse the phenomena described above.
Relevance to this unit:
Cymatics is the study of visible patterns produced by sound vibration. The most accessible demonstrations:
Sound is the bridge between the mechanical universe and human perception. Before it is music, before it is speech, before it is signal or noise, it is a pressure wave—a patterned disturbance in matter, governed by the same Newtonian mechanics that swing a pendulum and orbit a planet. By understanding the rigorous physics of how pressure waves propagate, bend, reflect, interfere, and resonate, we unlock the ability to manipulate the very fabric of auditory experience with scientific precision. And in the context of visual music theory, this physics becomes a vocabulary of form: frequency maps to spatial scale, amplitude to luminance, impedance to visual transparency, resonance to geometric structure, and the Doppler shift to the poetry of motion. In the next unit, we follow the wave as it enters the most sophisticated acoustic instrument ever evolved: the human ear.