1.1 Sound waves and their properties

Sound waves are mechanical vibrations that travel through a medium like air, carrying energy from one point to another. Their properties determine everything from the pitch of a musical note to how a concert hall sounds from the back row. For architectural acoustics, understanding these properties is the first step toward designing spaces where sound behaves the way you want it to.

Properties of sound waves

Sound waves propagate by causing particles in a medium to oscillate back and forth, transferring energy without the particles themselves traveling any significant distance. The three core properties you need to know are frequency, amplitude, and wavelength. Together, they determine how we perceive pitch, loudness, and tone quality.

Frequency and pitch

Frequency is the number of complete oscillation cycles per second, measured in Hertz (Hz). Pitch is the subjective way we perceive frequency: higher frequencies sound higher in pitch, lower frequencies sound lower.

• The human ear can typically detect frequencies between 20 Hz and 20,000 Hz (20 kHz)
• In music, the fundamental frequency determines the pitch of a note, while overtones shape its timbre (tone quality)
• A frequency below 20 Hz is called infrasound; above 20 kHz is ultrasound. Neither is audible to most people, but both can still affect architectural design through vibration and structural resonance.

Amplitude and loudness

Amplitude is the maximum displacement of particles from their resting (equilibrium) position during oscillation. Loudness is the subjective perception of that amplitude: greater amplitude means louder sound.

• Sound intensity is measured on the decibel (dB) scale, which is logarithmic
  • The threshold of human hearing is 0 dB, while a jet engine at close range is around 150 dB
  • Every 10 dB increase represents a tenfold increase in sound intensity
• Because the scale is logarithmic, small dB changes correspond to large differences in actual energy. A 3 dB increase roughly doubles the sound intensity, even though it doesn't sound dramatically louder.

Wavelength and speed of sound

Wavelength is the distance between two consecutive points in the same phase of a wave, such as two adjacent peaks or two adjacent troughs.

The speed of sound depends on the medium. In air at room temperature (20°C), it's approximately 343 m/s. The relationship between frequency ($f$), wavelength ($\lambda$), and speed of sound ($v$) is:

$v = f \times \lambda$

This equation is one you'll use constantly. If you know any two of the three variables, you can solve for the third. Wavelength matters in architectural acoustics because it determines how sound interacts with surfaces and openings of different sizes.

Types of sound waves

Sound waves are classified by how particles oscillate relative to the direction the wave travels. The type of wave affects how it behaves in different materials, which matters when you're analyzing sound transmission through walls, floors, and other building elements.

Longitudinal waves

In longitudinal waves, particles oscillate parallel to the direction of wave propagation. This creates alternating regions of compression (particles pushed together) and rarefaction (particles spread apart).

• Sound waves in air and other fluids are longitudinal waves
• Seismic P-waves (primary waves) are also longitudinal, which is why they travel fastest through the earth

Transverse waves

In transverse waves, particles oscillate perpendicular to the direction of wave propagation, forming crests and troughs.

• Sound waves in solids can be transverse, such as vibrations in a string or membrane
• Seismic S-waves (secondary waves) are transverse
• Transverse waves cannot propagate through fluids because fluids lack the shear stiffness needed to support perpendicular displacement

Plane vs. spherical waves

• Plane waves have flat, parallel wavefronts. They propagate in a single direction and are a useful approximation for sound waves far from a source, where the curvature of the wavefront becomes negligible.
• Spherical waves radiate outward in all directions from a point source, with wavefronts forming concentric spheres. Their intensity decreases with distance according to the inverse square law.
• In architectural acoustics, you'll use both models. Spherical waves better represent a speaker or instrument in a room, while plane wave assumptions simplify analysis of sound hitting a large wall.

Sound wave propagation

Propagation describes how sound energy moves through a medium and what happens when waves encounter surfaces, obstacles, and changes in the medium. These phenomena are central to room acoustics design.

Reflection and absorption

Reflection occurs when a sound wave hits a boundary and bounces back.

• The angle of incidence equals the angle of reflection (just like light off a mirror)
• Hard, smooth surfaces (glass, concrete, polished stone) reflect sound efficiently
• This is why bare rooms with hard walls produce strong echoes

Absorption converts sound energy into heat as the wave interacts with a material.

• Porous materials (acoustic foam, fiberglass, heavy curtains) absorb sound by allowing air to move through tiny passages, where friction dissipates energy as heat
• The absorption coefficient ranges from 0 (perfect reflection) to 1 (perfect absorption) and varies with frequency. A material might absorb high frequencies well but reflect low frequencies almost entirely.

Diffraction and interference

Diffraction is the bending of sound waves around obstacles or through openings.

• The key factor is the ratio of wavelength to obstacle/opening size. When the wavelength is large relative to the obstacle, the wave bends around it easily. When the wavelength is small, the obstacle casts more of an "acoustic shadow."
• Low-frequency waves (long wavelengths) diffract much more readily than high-frequency waves. This is why you can hear bass through a doorway but higher frequencies seem blocked.

Interference occurs when two or more sound waves overlap.

• Constructive interference: waves in phase combine to produce increased amplitude
• Destructive interference: waves out of phase combine to produce decreased amplitude, potentially canceling each other out

Refraction and diffusion

Refraction is the change in direction of a sound wave as it passes through a medium with varying properties (temperature, density, wind speed).

• Sound waves bend toward regions of lower speed (cooler air, for example)
• Outdoors, temperature gradients can cause sound to curve upward on hot days (when air near the ground is warmer) or downward on cool nights, significantly affecting how far sound carries

Diffusion is the scattering of sound energy in multiple directions by irregular surfaces.

• Diffusers use complex surface patterns (stepped, curved, or mathematically derived shapes) to break up and scatter reflections
• Good diffusion creates a more even sound field throughout a room and reduces distinct echoes without removing energy the way absorbers do

Sound wave measurements

Quantifying sound properties with standardized units lets you compare acoustic conditions, set design targets, and evaluate whether a space meets its performance goals.

Decibel scale for loudness

The decibel (dB) scale is logarithmic, expressing sound intensity relative to a reference level.

• The reference level (0 dB) corresponds to the threshold of human hearing: $10^{-12}$ W/m²
• Every 10 dB increase represents a tenfold increase in sound intensity and roughly a perceived doubling of loudness
• Common sound levels:
  • Rustling leaves: ~20 dB
  • Normal conversation: ~60 dB
  • Heavy traffic: ~80 dB
  • Jet engine at 100 m: ~120 dB (threshold of pain)

The logarithmic nature means that combining two identical 60 dB sources doesn't give you 120 dB. It gives you about 63 dB.

Hertz for frequency

Frequency is measured in Hertz (Hz), representing cycles per second.

• Human hearing range: 20 Hz to 20,000 Hz (20 kHz)
• Lower frequencies correspond to bass sounds; higher frequencies correspond to treble
• An octave represents a doubling of frequency. In Western music, each octave is divided into 12 semitones.
  • The A above middle C is 440 Hz
  • The A one octave higher is 880 Hz

Wavelength in meters

Wavelength ($\lambda$) is measured in meters and calculated using $\lambda = \frac{v}{f}$.

At a constant speed of sound, lower frequencies have longer wavelengths and higher frequencies have shorter ones. Here are some example wavelengths in air at 343 m/s:

• 20 Hz: $\lambda \approx 17.15$ m
• 1 kHz: $\lambda \approx 0.343$ m
• 20 kHz: $\lambda \approx 0.017$ m

These numbers have direct design implications. A 17-meter wavelength at 20 Hz is comparable to the dimensions of a large room, which is why low-frequency control is one of the hardest challenges in architectural acoustics.

Factors affecting sound waves

Environmental and material conditions shape how sound propagates. Predicting acoustic behavior in a real space requires accounting for these variables.

Temperature and humidity

• Temperature affects the speed of sound in air. Higher temperatures mean faster propagation. The speed increases by approximately 0.6 m/s for every 1°C rise in temperature.
• Humidity has a smaller effect, slightly increasing the speed of sound at higher moisture levels. Humidity also affects high-frequency absorption in air: drier air absorbs high frequencies more over long distances.
• Temperature and humidity gradients cause refraction, bending sound waves and altering their propagation paths. This is especially relevant in large venues and outdoor spaces.

Density and elasticity of medium

The speed of sound in a medium depends on both its density and its elasticity (how readily it returns to shape after deformation). More specifically, sound travels faster in materials that are stiffer (more elastic), and the relationship with density is more nuanced than "denser = faster."

• Speed of sound in air: ~343 m/s
• Speed of sound in water: ~1,480 m/s
• Speed of sound in steel: ~5,960 m/s

Steel is both denser and far stiffer than air, and stiffness dominates. Changes in density and elasticity at boundaries between materials cause reflection and refraction of sound waves.

Boundaries and obstacles

• Boundaries between different media (air-wall, air-ground) cause reflection and absorption. The amount of reflection depends on the acoustic impedance mismatch between the two media. The greater the mismatch, the more sound reflects.
  • Smooth, hard surfaces reflect sound efficiently; rough, soft surfaces scatter and absorb it
• Obstacles cause diffraction, scattering, and attenuation
  • Diffraction depends on the obstacle's size relative to the wavelength
  • Obstacles can create acoustic shadows (quiet zones behind them) or focus sound energy in specific areas

Harmonic content of sound waves

Most real-world sounds are not pure single-frequency tones. They're complex waves made up of multiple frequencies layered together. The specific mix of those frequencies gives each sound its unique character.

Fundamental frequency

The fundamental frequency ($f_0$) is the lowest frequency component of a complex sound wave. It determines the perceived pitch. When you hear a note and identify it as "A440," you're identifying its fundamental frequency of 440 Hz, even though many higher frequencies are present simultaneously.

Overtones and harmonics

Harmonics are frequency components at integer multiples of the fundamental:

• 1st harmonic = $f_0$ (the fundamental itself)
• 2nd harmonic = $2 \times f_0$
• 3rd harmonic = $3 \times f_0$, and so on

Overtones are all the frequency components above the fundamental. For harmonic sounds, the 1st overtone is the 2nd harmonic, the 2nd overtone is the 3rd harmonic, etc. The relative amplitudes of these harmonics shape the timbre of the sound.

Timbre and sound quality

Timbre is what lets you distinguish a violin from a trumpet even when both play the same note at the same loudness. It's determined by the relative amplitudes and phases of the harmonics present in the sound.

• Different instruments produce different harmonic profiles because of their physical construction (body shape, material, how the sound is generated)
• The human voice has a characteristic timbre that varies between individuals, which is why you can recognize someone by their voice alone
• In architectural acoustics, room surfaces and geometry can alter the harmonic balance of sounds through frequency-dependent absorption and reflection, changing the perceived timbre of music or speech

Interaction of sound waves

When multiple sound waves exist in the same space, they combine and interact. These interactions produce some of the most important (and sometimes problematic) acoustic phenomena in rooms.

Superposition principle

The superposition principle states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual wave displacements. Sound waves are linear under normal conditions, so this principle applies directly. The resulting sound pressure at any location is simply the sum of the pressures contributed by each wave at that moment.

Constructive vs. destructive interference

• Constructive interference occurs when waves are in phase (peaks align with peaks). The resulting amplitude equals the sum of the individual amplitudes, producing louder sound.
• Destructive interference occurs when waves are out of phase (peaks align with troughs). The resulting amplitude equals the difference between the individual amplitudes, producing quieter sound or even silence if the waves are equal in amplitude and perfectly out of phase.

Active noise cancellation technology exploits destructive interference by generating a wave that's the inverse of an unwanted sound, effectively canceling it out.

Standing waves and resonance

Standing waves form when two identical waves traveling in opposite directions interfere, creating a stationary pattern of nodes (points of minimal displacement) and antinodes (points of maximal displacement). The distance between two adjacent nodes (or two adjacent antinodes) equals half the wavelength.

Resonance occurs when a system oscillates with greater amplitude at specific frequencies called resonant frequencies. At these frequencies, the system efficiently absorbs and stores energy from the driving force.

In room acoustics, standing waves arise because of the room's geometry:

• Room modes are the resonant frequencies determined by a room's length, width, and height. For a rectangular room, the simplest mode along one dimension occurs when the room length equals half the wavelength.
• At room mode frequencies, certain locations will have exaggerated bass response (at antinodes) while others will have almost none (at nodes). This is why a subwoofer can sound completely different depending on where you stand in a room.
• Proper room design, including non-parallel walls, appropriate proportions, and targeted acoustic treatment (bass traps, tuned absorbers), helps control standing waves and minimize their negative effects on sound quality.