1.3 Sound wave characteristics and behavior

Sound Wave Propagation and Behavior

Sound waves are longitudinal waves that travel through a medium by creating alternating regions of high and low pressure. Understanding how they propagate, reflect, bend, lose energy, and shift in frequency is the foundation of acoustics. This section covers each of those behaviors and the physical principles behind them.

Propagation of sound waves

Sound travels as a longitudinal wave, meaning the particles in the medium oscillate back and forth parallel to the direction the wave is moving. This creates alternating zones of compression (high pressure) and rarefaction (low pressure) that propagate outward from the source.

Speed of sound in different media

The speed of sound depends heavily on the medium it's traveling through:

• Air: ~343 m/s at 20°C. Speed varies with temperature and humidity.
• Water: ~1480 m/s at 20°C. Much faster than air because water is denser and far less compressible.
• Solids: Generally fastest. Steel transmits sound at ~5000 m/s; wood at ~3300 m/s.

The pattern here is worth remembering: sound typically travels fastest in solids, then liquids, then gases. That's because the molecules in solids are packed tightly and coupled strongly, so vibrations pass between them quickly.

Factors affecting propagation speed

Three properties of the medium matter most:

• Temperature: In gases, higher temperature means faster molecular motion, which increases the speed of sound. In air, the speed rises by roughly 0.6 m/s for every 1°C increase.
• Density: Higher density generally increases speed, though this interacts with elasticity. Mercury transmits sound faster than water partly because of its greater density.
• Elasticity: More elastic (stiffer) materials transmit sound faster. Steel is far more elastic than rubber, so sound moves through steel much more quickly.

Acoustic impedance

Impedance describes how much a medium resists the propagation of sound waves. It's calculated as:

$Z = \rho c$

where $\rho$ is the density of the medium and $c$ is the speed of sound in that medium. Impedance is critical at boundaries between two media. When there's a large impedance mismatch between two materials (like air and water), most of the sound energy reflects back rather than transmitting through. When impedance values are similar, more energy passes across the boundary.

Reflection and refraction of sound

Reflection

When a sound wave hits a boundary or surface, some of its energy bounces back. The angle of incidence equals the angle of reflection, just like light reflecting off a mirror. Two types of reflection matter in acoustics:

• Specular reflection occurs off smooth, flat surfaces. The reflected wave stays coherent and directional.
• Diffuse reflection occurs off rough or irregular surfaces. The reflected energy scatters in many directions, which is why concert halls use textured wall panels to distribute sound evenly.

Refraction

Refraction is the bending of a sound wave when it passes from one medium to another (or through a region where the speed of sound changes). The wave changes direction because its speed changes. This relationship is described by Snell's law:

$\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}$

Here, $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, and $v_1$ and $v_2$ are the sound speeds in each medium. A practical example: on a warm day, air near the ground is hotter (faster sound speed) than air higher up, so sound waves bend upward and away from listeners. This creates sound shadows where it's harder to hear distant sources.

Diffraction

Sound waves bend around obstacles and spread through openings. This is why you can hear someone talking around a corner even when you can't see them. The key rule: diffraction is most pronounced when the wavelength is similar to or larger than the obstacle or opening. Low-frequency sounds (long wavelengths) diffract much more than high-frequency sounds, which is why bass travels around barriers more easily than treble.

Huygens' principle explains this: every point on a wavefront acts as a new source of secondary wavelets, and the sum of those wavelets determines the new wavefront shape.

Interference

When two or more sound waves overlap, they combine:

• Constructive interference: Waves in phase add together, producing a louder sound.
• Destructive interference: Waves out of phase cancel each other, reducing or eliminating sound.

Standing waves are a special case of interference. They form when sound reflects back and forth in an enclosed space, creating fixed patterns of nodes (no displacement) and antinodes (maximum displacement). This is how musical instruments produce resonant tones and why certain frequencies boom unnaturally in small rooms (room modes).

Sound wave attenuation factors

Attenuation is the gradual loss of sound energy as a wave travels. Sound gets quieter with distance for three main reasons:

• Geometric spreading: As a sound wave expands outward (like a growing sphere), its energy spreads over a larger area. This is the inverse-square law at work.
• Absorption: The medium itself converts some sound energy into heat through friction between vibrating particles.
• Scattering: Irregularities in the medium redirect portions of the wave in different directions.

Absorption in materials

Porous materials like acoustic foam and fiberglass are effective absorbers because sound waves enter the tiny air pockets and lose energy to friction. The absorption coefficient (ranging from 0 to 1) measures the fraction of incident sound energy a material absorbs. A coefficient of 0.9 means 90% of the energy is absorbed.

Several factors influence how much absorption occurs:

• Frequency: Higher frequencies generally attenuate faster than lower ones. This is why distant thunder sounds like a low rumble rather than a sharp crack.
• Material properties: Density, porosity, and stiffness all affect absorption. Soft, porous materials absorb well; hard, smooth surfaces reflect.
• Thickness: Thicker absorbing materials are more effective, especially at lower frequencies.

Transmission loss

Transmission loss (TL) measures how much sound a barrier blocks. The mass law for single-layer partitions states that doubling the mass per unit area of a wall increases TL by approximately 6 dB. This is why heavy concrete walls block more sound than thin drywall.

Reverberation time

Reverberation time ($T_{60}$) is the time it takes for sound in a room to decay by 60 dB after the source stops. The Sabine formula calculates it:

$T_{60} = \frac{0.161V}{A}$

where $V$ is the room volume in cubic meters and $A$ is the total absorption in the room (in sabins). A large, reflective room (like a cathedral) has a long $T_{60}$. A small, heavily treated recording studio has a short one.

Doppler effect in acoustics

The Doppler effect is the apparent change in frequency (perceived pitch) of a sound wave caused by relative motion between the source and the observer. You've experienced this if you've ever noticed an ambulance siren sounding higher-pitched as it approaches and lower-pitched as it drives away.

The formula:

$f' = f\frac{c \pm v_r}{c \mp v_s}$

where $f'$ is the observed frequency, $f$ is the emitted frequency, $c$ is the speed of sound, $v_r$ is the observer's speed, and $v_s$ is the source's speed. The sign conventions depend on the direction of motion (toward or away), so pay attention to how your textbook defines positive direction.

Three scenarios to know:

1. Source stationary, observer moving: As the observer approaches the source, they encounter wavefronts more frequently, so pitch increases. Moving away, pitch decreases.
2. Source moving, observer stationary: The source compresses wavefronts ahead of it (higher pitch) and stretches them behind it (lower pitch).
3. Both moving: The effects combine. You apply both the source and observer velocity terms in the formula.

Applications

• Speed enforcement: Radar guns measure the Doppler shift of reflected radio waves to calculate vehicle speed.
• Medical ultrasound: Doppler imaging detects blood flow velocity and direction by measuring frequency shifts in reflected ultrasound.
• Astronomy: The redshift and blueshift of light from stars and galaxies reveals whether they're moving toward or away from Earth, and how fast.

Limitations

• Wind and other medium motion can alter the observed Doppler shift in air, since the formula assumes a stationary medium.
• At velocities approaching the speed of light, the classical Doppler formula breaks down and you need the relativistic version from special relativity. For everyday acoustics, though, the classical formula works fine.