1.1 Fundamentals of sound and acoustics

Fundamentals of Sound and Acoustics

Sound waves are mechanical vibrations that travel through matter, transferring energy through particle oscillations. They produce alternating compressions and rarefactions in whatever medium they pass through. Unlike electromagnetic waves, sound cannot travel through a vacuum; it always needs a physical medium like air, water, or a solid.

This topic covers the core properties that define how sound behaves: frequency, wavelength, amplitude, speed, and how these relate to one another. You'll also see how sound waves differ from other wave types and what happens physically inside a medium as sound passes through it.

Nature of Sound as a Mechanical Wave

Sound is a pressure wave. When a source vibrates (a guitar string, a speaker cone, your vocal cords), it pushes and pulls on the surrounding particles. Those particles bump into their neighbors, which bump into their neighbors, and so on. The energy moves outward from the source, but the individual particles themselves just oscillate back and forth around their resting positions.

The regions where particles are pushed together are called compressions (high-pressure zones). The regions where particles are spread apart are called rarefactions (low-pressure zones). This alternating pattern of high and low pressure is what makes sound a pressure wave.

Because this process depends on particles interacting with each other, sound requires a physical medium. In the vacuum of space, there are no particles to carry the vibration, so sound simply can't propagate there.

Key Properties of Sound Waves

• Frequency ($f$): The number of complete oscillations per second, measured in Hertz (Hz). Frequency determines the pitch you perceive. A tuning fork vibrating at 440 Hz produces the note A4; doubling that to 880 Hz gives you A5, one octave higher.
• Wavelength ($\lambda$): The distance between two consecutive compressions (or two consecutive rarefactions). Wavelength is inversely proportional to frequency: higher frequencies have shorter wavelengths.
• Speed of sound ($c$): How fast the wave travels through the medium. In air at 20°C, sound travels at roughly 343 m/s. In water, it's about 1,480 m/s. In steel, around 5,960 m/s. Denser and stiffer media generally transmit sound faster.
• Amplitude: The maximum displacement of a particle from its equilibrium position. Larger amplitude means more energy and a louder sound.
• Sound pressure level (SPL): Measured in decibels (dB), which use a logarithmic scale. This matters because human hearing spans a huge range of intensities. A whisper is around 30 dB, normal conversation about 60 dB, and a rock concert can exceed 110 dB.

The fundamental wave equation ties three of these together:

$c = f \lambda$

If you know any two of these quantities, you can find the third. For example, a 1,000 Hz tone in air ($c = 343$ m/s) has a wavelength of $\lambda = 343 / 1000 = 0.343$ m, or about 34 cm.

Longitudinal vs. Transverse Waves

Sound in air and water takes the form of longitudinal waves, where particle motion is parallel to the direction the wave travels. Think of it like a slinky being pushed and pulled along its length.

Transverse waves, by contrast, have particle motion perpendicular to the wave's travel direction. Waves on a guitar string and light waves are transverse.

The type of wave a medium can support depends on its physical properties:

• Gases and liquids support only longitudinal waves, because they can't sustain shear forces (they flow instead of resisting sideways pushes).
• Solids support both longitudinal and transverse waves, since their rigid structure can resist both compression and shear.

This distinction becomes important when you study how sound behaves in different materials, especially at boundaries between media.

Relationships in Sound Propagation

Beyond frequency and wavelength, a few more quantities describe what's happening inside the medium as sound passes through:

• Sound pressure ($p$): The local deviation from the ambient (background) pressure, measured in Pascals (Pa). When a compression passes by, pressure rises above ambient; during a rarefaction, it drops below.
• Particle velocity ($v$): A vector quantity describing how fast and in which direction the medium's particles are oscillating. This is not the same as the speed of sound; particle velocities are typically very small compared to the wave speed.
• Acoustic impedance ($Z$): A measure of how much a medium resists the propagation of sound waves. It's defined as:

$Z = \frac{p}{v}$

For a plane wave (a wave with flat, parallel wavefronts, which is a good approximation far from the source), pressure and particle velocity are in phase, meaning they reach their peaks and zero-crossings at the same time.

Acoustic impedance is especially important at boundaries between two different media. When sound hits a boundary (say, air meeting a wall), the difference in impedance between the two materials determines how much energy is reflected back and how much is transmitted through. Large impedance mismatches cause strong reflections, which is why you hear echoes off hard surfaces but not off curtains.