2.1 Properties of Sound Waves

Sound waves are vibrations that travel through matter, carrying energy from one place to another. Understanding their key properties, like frequency, wavelength, and amplitude, is essential for explaining how we hear, how instruments produce music, and how waves behave when they move between different materials. These concepts form the foundation for everything else in acoustics.

Sound Wave Characteristics

Fundamental Properties of Sound Waves

Sound waves are mechanical waves, meaning they need a medium (air, water, steel, etc.) to travel through. They propagate by causing particles in the medium to vibrate, passing energy along from one particle to the next.

Every sound wave can be described by three core properties:

• Frequency ($f$): The number of complete oscillations per second, measured in Hertz (Hz). One Hz means one full cycle per second.
• Wavelength ($\lambda$): The distance between two consecutive compressions (or two consecutive rarefactions). Shorter wavelengths correspond to higher frequencies.
• Amplitude: The maximum displacement of a particle from its equilibrium (rest) position. Larger amplitude means more energy in the wave.

These three properties are connected by the wave equation:

$v = f\lambda$

where $v$ is the wave speed, $f$ is frequency, and $\lambda$ is wavelength. If you know any two of these, you can find the third.

One more relationship worth remembering: the energy carried by a sound wave is proportional to the square of its amplitude. Double the amplitude and you get four times the energy.

When you graph a sound wave as pressure variation vs. distance (or time), you get a sinusoidal curve. The peaks represent compressions (high pressure) and the valleys represent rarefactions (low pressure).

Sound Wave Behavior in Different Media

The speed of sound depends heavily on the medium it travels through. A few benchmarks to know:

Sound generally travels faster in denser, stiffer materials because the particles are closer together and can transmit vibrations more quickly.

There's also an important distinction in what types of waves different media can support:

• Fluids (gases and liquids) transmit only longitudinal sound waves.
• Solids can propagate both longitudinal and transverse sound waves. In the same solid, longitudinal waves travel faster than transverse waves.

When a sound wave hits a boundary between two different media, several things can happen: reflection (bouncing back), refraction (bending as it enters the new medium), and diffraction (spreading around obstacles). How much sound reflects vs. transmits depends on the acoustic impedance mismatch between the two materials. A large mismatch means most of the sound reflects back.

Sound also loses energy as it travels through attenuation, which happens due to absorption by the medium, scattering off irregularities, and simple geometric spreading (the wave's energy spreads over a larger area as it moves outward).

Frequency, Pitch, Amplitude, and Loudness

Frequency and Pitch Perception

Pitch is the subjective perception of how "high" or "low" a sound seems. It maps closely to frequency: higher frequency means higher pitch.

The human hearing range spans roughly 20 Hz to 20,000 Hz. Below 20 Hz is infrasound; above 20,000 Hz is ultrasound. Most everyday sounds fall well within this range, and your upper limit decreases with age.

Pitch perception doesn't scale linearly with frequency. Instead, it follows a logarithmic relationship. Going from 100 Hz to 200 Hz sounds like the same "jump" as going from 1,000 Hz to 2,000 Hz (both are a doubling, or one octave). This is why musical scales are built on frequency ratios rather than fixed frequency intervals.

Pitch perception isn't purely about frequency, though. Intensity, duration, and the presence of harmonics can all shift how you perceive pitch. The Fletcher-Munson curves (equal-loudness contours) show that human ears are not equally sensitive at all frequencies. We hear mid-range frequencies (around 1,000–5,000 Hz) much more easily than very low or very high ones at the same intensity level.

Amplitude and Loudness Measurement

Loudness is the subjective perception of sound intensity, and it correlates with amplitude. A larger amplitude wave pushes your eardrum further, and you perceive it as louder.

Sound intensity is measured on the decibel (dB) scale, which is logarithmic. Here's what that means in practice:

• A 10 dB increase corresponds to a 10× increase in sound intensity.
• A 20 dB increase corresponds to a 100× increase in intensity.

This logarithmic scaling reflects how human hearing actually works. The Weber-Fechner law describes this: perceived loudness increases roughly with the logarithm of actual intensity. That's why going from 40 dB to 50 dB feels like a similar change as going from 70 dB to 80 dB, even though the absolute intensity differences are vastly different.

Loudness perception is also influenced by frequency (as the Fletcher-Munson curves show), duration of the sound, and background noise levels.

Longitudinal vs. Transverse Waves

Longitudinal Wave Characteristics

In a longitudinal wave, particles vibrate parallel to the direction the wave travels. Picture a slinky being pushed and pulled along its length. The regions where coils bunch together are compressions (high pressure), and the regions where they spread apart are rarefactions (low pressure).

• Propagate through all states of matter: solids, liquids, and gases
• The primary wave type for sound in fluids
• Travel faster than transverse waves in the same medium
• Examples: sound waves in air, P-waves (primary waves) in earthquakes

Transverse Wave Characteristics

In a transverse wave, particles vibrate perpendicular to the direction the wave travels. Think of shaking a rope up and down while the wave moves horizontally along it. The high points are crests and the low points are troughs.

• Generally limited to solids and the surfaces of liquids (fluids can't sustain the shear forces needed for transverse waves in their bulk)
• Travel slower than longitudinal waves in the same medium
• Examples: waves on a string, S-waves (secondary waves) in earthquakes, electromagnetic waves (though those don't need a medium)

The fact that fluids don't support transverse waves is actually how seismologists figured out that Earth's outer core is liquid: S-waves can't pass through it.

Sound Wave Generation and Propagation

Sound Generation Mechanisms

Any vibrating object can generate sound. Vocal cords, speaker diaphragms, tuning forks, and guitar strings all work by pushing and pulling on the surrounding medium, creating alternating compressions and rarefactions that spread outward.

The frequency of the vibrating source determines the frequency of the sound produced, and the amplitude of the vibration controls the amplitude of the resulting wave.

Two concepts that matter for real sound sources:

• Resonance occurs when an object vibrates at its natural frequency, amplifying the sound. This is why a guitar body amplifies the vibrations of its strings, and why blowing across a bottle at just the right angle produces a clear tone.
• Harmonics are integer multiples of a fundamental frequency that a vibrating object produces simultaneously. They're what give different instruments their distinctive timbre (tone quality). A piano and a violin playing the same note at the same loudness still sound different because of their unique harmonic profiles.

Propagation Through Various Media

How sound travels depends on the medium's physical properties:

• In gases and liquids, sound propagates through molecular collisions. One molecule bumps the next, passing the vibration along.
• In solids, sound travels through vibrations of the atomic/molecular lattice structure, which is more rigid and efficient.

The speed of sound in any medium depends on its elastic modulus (stiffness) and density. Stiffer materials transmit sound faster; higher density alone tends to slow it down. Solids are typically both stiffer and denser than fluids, but stiffness wins out, which is why sound is fastest in materials like steel.

Temperature also matters. In air, the speed of sound increases with temperature because warmer air molecules move faster and collide more readily. At 20°C, sound in air travels at about 343 m/s. At 0°C, it drops to about 331 m/s.

When sound crosses from one medium to another, impedance mismatches at the interface determine how much energy reflects back and how much transmits through. This principle is central to applications like ultrasound imaging, where gel is used to reduce the impedance mismatch between the transducer and skin.