17.2 Speed of Sound, Frequency, and Wavelength

Properties of Sound Waves

Pitch and Frequency Relationship

Pitch is how high or low a sound seems to you, and it maps directly to the frequency of the sound wave. Higher frequency means higher pitch; lower frequency means lower pitch.

• Doubling the frequency raises the pitch by one octave. Halving the frequency drops it by one octave.
• High-pitched sounds: a whistle, bird chirps
• Low-pitched sounds: a bass guitar, rumbling thunder

The human hearing range spans roughly 20 Hz to 20,000 Hz.

• Frequencies below 20 Hz are called infrasound. Elephants and whales communicate using infrasound.
• Frequencies above 20,000 Hz are called ultrasound. Bats use ultrasound for echolocation, and it's also used in medical imaging.

Loudness is a separate property from pitch. It relates to the amplitude of the sound wave and is measured in decibels (dB).

Speed of Sound Calculation

The speed of sound connects to frequency and wavelength through one core equation:

$v = f\lambda$

• $v$ = speed of sound (m/s)
• $f$ = frequency (Hz)
• $\lambda$ = wavelength (m)

You can rearrange this equation to solve for any of the three variables. Here's how to use it:

1. Finding speed: Multiply frequency by wavelength.

   • A sound wave has $f = 500$ Hz and $\lambda = 0.68$ m. Speed: $v = 500 \times 0.68 = 340$ m/s.

2. Finding wavelength: Divide speed by frequency ($\lambda = v / f$).

   • A sound wave has $f = 1000$ Hz and travels at $v = 340$ m/s. Wavelength: $\lambda = 340 / 1000 = 0.34$ m.

3. Finding frequency: Divide speed by wavelength ($f = v / \lambda$).

   • A sound wave has $\lambda = 0.5$ m and travels at $v = 340$ m/s. Frequency: $f = 340 / 0.5 = 680$ Hz.

One thing to keep straight: the speed of sound depends on the medium, not on the frequency or wavelength of the wave itself. If you change the frequency, the wavelength adjusts so that $v$ stays the same (in the same medium at the same temperature).

Factors Affecting Sound Propagation

Materials and Sound Propagation

Sound travels at different speeds through different materials. The general rule: solids > liquids > gases. This is because molecules in solids are packed more tightly and transmit vibrations more efficiently.

Some reference values at 20°C:

• Air: ~343 m/s
• Water: ~1,482 m/s
• Steel: ~5,960 m/s

Steel transmits sound about 17 times faster than air. That's why you can hear a train coming by pressing your ear to the rail long before you'd hear it through the air.

When sound encounters a boundary between materials, three things can happen:

• Reflection: The wave bounces off a surface. An echo is a familiar example.
• Refraction: The wave bends as it passes from one medium to another, changing speed. This affects underwater sound communication.
• Absorption: Sound energy is converted into heat. This is the principle behind soundproofing materials.

Temperature Effects on Sound Speed

In gases, the speed of sound increases with temperature. Higher temperatures mean gas molecules move faster on average, which lets them pass vibrations along more quickly.

For air specifically, you can approximate the speed of sound with this equation:

$v = 331.3 + 0.606T$

• $v$ = speed of sound (m/s)
• $T$ = temperature in degrees Celsius (°C)

For every 1°C increase, the speed of sound in air rises by about 0.6 m/s.

• At 0°C: $v = 331.3 + 0.606(0) = 331.3$ m/s
• At 20°C: $v = 331.3 + 0.606(20) = 343.4$ m/s

This is why sound can travel noticeably faster on a hot summer day than on a cold winter night.

Sound Wave Characteristics

Longitudinal Waves in Sound

Sound waves are longitudinal waves, meaning the particles in the medium vibrate parallel to the direction the wave travels. This creates alternating regions of:

• Compressions: zones of high pressure where particles are pushed closer together
• Rarefactions: zones of low pressure where particles are spread farther apart

The wavelength is the distance between two consecutive compressions (or two consecutive rarefactions).

Resonance in Sound Systems

Resonance occurs when an object is driven to vibrate at its natural frequency by an external force. At resonance, even a small driving force can produce large-amplitude vibrations, effectively amplifying the sound.

This is central to how musical instruments work. A guitar body, for instance, resonates with the vibrating strings, amplifying the sound so you can actually hear it across a room. Resonance also matters in acoustic design, where engineers shape rooms to enhance or control which frequencies get amplified.