Sound

Sound is a mechanical wave that our ears can detect. Speech, music, echo sounders and ultrasound scans are all based on it. Exam problems deal with the speed of sound, echoes, pitch and wavelength. This topic continues the section on mechanical oscillations and waves: the wave equation v = λν and the ideas of period and frequency are applied directly here. So we first recall the wave parameters and then look at the special properties of sound.

§1. The nature of sound

A source of sound is a vibrating object: a guitar string, a loudspeaker cone, the vocal cords. The vibrating object alternately compresses and rarefies the surrounding layers of air, and these compressions and rarefactions travel through the medium. So sound is a longitudinal wave: the particles vibrate along the direction in which the wave travels. The particles themselves are not carried away; only energy is transferred.

Sound needs matter to travel: it cannot pass through a vacuum. The bell-jar experiment shows this: when the air is pumped out, the ringing bell can no longer be heard. The Moon has no atmosphere, so astronauts talk by radio.

• < 16 Hz — infrasound (earthquakes, storms, large machines; elephants use it to communicate).
• > 20 kHz — ultrasound (bats, dolphins; medical scans, echo sounders, flaw detection).

With age the ability to hear high frequencies declines. The human ear is most sensitive to sounds of roughly 2–5 kHz, and hears very low and very high frequencies only when they are loud. Many animals have a different hearing range from ours: dogs and cats can also hear part of the ultrasound range.

Worked example

What range of wavelengths in air do audible sounds have (v = 340 m/s)?

1. Lowest frequency: λ = v/ν = 340/20 = 17 m (for 20 Hz).
2. Highest frequency: λ = 340/20 000 = 0.017 m = 1.7 cm.

§2. The speed of sound

The speed of sound depends on the elasticity and density of the medium: in solids the particles are strongly bound, so a compression is passed on quickly. The speed is greatest in solids and smallest in gases.

In air, a higher temperature makes the molecules move faster and the speed of sound increases: approximately v ≈ 331 + 0.6t (m/s, t in °C). When sound passes from one medium to another, the frequency does not change — it is set by the source; the speed and wavelength change: λ = v/ν.

Worked example

A 680 Hz sound passes from air into water. Find the wavelengths.

1. In air λ = 340/680 = 0.5 m.
2. In water λ = 1500/680 ≈ 2.2 m (the frequency is the same).

Worked example

One end of a steel pipe 1020 m long is struck. With what interval does a person at the other end hear the two sounds?

1. Through the air: t₁ = 1020/340 = 3 s.
2. Through the steel: t₂ = 1020/5000 ≈ 0.2 s.
3. Interval: Δt = 3 − 0.2 ≈ 2.8 s — the sound through the steel arrives first, then the one through the air.

§3. Echoes and echolocation

Sound reflects from an obstacle — an echo. The sound travels there and back, so the path is doubled:

This is how an echo sounder measures the depth of the sea and how bats and dolphins “see” their prey — echolocation. To hear an echo separately, a person needs a delay of at least 0.1 s, so the obstacle must be at least 17 m away; otherwise the echo merges with the original sound and prolongs it (as happens in halls). Concert halls cover their walls with soft, porous materials to reduce unwanted echoes.

Worked example

An echo sounder’s signal returns after 4 s. How deep is the sea?

1. s = vt/2 = 1500·4/2.
2. s = 3000 m.

Lightning and thunder: light arrives almost instantly, while sound lags behind; s ≈ 340·t. A delay of 3 s means ≈ 1 km. Here the path is not halved — the sound travels in one direction only.

§4. Characteristics of sound

• Pitch (a low or high voice) depends on frequency: the higher the frequency, the higher the pitch.
• Loudness depends on amplitude: the larger the amplitude, the louder the sound.
• Timbre is the set of overtones sounding together with the fundamental; a violin and a piano play the same note with a different “colour”, and timbre is how we recognise people by their voices.

The intensity of sound I is the energy passing through unit area per unit time (W/m²). The ear responds to an enormous range, so a logarithmic scale is used — the intensity level in decibels:

Every +10 dB multiplies the intensity by 10. A whisper is ~20 dB, normal conversation ~60 dB, and around 120 dB is the threshold of pain; long exposure to noise above 85 dB damages hearing.

Worked example

Conversation has an intensity level of 60 dB. What is its intensity?

1. I = I₀·10^(L/10) = 10⁻¹²·10⁶.
2. I = 10⁻⁶ W/m².

§5. Resonance and musical instruments

Every object has its own natural frequency. If the frequency of an outside sound matches it, the object starts to vibrate strongly — acoustic resonance. A tuning fork is mounted on a box, and the body of a guitar or violin acts as a resonator and amplifies the sound. If one of two identically tuned tuning forks is struck, the other begins to sound by itself. Shortening a string or an air column raises its natural frequency — which is why a guitarist raises a note by pressing the string with a finger. Resonance can be useful or harmful: if a loud sound matches the natural frequency of a glass or a dish, it vibrates violently, so engineers avoid resonance when designing machines and buildings.

§6. Uses of ultrasound and infrasound

• Medicine: an ultrasound scan builds an image of internal organs from signals reflected at tissue boundaries; unlike X-rays, it is not ionising radiation.
• Engineering: flaw detection finds cracks inside metal, and ultrasound is used to clean and weld parts.
• At sea: echo sounders and sonar locate underwater objects — radio waves die out quickly in water, but sound travels far.
• Infrasound travels long distances with little loss; recording it allows earthquakes and explosions to be monitored.

Worked example

An ultrasound scanner works at 2 MHz, and the speed of sound in tissue is ~1500 m/s. What is the wavelength?

1. λ = v/ν = 1500/(2·10⁶).
2. λ = 7.5·10⁻⁴ m = 0.75 mm — the shorter the wavelength, the finer the detail visible, which is why medicine uses megahertz frequencies.

An approaching source sounds higher and a receding one lower: this is the Doppler effect (the siren of a passing ambulance).

Check yourself

1. How far does sound travel in air in 3 s?

2. What is the wavelength of a 170 Hz sound in air?

3. Thunder is heard 5 s after the lightning. How far away is it?

4. An echo from a cliff returns after 2 s. How far away is the cliff?

5. Why can’t an explosion be heard on the Moon?

6. Increasing which quantity makes a sound higher?

7. The noise level rises from 60 dB to 80 dB. By what factor has the intensity increased?

8. What is the approximate speed of sound in air at 25 °C?

Common mistakes

• Forgetting to halve the path in an echo problem — or halving it in a lightning problem.
• Thinking the frequency changes when sound enters a new medium.
• Confusing pitch (frequency) with loudness (amplitude).
• Treating sound as a transverse wave (in a gas it is longitudinal).
• Adding decibels like ordinary quantities: +10 dB multiplies the intensity by 10.