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Waves & Sound

Q: Why does sound travel faster in solids than in air?

Sound is a mechanical wave, so it needs a medium. Solids are generally stiffer and more elastic than gases, so their particles pass on vibrations more quickly.

Q: What is the difference between transverse and longitudinal waves?

In a transverse wave, particles oscillate at right angles to the direction the wave travels, as on a string. In a longitudinal wave, particles oscillate along the direction of travel, as in sound in air.

Q: Why do you hear the pitch of a passing siren change?

The Doppler effect shifts the frequency. The pitch is higher as the source approaches, because the waves are compressed, and lower as it moves away.

Q: Do waves transport matter?

Not usually. In most waves, particles oscillate around fixed positions and only the energy and the pattern of the disturbance move forward.

Q: Why is sound louder when waves add in phase?

In phase, the displacements add, which increases the amplitude and therefore the intensity of the sound. Out of phase, they cancel and the sound is quieter.

Wave properties, superposition, standing waves, sound and the Doppler effect. 57 topics, from beginner to advanced. Open the interactive version with labs and quizzes

A wave transfers energy without transferring matter. This area distinguishes transverse and longitudinal waves, mechanical and electromagnetic waves, and defines wavelength, frequency, period and speed. It goes on to superposition, interference, standing waves in strings and pipes, sound intensity, and the Doppler effect.

How the topics connect

Waves carry energy without carrying matter. This area covers the basic wave properties, then superposition and interference, standing waves in strings and pipes, sound and the Doppler effect. Waves also link to optics and quantum physics, where light and matter show the same wave behaviour.

Real-world uses

Music and acoustics

Design concert halls, speakers and musical instruments using standing-wave patterns and sound intensity.

Medical ultrasound

Use the reflection and Doppler shift of high-frequency sound to image organs and measure blood flow.

Seismology

Infer the structure of the Earth's interior from the speeds and paths of seismic P and S waves.

Radar and speed checks

Measure the speed of a moving car or storm from the frequency shift of a reflected wave.

Music and acoustics

Instrument makers choose string tensions and pipe lengths to produce the right harmonics, and architects shape rooms to control echo.

Medical and geophysical imaging

Ultrasound scanners and seismologists use wave reflections and travel times to see inside bodies and the Earth.

Beginner

What is a wave?

A disturbance that carries energy without carrying matter.

Key relationwave

A wave is a pattern of disturbance that travels through space, carrying energy and momentum but not matter. The medium's particles only oscillate around their positions.

Worked example

A stadium wave moves around the arena while each person only stands and sits.

Common mistake

Waves transport energy, not material.

Beginner

Mechanical vs electromagnetic waves

Mechanical waves need a medium; EM waves do not.

Mechanical waves (sound, water, seismic, string) need a material medium. Electromagnetic waves (light, radio, X-rays) are oscillating electric and magnetic fields and can cross a vacuum.

Worked example

Sunlight reaches us through empty space, but sound cannot.

Common mistake

Sound doesn't travel in a vacuum.

Beginner

Transverse waves

Oscillation perpendicular to propagation, like waves on a string.

Key relationwave

In a transverse wave the particles move perpendicular to the direction the wave travels: ripples on a rope, light, S-waves. They can be polarised.

Worked example

Flicking one end of a rope up and down sends a wave along it.

Common mistake

Don't confuse the particle motion with the wave velocity.

Beginner

Longitudinal waves

Oscillation parallel to propagation, like sound.

Particles oscillate parallel to the propagation direction, creating compressions and rarefactions: sound, P-waves, a slinky push. They can't be polarised.

Worked example

A loudspeaker cone pushes air back and forth.

Common mistake

Wavelength is the distance between successive compressions.

Beginner

Wavelength, frequency, speed

Fundamental wave relation.

Key relationv = fλ

v = fλ: the wave advances one wavelength in one period. The speed is set by the medium; frequency by the source.

Worked example

A 440 Hz tone in air (343 m/s) has λ = 0.78 m.

Common mistake

When the wave enters a new medium, f stays the same and λ changes.

Beginner

Amplitude and intensity

Intensity is proportional to amplitude squared.

Key relationI ∝ A²

Amplitude is the maximum displacement. Intensity (power per area) is proportional to amplitude squared (and frequency squared). Doubling the amplitude gives four times the intensity.

Worked example

Sound twice as loud in amplitude carries four times the power.

Common mistake

Intensity isn't amplitude.

Intermediate

Wave equation (travelling)

A sinusoidal wave function of space and time.

Key relationy = A sin(kx − ωt)

y(x, t) = A sin(kx − ωt) describes a wave moving in +x at v = ω/k. Using (kx + ωt) gives motion toward −x.

Worked example

A = 0.1 m, k = 2π, ω = 4π: v = 2 m/s.

Common mistake

Sign of ωt gives the direction.

Intermediate

Wave number

Number of radians per metre.

Key relationk = 2π/λ

k = 2π/λ in rad/m is the spatial analogue of ω: it counts how many radians of phase per metre.

Worked example

λ = 0.5 m gives k = 12.6 rad/m.

Common mistake

k isn't the same as 1/λ (that's the wavenumber used in spectroscopy).

Intermediate

Wave speed on a string

Depends on tension and mass per unit length.

Key relationv = √(T/μ)

v = √(T/μ), where T is tension and μ mass per unit length. Tighter and lighter strings carry waves faster.

Worked example

T = 100 N, μ = 0.01 kg/m: v = 100 m/s.

Common mistake

Frequency doesn't change the speed.

Intermediate

Wave speed in a medium

Depends on elasticity and density.

Key relationv = √(B/ρ)

Generally v = √(elastic modulus / density): for sound in a fluid v = √(B/ρ), in a solid rod √(Y/ρ). Stiffer, lighter materials are faster.

Worked example

Sound in steel 5000 m/s, in water 1480 m/s, in air 343.

Common mistake

More elastic gives higher speed.

Beginner

Speed of sound

About 343 m/s in air at 20 °C.

Key relationv = 331 + 0.6T

In air v ≈ 331 + 0.6T (°C) m/s. It rises with temperature and does not depend on pressure at constant temperature.

Worked example

At 20 °C: 343 m/s.

Common mistake

Higher humidity raises it slightly.

Intermediate

Speed of sound in gases

Depends on temperature and molar mass.

Key relationv = √(γRT/M)

v = √(γRT/M). Heavier molecules are slower, hotter gases faster. Helium raises the speed to about 1000 m/s, giving a squeaky voice.

Worked example

Air at 300 K: √(1.4·8.31·300/0.029) ≈ 347 m/s.

Common mistake

Use T in kelvin.

Beginner

Reflection of waves

Waves bounce at boundaries.

When a wave hits a boundary it bounces back, with the angle of incidence equal to the angle of reflection. Echoes are reflections of sound.

Worked example

A canyon echo returns after 2d/v.

Common mistake

Frequency doesn't change on reflection.

Intermediate

Phase change on reflection

Fixed ends invert the pulse; free ends do not.

A pulse on a string reflects inverted from a fixed end and upright from a free end. For light, reflection from a denser medium gives a π phase shift.

Worked example

Waves on a rope tied to a wall come back upside down.

Common mistake

Phase changes matter for interference.

Intermediate

Refraction of waves

Waves change direction when speed changes.

Key relationn₁sinθ₁ = n₂sinθ₂

When a wave enters a medium with a different speed, its direction bends and its wavelength changes while frequency stays the same. Snell: sinθ₁/sinθ₂ = v₁/v₂.

Worked example

Water waves bending toward the shore as depth decreases.

Common mistake

Bends toward the normal in the slower medium.

Intermediate

Diffraction

Bending of waves around obstacles and through gaps.

Waves spread when passing through gaps or around edges. The effect is strong when the gap is comparable to the wavelength.

Worked example

You hear someone round the corner but can't see them: sound λ ≈ 1 m, light λ ≈ 500 nm.

Common mistake

Narrower slit gives more spreading.

Beginner

Principle of superposition

Displacements of overlapping waves add.

Key relationy = y₁+y₂

When waves overlap, the resulting displacement is the sum of the individual ones. They pass through each other unchanged.

Worked example

Two pulses on a string cross and continue.

Common mistake

Applies to linear media only.

Beginner

Constructive and destructive interference

Waves in phase add; in antiphase cancel.

If waves arrive in phase they add (bright, loud); if out of phase by π they cancel. Complete cancellation needs equal amplitudes.

Worked example

Two speakers produce loud and quiet spots in a room.

Common mistake

Energy isn't destroyed; it's redistributed.

Intermediate

Path difference and phase

Phase difference equals 2π times path difference over λ.

Key relationΔφ = 2πΔx/λ

A path difference Δx corresponds to a phase difference 2πΔx/λ. Constructive when Δx = nλ, destructive when Δx = (n + ½)λ.

Worked example

Δx = 1.5λ gives destructive.

Common mistake

Include any phase change on reflection.

Intermediate

Beats

Two close frequencies produce loud–soft pulsing.

Key relationf_b = ∣f₁−f₂∣

Two waves of slightly different frequencies produce amplitude variations at the beat frequency f_b = |f₁ − f₂|. Musicians use beats to tune instruments.

Worked example

Tuning a guitar string against a reference: beats slow down as they match.

Common mistake

The perceived pitch is the average frequency.

Intermediate

Standing waves

Interference of opposite waves gives fixed nodes and antinodes.

Key relationstanding

Two identical waves travelling in opposite directions form a pattern that doesn't move, with fixed nodes (zero amplitude) and antinodes (maximum). y = 2A sin(kx) cos(ωt).

Worked example

A guitar string pinned at both ends.

Common mistake

Standing waves don't transport energy along.

Intermediate

Nodes and antinodes

Points of no motion and maximum motion.

Nodes are points that never move; antinodes oscillate with the maximum amplitude. Adjacent nodes are λ/2 apart.

Worked example

A string of 1 m vibrating in the third harmonic has 4 nodes.

Common mistake

A closed end is a displacement node, an open end an antinode (for air).

Intermediate

Modes of a stretched string

Allowed frequencies are integer multiples of the fundamental.

Key relationf_n = nv/2L

A string fixed at both ends supports wavelengths λ = 2L/n and frequencies fₙ = n v/2L = n f₁. The fundamental is the lowest mode.

Worked example

L = 0.65 m, v = 400 m/s: f₁ = 308 Hz.

Common mistake

Only integer multiples are allowed.

Intermediate

Harmonics and overtones

Higher modes of vibration give timbre.

The nth harmonic is n times the fundamental; the first overtone is the second harmonic. The mix of harmonics gives an instrument its timbre.

Worked example

A violin and a flute playing 440 Hz sound different.

Common mistake

"Overtone" counting starts from the second harmonic.

Intermediate

Open organ pipe

Open at both ends supports all harmonics.

Key relationf_n = nv/2L

Open at both ends: antinodes at both ends, fₙ = nv/2L with all harmonics (n = 1, 2, 3...).

Worked example

L = 0.5 m, v = 343: f₁ = 343 Hz.

Common mistake

Similar to a string, as with flutes.

Intermediate

Closed organ pipe

Closed at one end supports only odd harmonics.

Key relationf_n = (2n−1)v/4L

Closed at one end: node at the closed end and antinode at the open end, so only odd harmonics: fₙ = (2n − 1)v/4L; fundamental v/4L is half of an open pipe's.

Worked example

L = 0.5 m: f₁ = 171 Hz, then 514, 857.

Common mistake

Sounds an octave lower than an open pipe of the same length.

Advanced

End correction

Antinode lies slightly beyond an open end.

Key relatione ≈ 0.6r

The antinode sits slightly beyond the open end of a pipe, so the effective length is L + 0.6r (one open end). It matters for accurate resonance calculations.

Worked example

r = 2 cm adds 1.2 cm.

Common mistake

Ignoring it gives errors for fat pipes.

Advanced

Kundt's tube

Measures sound speed using dust patterns.

A glass tube containing fine powder, with a vibrating rod at one end. The powder collects at nodes, whose spacing λ/2 gives the sound speed.

Worked example

Powder nodes 0.5 m apart at 340 Hz give λ = 1.0 m, so v = 340 m/s.

Common mistake

Used in classic labs.

Intermediate

Resonance tube experiment

Finds speed of sound from resonance lengths.

Key relationv = 2f(l₂−l₁)

A tuning fork over a tube with adjustable water level. Resonance happens at successive lengths l₁ and l₂, and v = 2f(l₂ − l₁), which eliminates end correction.

Worked example

f = 512 Hz, l₂ − l₁ = 33.5 cm: v = 343 m/s.

Common mistake

Use the difference of two resonances.

Beginner

Sound as pressure waves

Compressions and rarefactions in air.

Sound is a longitudinal pressure variation travelling through a medium. A loudspeaker or vocal cord compresses and rarefies the air layer by layer.

Worked example

Normal conversation: pressure variations ~0.02 Pa out of 10⁵ Pa.

Common mistake

Air itself isn't carried across the room.

Beginner

Pitch and frequency

Higher frequency means higher pitch.

Pitch is the perception of frequency: higher frequency, higher pitch. Doubling the frequency raises the pitch by an octave.

Worked example

Middle A is 440 Hz.

Common mistake

Pitch is subjective, but closely follows frequency.

Beginner

Loudness and intensity

Perceived loudness relates to intensity logarithmically.

Loudness is perceived and roughly logarithmic in intensity; physical intensity I = P/A in W/m². The ear can handle 10⁻¹² to 1 W/m².

Worked example

A jet engine at 100 m has I ≈ 1 W/m².

Common mistake

Loudness also depends on frequency.

Intermediate

Decibel scale

Logarithmic unit of sound level.

Key relationβ = 10 log(I/I₀)

Sound level β = 10 log₁₀(I/I₀) with I₀ = 10⁻¹² W/m². Each 10 dB is a tenfold intensity increase; 3 dB roughly doubles it.

Worked example

A conversation of 60 dB is 10⁶ I₀.

Common mistake

Two 60 dB sources give 63 dB, not 120.

Intermediate

Intensity and distance

Spherical spreading follows the inverse-square law.

Key relationI = P/4πr²

From a point source I = P/4πr², so doubling the distance gives one quarter of the intensity (6 dB drop).

Worked example

A 100 W source at 10 m: 0.08 W/m².

Common mistake

Valid in open space without reflections.

Beginner

Timbre

The tone quality that distinguishes instruments.

Timbre (tone quality) is determined by which harmonics are present and how they change over time, letting you tell a piano from a violin.

Worked example

Same note with different harmonic mixes.

Common mistake

Not the same as pitch or loudness.

Beginner

Range of hearing

Humans hear about 20 Hz to 20 kHz.

Humans hear about 20 Hz to 20 kHz, most sensitive at 2–5 kHz; the upper limit falls with age. The threshold of pain is about 120 dB.

Worked example

Dogs hear up to 45 kHz.

Common mistake

Hearing damage begins at long exposure over 85 dB.

Intermediate

Ultrasound

Sound above 20 kHz used in imaging and cleaning.

Sound above 20 kHz used in medical imaging (1–15 MHz), cleaning, and sonar. Imaging uses echoes of short pulses from tissue boundaries.

Worked example

Foetal scanning.

Common mistake

Higher frequency gives finer detail but less penetration.

Intermediate

Infrasound

Sound below 20 Hz from earthquakes and large animals.

Frequencies below 20 Hz are produced by earthquakes, volcanoes, whales and elephants. They travel long distances with little absorption.

Worked example

Elephants communicate over kilometres.

Common mistake

Can cause unease at high levels.

Beginner

Sonar and echoes

Distance from the echo time.

Key relationd = vt/2

Distance from the echo delay: d = vt/2. Bats, submarines and depth sounders use it.

Worked example

A 0.4 s delay in water (1500 m/s) means 300 m.

Common mistake

Divide by 2 because of the round trip.

Intermediate

Doppler effect

Frequency shifts when source or observer moves.

Key relationf' = f(v±v_o)/(v∓v_s)

If source and observer move relative to each other, the observed frequency changes: f' = f(v ± v_o)/(v ∓ v_s). Approaching raises, receding lowers pitch.

Worked example

An ambulance siren dropping in pitch as it passes.

Common mistake

Signs: approach means higher frequency.

Intermediate

Doppler with moving source

Approaching sources raise pitch; receding ones lower it.

Key relationdoppler

For a source moving at v_s toward a stationary observer f' = fv/(v − v_s); wavefronts bunch up ahead and spread behind.

Worked example

A 500 Hz siren at 30 m/s approaching: 548 Hz; receding: 460 Hz.

Common mistake

The medium speed v is unchanged.

Intermediate

Doppler with moving observer

An approaching observer detects more wavefronts per second.

For an observer moving at v_o toward the source f' = f(v + v_o)/v. The observer meets more wavefronts per second.

Worked example

Driving at 20 m/s toward a 500 Hz source: 529 Hz.

Common mistake

Not symmetrical with a moving source at high speed.

Advanced

Doppler in light

Redshift and blueshift reveal motion of stars.

Key relationΔλ/λ = v/c

For light, redshift z = Δλ/λ ≈ v/c gives the radial speed of stars and galaxies. Expansion of the universe shows up as cosmological redshift; police radar uses the Doppler shift too.

Worked example

A galaxy with z = 0.1 recedes at about 30,000 km/s.

Common mistake

At relativistic speeds use the exact formula.

Advanced

Shock waves and sonic boom

Source faster than sound generates a cone of compression.

Key relationsinθ = v/v_s

A source moving faster than the wave speed outruns its waves, forming a cone with half angle sinθ = v/v_s. The sudden pressure jump is heard as a boom.

Worked example

Concorde at Mach 2: θ = 30°.

Common mistake

The boom is continuous along the track, not a one-time event.

Intermediate

Mach number

Speed relative to the speed of sound.

Key relationM = v/c_s

M = v/c_s measures speed relative to the sound speed. M < 1 subsonic, M > 1 supersonic, M > 5 hypersonic.

Worked example

A jet at 680 m/s at altitude (c_s = 295 m/s): M ≈ 2.3.

Common mistake

Sound speed varies with temperature, so M for the same speed changes.

Advanced

Energy transport in waves

Power carried by a string wave.

Key relationP = ½μω²A²v

A sinusoidal wave on a string carries average power P = ½μω²A²v: proportional to the square of both amplitude and frequency.

Worked example

Doubling frequency at fixed amplitude gives four times the power.

Common mistake

Energy flows in the direction of propagation.

Advanced

Wave packets and group velocity

Envelope of a combined wave moves at group velocity.

Key relationv_g = dω/dk

A pulse is a superposition of many frequencies. The envelope moves at the group velocity v_g = dω/dk, while the crests travel at the phase velocity ω/k. Energy and information move at v_g.

Worked example

Deep water waves: v_g is half v_p.

Common mistake

In dispersive media they differ.

Advanced

Dispersion

Different wavelengths travel at different speeds.

In a dispersive medium wave speed depends on frequency, so a pulse spreads. Glass prisms disperse white light; optical fibres broaden pulses.

Worked example

Rainbow colours.

Common mistake

Non-dispersive: sound in air at normal frequencies.

Intermediate

Waves in water

Surface waves combine transverse and longitudinal motion.

Surface waves are a combination of transverse and longitudinal motion — particles move in circles. Deep water waves have v = √(gλ/2π); shallow water waves √(gh).

Worked example

A buoy bobs in circles as swells pass.

Common mistake

Wave speed depends on wavelength in deep water.

Advanced

Tsunami and shallow water waves

Speed depends on depth.

Key relationv = √(gh)

In the deep ocean a tsunami has a small amplitude (< 1 m) and huge wavelength (200 km); its speed √(gh) ≈ 200 m/s at 4 km depth. Near the shore the wave slows and the amplitude grows enormously.

Worked example

The 2004 Indian Ocean tsunami crossed the ocean in hours.

Common mistake

Energy conservation: slower speed gives a higher wave.

Intermediate

Seismic waves

P waves are longitudinal; S waves are transverse and slower.

Earthquakes generate P-waves (longitudinal, fast, through liquids too) and S-waves (transverse, slower, solids only), plus surface waves. The delay between them gives the distance to the epicentre; S-wave shadow showed the liquid core.

Worked example

The P-S gap of 30 s ≈ 250 km.

Common mistake

Surface waves cause the most damage.

Advanced

Acoustics of rooms

Reverberation time depends on volume and absorption.

Key relationT = 0.161V/A

Reverberation time T = 0.161V/A (Sabine) depends on room volume and total absorption. Concert halls aim for 1.5–2.2 s.

Worked example

V = 3000 m³ and absorption 1500 m²: T = 0.161·3000/1500 ≈ 0.32 s.

Common mistake

Too much reverberation blurs speech.

Intermediate

Musical scales and intervals

Notes relate by simple frequency ratios.

Notes sound consonant when their frequencies are ratios of small integers: octave 2:1, fifth 3:2, fourth 4:3. Equal temperament divides the octave into 12 steps of 2^{1/12}.

Worked example

A440 and E at 660 Hz are a perfect fifth.

Common mistake

Equal temperament slightly detunes pure intervals.

Advanced

Fourier analysis

Any periodic wave is a sum of sines.

Any periodic wave can be written as a sum of sinusoids at multiples of the fundamental. A square wave contains odd harmonics with amplitudes 1/n.

Worked example

Audio spectrum analysers show these components.

Common mistake

Non-periodic signals need the Fourier transform.

Advanced

Wave equation (PDE)

The partial differential equation governing waves.

Key relation∂²y/∂x² = (1/v²)∂²y/∂t²

∂²y/∂x² = (1/v²)∂²y/∂t² governs non-dispersive waves. Any function f(x − vt) or g(x + vt) solves it.

Worked example

Plucked string: d'Alembert's solution.

Common mistake

It's linear, so superposition holds.

Advanced

Solitons

Waves that keep their shape while travelling.

Solitary waves that keep their shape and speed over long distances because nonlinearity balances dispersion; they pass through each other unchanged. Examples: tidal bores, optical fibre pulses.

Worked example

John Scott Russell followed a canal wave for miles in 1834.

Common mistake

Found in the KdV equation.

Intermediate

Noise cancellation

Anti-phase sound cancels unwanted noise.

A microphone samples noise and a speaker emits the same waveform with opposite phase, so the sum cancels. It works best for steady low-frequency sound such as engine hum.

Worked example

Aeroplane headphones.

Common mistake

Needs precise phase matching.

Frequently asked questions

Why does sound travel faster in solids than in air?

Sound is a mechanical wave, so it needs a medium. Solids are generally stiffer and more elastic than gases, so their particles pass on vibrations more quickly.

What is the difference between transverse and longitudinal waves?

In a transverse wave, particles oscillate at right angles to the direction the wave travels, as on a string. In a longitudinal wave, particles oscillate along the direction of travel, as in sound in air.

Why do you hear the pitch of a passing siren change?

The Doppler effect shifts the frequency. The pitch is higher as the source approaches, because the waves are compressed, and lower as it moves away.

Do waves transport matter?

Not usually. In most waves, particles oscillate around fixed positions and only the energy and the pattern of the disturbance move forward.

Why is sound louder when waves add in phase?

In phase, the displacements add, which increases the amplitude and therefore the intensity of the sound. Out of phase, they cancel and the sound is quieter.

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