Waves & Sound — Topic Summary

1 Wave Basics

A wave is a disturbance that transfers energy from one place to another without transferring matter. The medium vibrates, but it does not travel with the wave.

Mechanical vs Electromagnetic

Type	Requires a medium?	Examples
Mechanical	Yes — needs matter to propagate	Sound, water waves, seismic waves
Electromagnetic	No — can travel through a vacuum	Light, radio waves, X-rays, microwaves

Transverse vs Longitudinal

Type	Particle motion	Examples
Transverse	Perpendicular to direction of wave travel	Light, waves on a string, water surface
Longitudinal	Parallel to direction of wave travel	Sound, compression waves in a spring

💡Wavefronts are surfaces connecting all points of the wave that are in the same phase (e.g. all crests). For a point source, wavefronts are spherical. Far from the source they appear flat — these are plane waves.

2 Wave Properties

Property	Symbol	Definition	Unit
Amplitude	A	Maximum displacement from equilibrium	m (or same as displacement)
Wavelength	λ	Distance for one complete cycle (crest to crest)	m
Period	T	Time for one complete cycle	s
Frequency	f	Number of complete cycles per second	Hz (= 1/s)
Wave speed	v	Speed at which the wave pattern moves	m/s

Key Relationships

Period & Frequency

T = 1 / f (and f = 1 / T)

Wave speed

v = f × λ

⚠️Wave speed is set by the medium, not the source. Changing the source frequency changes λ (wavelength), but v stays constant within the same medium.

3 Sound Waves

Sound is a longitudinal mechanical wave. As the wave travels, it creates alternating regions of high pressure (compressions) and low pressure (rarefactions).

Speed of Sound in Different Media

Sound travels faster in denser, stiffer materials. The order from fastest to slowest:

🔊Solids > Liquids > Gases — sound molecules are closer together in solids, so the pressure disturbance propagates more quickly.

Medium	Approximate speed
Air (20 °C)	≈ 340 m/s
Water	≈ 1500 m/s
Steel	≈ 5000 m/s

The speed of sound in air increases with temperature — warmer air molecules move faster.

4 Simple Harmonic Motion & the Pendulum

Simple Harmonic Motion (SHM) occurs when an object experiences a restoring force proportional to its displacement and directed back toward equilibrium. The result is sinusoidal oscillation.

Restoring force (SHM)

F = −kx (k = spring constant, x = displacement)

The Pendulum

A simple pendulum (mass on a string, small angle) undergoes SHM. Its period depends only on the string length and gravity — not on mass or amplitude (for small swings).

Pendulum period

T = 2π √(L / g)

Spring-mass period

T = 2π √(m / k)

💡Period is independent of amplitude (for small angles). This is why pendulum clocks are reliable — swinging wider or narrower doesn't change the beat.

Energy in SHM

At the equilibrium position: maximum speed, zero potential energy
At the extremes (amplitude): zero speed, maximum potential energy
Total mechanical energy is conserved throughout the motion

5 Interference

When two waves overlap in the same medium, their displacements add together (superposition principle).

Constructive Interference

When waves arrive in phase (crests meet crests, troughs meet troughs), their amplitudes add. The result is a larger amplitude.

Constructive (path diff.)

path difference = nλ (n = 0, 1, 2, 3, …)

Destructive Interference

When waves arrive out of phase (crest meets trough), they cancel. The result is a smaller or zero amplitude.

Destructive (path diff.)

path difference = (n + ½)λ (n = 0, 1, 2, …)

💡Interference is what makes noise-cancelling headphones work — they generate a wave that destructively interferes with ambient noise.

6 The Doppler Effect

When a source of waves and an observer are moving relative to each other, the observed frequency differs from the emitted frequency. The wave speed through the medium does not change.

Source moving toward observer: wavefronts compressed → higher observed frequency (higher pitch)
Source moving away from observer: wavefronts stretched → lower observed frequency (lower pitch)

Doppler formula

f_obs = f_s × (v ± v_obs) / (v ∓ v_s)

v = wave speed in medium · v_obs = observer speed · v_s = source speed. Use + on top when observer approaches, − on bottom when source approaches.

⚠️The Doppler effect changes observed frequency, NOT the speed of the wave. Sound always travels at ~340 m/s in air regardless of source motion.

7 Standing Waves

When a wave reflects back on itself and the two travelling waves overlap, a standing wave forms. The pattern appears stationary — it does not travel.

Nodes: points of zero displacement (destructive interference always occurs here)
Antinodes: points of maximum displacement (constructive interference)

Harmonics on a String / Open Pipe

Both ends are antinodes (open pipe) or nodes (fixed string). All harmonics are present.

Harmonic frequencies

f_n = n × v / (2L) n = 1, 2, 3, …

Closed Pipe (one open end)

One node (closed end) and one antinode (open end). Only odd harmonics are present.

Harmonic frequencies

f_n = n × v / (4L) n = 1, 3, 5, … (odd only)

System	Harmonics	Formula
String (fixed both ends)	All (1, 2, 3, …)	f_n = nv / (2L)
Open pipe (both ends open)	All (1, 2, 3, …)	f_n = nv / (2L)
Closed pipe (one end closed)	Odd only (1, 3, 5, …)	f_n = nv / (4L)

8 Sound Intensity & Decibels

Sound intensity (I) is the power transmitted per unit area (W/m²). As sound spreads outward from a point source, the intensity decreases with the square of the distance.

Inverse square law

I ∝ 1 / r² (double distance → intensity drops to 1/4)

Intensity level (dB)

L = 10 × log(I / I₀) dB

💡The threshold of hearing: I₀ = 10⁻¹² W/m². Every increase of 10 dB represents a tenfold increase in intensity.

Sound	Approximate level (dB)
Threshold of hearing	0 dB
Whisper	30 dB
Normal conversation	60 dB
Rock concert	110 dB
Jet engine	140 dB

9 Common Mistakes to Avoid

Mistake	What to do instead
Confusing T and f	T = 1/f. If f is large (many cycles per second), T is small (short time per cycle).
Forgetting v = fλ applies to all waves	This relationship holds for light, sound, water waves — everything. v is set by the medium.
Thinking the Doppler effect changes wave speed	Doppler only changes the observed frequency. The wave still travels at v through the medium.
Pendulum period depends on mass or amplitude	T = 2π√(L/g) — only L and g matter. Mass and amplitude (small angles) have no effect.
Assuming all pipes produce all harmonics	Closed pipes (one sealed end) produce only odd harmonics. Open pipes and strings produce all.
Constructive interference always doubles amplitude	Only when two waves have equal amplitude and are perfectly in phase. Partial constructive interference gives a partial increase.