Wave model

Lesson C.2 Wave Model

Lesson Notes, Worked Examples and Exam Practice

These notes introduce waves as energy transfers, compare transverse and longitudinal waves, develop the key wave quantities (amplitude, wavelength, period, frequency, wave speed), and compare sound waves (mechanical) with electromagnetic (EM) waves.

Syllabus map

C.2.1 Transverse and longitudinal waves
C.2.2 Describing waves
C.2.3 Sound waves and electromagnetic waves

IB Formula Reference and Notation

Wave equation: $v=f\lambda$ .
Frequency and period: $f=\frac{1}{T}$ .
Useful equivalent form: $v=\frac{\lambda}{T}$ .

$v$ is wave speed in $\mathrm{m\,s^{-1}}$ ; $f$ is frequency in hertz; $\lambda$ is wavelength in metres; and $T$ is period in seconds.

To match the notation used in C.1, $x$ denotes signed particle displacement from equilibrium and $x_0$ denotes displacement amplitude. Distance along the medium is a separate quantity. In crest-travel calculations, $d$ is the distance travelled by a crest.

C.2.1 Transverse and Longitudinal Waves

Learning objectives

Define a wave as an energy transfer without net transport of matter.
Describe the difference between transverse and longitudinal waves.
Identify examples of each type.

What is a wave?

Core definition

A wave is a disturbance that transfers energy from one place to another through space or through a medium, without transferring matter overall.

Energy transfer without matter transport

In the ideal wave model, particles oscillate locally while the disturbance travels. Surface water waves involve both vertical and horizontal particle motion, so they are not purely transverse. Any bulk flow or drift is distinct from the basic wave motion.
In sound, air particles oscillate back and forth about their local equilibrium positions while the sound disturbance carries energy through the air.
This is why we say waves transfer energy (and information), not matter.

Transverse waves

Definition

A transverse wave is a wave in which the oscillations are perpendicular to the direction of energy transfer.

Transverse wave showing oscillations perpendicular to the direction of travel

Key features:

A transverse displacement diagram has crests (maximum positive displacement) and troughs (maximum negative displacement).
Examples include waves on a string, electromagnetic waves and seismic S-waves. Surface water waves have a transverse component, together with a longitudinal component.

Longitudinal waves

Definition

A longitudinal wave is a wave in which the oscillations of the medium are parallel to the direction of energy transfer.

Longitudinal waves consist of alternating regions:

Compression: particles closer together (higher pressure/density)
Rarefaction: particles further apart (lower pressure/density)

Longitudinal sound wave showing compressions and rarefactions

Examples: sound waves in air, seismic P-waves, compression waves in a slinky.

Common misconception

Students often think “transverse = water” and “longitudinal = sound”. In reality, the type depends on the direction of oscillation relative to travel direction, not on the medium.

C.2.2 Describing Waves

Learning objectives

Use the terms amplitude, wavelength, period, and frequency correctly.
Relate wavelength, frequency and speed using $v=f\lambda$ .
Use $f=\frac{1}{T}$ and determine speed from wave graphs.

Amplitude and wavelength

Definitions

Amplitude, $x_0$ , is the maximum magnitude of particle displacement from equilibrium. For otherwise identical waves in the same medium, a greater amplitude means a greater rate of energy transfer. For electromagnetic waves, amplitude describes the maximum field variation rather than particle displacement.

Wavelength, $\lambda$ , is the distance between two consecutive points in the same phase (e.g. crest-to-crest, trough-to-trough, compression-to-compression).

Reading a diagram (self-study scaffold)

On a displacement graph, amplitude is the distance along the displacement axis from equilibrium to a crest or trough. It is not the peak-to-peak displacement, which is $2x_0$ .
Wavelength is a horizontal measurement between repeating features.
Two oscillating points are in the same phase when they are at the same stage of the cycle: they have the same displacement and are moving in the same direction.

Time period and frequency

The time period $T$ is the time for one complete oscillation at a fixed point. The frequency $f$ is the number of oscillations per second.

$\displaystyle f=\frac{1}{T}$

Wave speed

Wave speed $v$ describes how fast the disturbance travels through the medium. The key relationship is:

$\displaystyle v=f\lambda$

Since $f=\frac{1}{T}$ , another useful form is:

$\displaystyle v=\frac{\lambda}{T}$

Why the wave equation works: In one period, a crest advances by one wavelength. Therefore $v=\lambda/T$ . Since $1/T=f$ , this gives $v=f\lambda$ . Keep units consistent: convert centimetres to metres and milliseconds to seconds before substituting.

IB exam tip

Do not confuse:

Speed of the wave (energy transfer) with
Speed of particles in the medium (oscillation speed).

The wave can move to the right while particles move up and down (transverse) or back and forth (longitudinal).

Displacement–distance graphs

A displacement–distance graph is a “snapshot” of the wave at a single instant. It shows how displacement varies along the medium.

How to use a displacement–distance graph

Amplitude $x_0$ is the maximum magnitude of displacement from the equilibrium line.
Wavelength $\lambda$ is the horizontal distance between repeating points in phase.
It does not directly show time period unless you also know the speed or frequency.

Displacement–time graphs

A displacement–time graph follows one fixed position in the medium. It shows how the particle displacement changes as time passes. Read $x_0$ from the maximum magnitude of displacement and $T$ from the time between consecutive peaks, troughs or other equivalent points in the cycle.

The gradient of a displacement–time graph gives the particle velocity. It does not give the speed at which a crest travels along the medium. A displacement–distance graph alone cannot give the period, and a displacement–time graph alone cannot give the wavelength.

Compare the two graph types

Feature	Displacement–distance	Displacement–time
What is fixed?	One instant in time	One position in the medium
Horizontal axis	Distance along the medium	Time
Vertical axis	Particle displacement	Particle displacement
Read directly	Amplitude and wavelength	Amplitude and period

Longitudinal-wave reminder: A sinusoidal displacement graph can describe a longitudinal wave. The shape of the graph does not mean that the particles move perpendicular to the wave direction; the axes determine what the graph represents.

Finding the speed of a wave

There are two common IB approaches:

From definitions: measure $\lambda$ and use $v=f\lambda$ .
From motion: if you know how far a crest travels in time $t$ , then $v=\frac{d}{t}$ .

Worked example 1 — Wave speed from $f$ and $\lambda$

A wave has frequency $8.0\,\mathrm{Hz}$ and wavelength $0.75\,\mathrm{m}$ . Calculate the wave speed.

Show answer and explanation

$\displaystyle v=f\lambda=(8.0)(0.75)=6.0\,\mathrm{m\,s^{-1}}$ Answer: $v=6.0\,\mathrm{m\,s^{-1}}$

Worked example 2 — Finding frequency from a time period

A point on a string completes one oscillation in $0.40\,\mathrm{s}$ . Determine the frequency.

Show answer and explanation

$\displaystyle f=\frac{1}{T}=\frac{1}{0.40}=2.5\,\mathrm{Hz}$ Answer: $f=2.5\,\mathrm{Hz}$

Worked example 3 — Wave speed using $v=\frac{\lambda}{T}$

A wave has wavelength $1.20\,\mathrm{m}$ and time period $0.30\,\mathrm{s}$ . Calculate the wave speed.

Show answer and explanation

$\displaystyle v=\frac{\lambda}{T}=\frac{1.20}{0.30}=4.0\,\mathrm{m\,s^{-1}}$ Answer: $v=4.0\,\mathrm{m\,s^{-1}}$

C.2.3 Sound Waves and Electromagnetic Waves

Learning objectives

Describe the properties of sound waves.
Explain differences between mechanical and electromagnetic waves.
Describe key properties of electromagnetic waves.

Sound waves (mechanical waves)

Key properties of sound

Sound in air is a longitudinal mechanical wave.
It requires a medium (solid, liquid, or gas) to travel.
It consists of compressions and rarefactions in the medium.
Sound cannot travel through a vacuum.

What affects sound speed?

Sound speed depends on both elastic properties and density. It is usually greater in solids than in liquids and gases, but this is a general trend rather than an absolute rule.
For a gas of fixed composition, higher temperature increases sound speed. The sound speed is determined by the state and properties of the medium, rather than by the speed of the vibrating source.

Pitch, amplitude and sound speed

A higher frequency produces a higher perceived pitch. A larger amplitude generally gives a more intense sound and greater perceived loudness. Pitch and loudness describe different properties.

At a fixed sound speed, $\lambda=v/f$ , so a higher frequency gives a shorter wavelength. Increasing frequency does not automatically increase wave speed: for ordinary sound in a given medium under the same conditions, speed is approximately independent of frequency.

Mechanical waves vs electromagnetic waves

Feature	Mechanical waves (e.g. sound)	Electromagnetic waves (e.g. light)
Medium required?	Yes	No (can travel in vacuum)
Nature	Vibrations of particles in a medium	Oscillating electric and magnetic fields
Type	Can be transverse or longitudinal	Transverse
Speed depends on	Medium properties (density, elasticity, temperature)	In vacuum, $c$ ; in ordinary transparent materials, the wave speed depends on the material and frequency

Electromagnetic waves

Electromagnetic wave with perpendicular electric and magnetic fields

Key properties

EM waves are transverse.
They are made of oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel.
They travel through vacuum at speed $c\approx3.00\times10^8\,\mathrm{m\,s^{-1}}$ .
They transport energy and momentum.

The electromagnetic spectrum

In order of increasing frequency: radio waves → microwaves → infrared → visible light → ultraviolet → X-rays → gamma rays. Wavelength decreases in the same order.

All electromagnetic waves travel at the same speed in vacuum, $c\approx3.00\times10^8\,\mathrm{m\,s^{-1}}$ , so $c=f\lambda$ . Their different frequencies and wavelengths do not give them different vacuum speeds.

Self-study scaffold

If you are unsure about “fields”: an electric field exerts a force on a charge. A magnetic field can exert a force on a moving charge when its velocity has a component perpendicular to the field. In EM waves, these fields vary with time and carry energy through space.

IB-Style Exam Practice (C.2)

Instructions

Show clear working and include units.
Use $v=f\lambda$ and $f=\frac{1}{T}$ where appropriate.
Answer in 2–3 significant figures unless otherwise stated.

Question 1 (SL) — Classifying wave type

(a) State the difference between transverse and longitudinal waves.
(b) A sound wave travels through air. Identify whether it is transverse or longitudinal, and justify your answer using the terms compression and rarefaction.

Show answer and explanation

(a) In a transverse wave, oscillations are perpendicular to the direction of energy transfer. In a longitudinal wave, oscillations are parallel to the direction of energy transfer.
(b) Sound in air is longitudinal. Air particles oscillate back and forth parallel to the direction the wave travels, producing alternating compressions (high pressure) and rarefactions (low pressure).

Question 2 (SL) — Wave speed calculation

A wave has wavelength $0.50\,\mathrm{m}$ and frequency $12\,\mathrm{Hz}$ . Calculate its speed.

Show answer and explanation

$\displaystyle v=f\lambda=(12)(0.50)=6.0\,\mathrm{m\,s^{-1}}$ Answer: $v=6.0\,\mathrm{m\,s^{-1}}$

Question 3 (SL) — Using a time period

A point on a string oscillates with time period $0.20\,\mathrm{s}$ . The wavelength of the wave is $1.6\,\mathrm{m}$ . Determine the wave speed.

Show answer and explanation

$\displaystyle f=\frac{1}{T}=\frac{1}{0.20}=5.0\,\mathrm{Hz}$ $\displaystyle v=f\lambda=(5.0)(1.6)=8.0\,\mathrm{m\,s^{-1}}$ Answer: $v=8.0\,\mathrm{m\,s^{-1}}$

Question 4 (SL) — Mechanical vs electromagnetic

State two differences between mechanical waves and electromagnetic waves.

Show answer and explanation

Mechanical waves require a medium; electromagnetic waves can travel through vacuum.
Mechanical waves are vibrations of particles in a medium; electromagnetic waves are oscillations of electric and magnetic fields.

Question 5 (SL) — Speed from “crest travel” data

A crest of a water wave travels $3.6\,\mathrm{m}$ in $1.5\,\mathrm{s}$ . (a) Determine the wave speed. (b) If the wavelength is $0.60\,\mathrm{m}$ , determine the frequency.

Show answer and explanation

(a) $\displaystyle v=\frac{d}{t}=\frac{3.6}{1.5}=2.4\,\mathrm{m\,s^{-1}}$ (b) $\displaystyle f=\frac{v}{\lambda}=\frac{2.4}{0.60}=4.0\,\mathrm{Hz}$ Answers: $v=2.4\,\mathrm{m\,s^{-1}}$ , $f=4.0\,\mathrm{Hz}$

Key ideas from C.2

A wave transfers energy without net transfer of matter.
Transverse: oscillations perpendicular to travel; Longitudinal: oscillations parallel to travel.
Wave speed: $v=f\lambda$ and $v=\frac{\lambda}{T}$ .
Sound in air is longitudinal and requires a medium.
EM waves are transverse, require no medium, and travel at $c$ in vacuum.

Wonder, Truth and Hope

Memory verse — Psalm 104:24 (KJV):
“O LORD, how manifold are thy works! in wisdom hast thou made them all: the earth is full of thy riches.”

Creation: Discovering order in God’s world

Waves help us understand how sound reaches our ears and how light makes the world visible. Their patterns invite careful observation and thoughtful questions. Psalm 104:24 celebrates God’s wisdom in creation. As we study, we can respond with gratitude, curiosity and a desire to use our knowledge well.

The Fall: Learning with humility and honesty

Sin can affect our attitudes towards learning and towards other people. Pride may make us hide mistakes, and fear may tempt us to copy another person’s work. God calls us to truth. Admit what you do not yet understand, ask for help and treat classmates with patience. First John 1:9 reminds us that God offers forgiveness when we confess our sins.

Redemption: Receiving new life through Jesus Christ

Jesus Christ died for our sins and rose again so that we could be reconciled to God. His love is not earned through examination results. Turn to Him in repentance and faith, and receive Him as your personal God, Saviour and Lord. Ask Him to shape your character as well as your understanding.

Restoration: Growing in faith and serving others

Revelation 21:3–5 gives believers hope of life with God, where suffering and death will be overcome. Grow in your knowledge of Him by reading and studying the Bible, spending time in prayer and connecting with other believers in a local church. Use your learning to serve others, explain difficult ideas patiently and encourage those who struggle.

Reflection: How can your study of waves lead you to greater gratitude for God’s creation and greater care for the people around you?

Prayer: Lord Jesus, thank You for Your creation and Your saving love. Forgive my sins and help me trust You. Give me understanding, honesty and perseverance in my studies. Help me grow through Your Word, prayer and fellowship, and use what I learn to serve others. Amen.

IB Physics HL Book 2025; Papers 1 & 2 Practice 5 Sections65 Lessons54 Weeks

Curriculum

Wave model

Lesson C.2 Wave Model

Syllabus map

IB Formula Reference and Notation

C.2.1 Transverse and Longitudinal Waves

Learning objectives

What is a wave?

Core definition

Energy transfer without matter transport

Transverse waves

Definition

Longitudinal waves

Definition

Common misconception

C.2.2 Describing Waves

Learning objectives

Amplitude and wavelength

Definitions

Reading a diagram (self-study scaffold)

Time period and frequency

Wave speed

IB exam tip

Displacement–distance graphs

How to use a displacement–distance graph

Displacement–time graphs

Compare the two graph types

Finding the speed of a wave

Worked example 1 — Wave speed from and

Worked example 2 — Finding frequency from a time period

Worked example 3 — Wave speed using

C.2.3 Sound Waves and Electromagnetic Waves

Learning objectives

Sound waves (mechanical waves)

Key properties of sound

Pitch, amplitude and sound speed

Mechanical waves vs electromagnetic waves

Electromagnetic waves

Key properties

The electromagnetic spectrum

Self-study scaffold

IB-Style Exam Practice (C.2)

Instructions

Question 1 (SL) — Classifying wave type

Question 2 (SL) — Wave speed calculation

Question 3 (SL) — Using a time period

Question 4 (SL) — Mechanical vs electromagnetic

Question 5 (SL) — Speed from “crest travel” data

Key ideas from C.2

Wonder, Truth and Hope

Creation: Discovering order in God’s world

The Fall: Learning with humility and honesty

Redemption: Receiving new life through Jesus Christ

Restoration: Growing in faith and serving others

Related

Leave A Reply Cancel reply

Company

Links

Support

Recommend

alphy school publishers

Customise Consent Preferences

Log in with your account

Modal title

Worked example 1 — Wave speed from $f$ and $\lambda$

Worked example 3 — Wave speed using $v=\frac{\lambda}{T}$