Waves & energy transfer through media

What you will learn

describe how energy is transferred through media by conduction, convection, and radiation,
distinguish transverse and longitudinal waves and identify examples of each,
use the wave equation $v = f\lambda$ to calculate speed, frequency, or wavelength,
order the electromagnetic spectrum and give uses of each region,
describe reflection, refraction, diffraction, and why sound travels faster in solids than in gases.

Worked example 0 Real-world example: radio station tuning

A local radio station broadcasts at $101.1$ MHz. Radio waves travel at the speed of light, $c = 3.0 \times 10^{8}$ m/s. What is the wavelength?

Convert frequency: $f = 101.1 \times 10^{6}$ Hz.
Wave equation: $\lambda = \dfrac{v}{f} = \dfrac{3.0 \times 10^{8}}{1.011 \times 10^{8}} \approx 2.97$ m.
That is why an FM antenna is a metre or two long — it needs to match the wavelength.

Key idea: every wave technology is designed around the wavelength and frequency of the signal it uses.

1. Energy transfer: three modes

Mode	How energy moves	Needs matter?	Example
Conduction	through direct contact — particles vibrate and pass energy to neighbours	yes (best in solids)	metal spoon heating in soup
Convection	in fluids: hot fluid rises, cool fluid sinks, carrying energy with it	yes (fluids only)	boiling water, sea breezes
Radiation	by electromagnetic waves; no medium needed	no	sunlight reaching Earth

Waves of particles (e.g. moving air in sound) transfer kinetic energy; electromagnetic waves carry energy through the fields themselves.

2. Transverse vs longitudinal waves

Transverse wave: particles vibrate perpendicular to the direction of energy transfer. Examples: water ripples, light, EM waves, wave on a string.

Longitudinal wave: particles vibrate parallel to the direction of energy transfer — compressions and rarefactions. Example: sound, seismic P-waves.

Top: a transverse wave with crests and troughs. Bottom: a longitudinal wave with alternating compressions and rarefactions. Wavelength is one full cycle in each.

3. Wave properties and the wave equation

Every periodic wave has:

Amplitude $A$ — maximum displacement from equilibrium (loudness, brightness).
Wavelength $\lambda$ — distance between consecutive crests (or compressions).
Frequency $f$ — cycles per second (Hz).
Period $T$ — time for one cycle; $T = \dfrac{1}{f}$ .
Speed $v$ — how fast the wave pattern moves through the medium.

Wave relations

Wave equation

v = f \lambda

Units: $v$ in m/s, $f$ in Hz, $\lambda$ in m.

Period and frequency

T = \dfrac{1}{f}, \qquad f = \dfrac{1}{T}.

Worked example 1 Sound in air

A tuning fork vibrates at $440$ Hz. Sound speed in air is about $340$ m/s. Find the wavelength.

$\lambda = \dfrac{v}{f} = \dfrac{340}{440} \approx 0.77$ m.
About $77$ cm — roughly an arm’s length.

Worked example 2 Ocean wave speed

Ocean waves reach a beach with crests $12$ m apart every $8$ s. Find the speed.

Wavelength $\lambda = 12$ m; period $T = 8$ s; frequency $f = 1/8 = 0.125$ Hz.
$v = f\lambda = 0.125 \times 12 = 1.5$ m/s.

4. The electromagnetic (EM) spectrum

All EM waves travel at $c = 3.0 \times 10^{8}$ m/s in vacuum. They differ in frequency and wavelength.

Region	Typical wavelength	Example use
Radio	m to km	broadcasting, phones, Wi-Fi
Microwave	cm	microwave ovens, radar, satellites
Infrared	$\sim 10\,\mu m$	heat imaging, remote controls
Visible light	400-700 nm	vision, photography
Ultraviolet	$\sim 100$ nm	sterilisation, suntan (DNA damage)
X-ray	$\sim 0.1$ nm	medical imaging
Gamma	$< 0.01$ nm	cancer treatment, sterilising food

As wavelength decreases, frequency and energy per photon increase. UV, X-rays and gamma are “ionising” — they can knock electrons from atoms and damage DNA.

Worked example 3 Photon wavelength

Visible green light has frequency $5.6 \times 10^{14}$ Hz. Find the wavelength in nanometres.

$\lambda = \dfrac{c}{f} = \dfrac{3.0 \times 10^{8}}{5.6 \times 10^{14}} = 5.36 \times 10^{-7}$ m.
$= 536$ nm — green.

5. Wave behaviour: reflection, refraction, diffraction

Reflection: wave bounces off a barrier. Angle of incidence $=$ angle of reflection (mirrors, echoes).
Refraction: wave speed changes when it enters a new medium, causing direction to change (light bending as it enters water; a straw looking bent in a glass).
Diffraction: waves bend around obstacles or spread through gaps. Bigger effect when the gap is close to the wavelength (radio diffracts around hills; light barely diffracts around ordinary objects).

6. Sound in different media

Sound is a longitudinal wave that needs particles. Its speed depends on how tightly particles are bound.

Medium	Speed of sound (approx.)
Air (20 $^{\circ}\text{C}$ )	343 m/s
Water	1480 m/s
Steel	5000 m/s

Sound cannot travel through a vacuum — no particles to vibrate.

Practice: Year 9

Fluency

Types and properties

Classify as transverse or longitudinal: (a) light, (b) sound, (c) water surface waves, (d) wave on a rope.
Define amplitude, frequency, wavelength, and period.
State the wave equation in symbols and in words.
List the regions of the EM spectrum in order of increasing frequency.
State the three modes of energy transfer and which require matter.
Why can sound not travel through a vacuum?

Fluency

Wave equation calculations

A wave has frequency $50$ Hz and wavelength $4$ m. Find the speed.
Find the wavelength of a $2.4$ GHz Wi-Fi signal ( $c = 3 \times 10^{8}$ m/s).
A radio station broadcasts at wavelength $3$ m. What is the frequency?
A wave on a string travels at $12$ m/s with wavelength $0.4$ m. Find the frequency and period.
Sound travels at $1480$ m/s in water. A whale’s call has frequency $20$ Hz. Wavelength?
A red laser has wavelength $650$ nm. Find the frequency.

Reasoning

Apply the ideas

Explain why a low bass note reaches your ear from around a corner but a high treble note does not.
Give one everyday use of each of: microwaves, infrared, X-rays.
Why is UV light harmful to skin while visible light is not?
Describe what happens to the speed, wavelength, and frequency of a light wave when it passes from air into water.
A student claims sound travels faster in air than in steel because steel is heavier. Evaluate this claim.

Problem solving

Wave scenarios

You see lightning and hear thunder 6 seconds later. Estimate how far away the strike was (sound speed $343$ m/s).
A bat’s ultrasonic call is $40$ kHz and travels at $343$ m/s. Calculate the wavelength, and suggest why bats use such a high frequency for echolocation.
A swimming-pool lane rope is shaken to produce a wave at $2$ Hz with wavelength $1.5$ m. Find the wave speed. How long to travel the $25$ m pool?
An AM radio station at $1.0 \times 10^{6}$ Hz diffracts well around hills; an FM station at $1.0 \times 10^{8}$ Hz does not. Explain using wavelength.

Challenge

Reasoning

Harder reasoning

Earthquakes produce both P-waves (longitudinal) and S-waves (transverse). S-waves cannot travel through the outer core. Explain how this is evidence that Earth’s outer core is liquid.
Ultrasound imaging sends pulses into the body at $\sim 3$ MHz and detects echoes. Sound speed in soft tissue is about $1540$ m/s. Calculate the wavelength and comment on the smallest feature that can reasonably be resolved.
Two loudspeakers emit the same frequency sound waves. At certain points listeners hear a loud sound; at others almost silence. Explain using superposition (constructive and destructive interference).
A radar station sends a pulse at $10$ GHz; the echo returns $1.2 \times 10^{-4}$ s later. Find the distance to the target. State the two wave ideas you used (speed of EM waves, and distance-time).

Answers

Answer key

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Year 9 answers

Fluency

Types and properties

(a) transverse, (b) longitudinal, (c) mostly transverse (water surface moves up/down), (d) transverse.
Amplitude: max displacement from equilibrium. Frequency: cycles per second (Hz). Wavelength: distance between consecutive crests (m). Period: time for one full cycle (s).
$v = f \lambda$ : the speed of a wave equals its frequency multiplied by its wavelength.
Radio -> microwave -> infrared -> visible -> ultraviolet -> X-ray -> gamma.
Conduction (needs matter, best in solids), convection (needs fluid), radiation (no medium needed).
Sound is a mechanical wave needing particles to vibrate. Vacuum has (almost) no particles, so no vibration can propagate.

Fluency

Wave equation calculations

$v = 50 \times 4 = 200$ m/s.
$\lambda = \dfrac{3 \times 10^{8}}{2.4 \times 10^{9}} = 0.125$ m $= 12.5$ cm.
$f = \dfrac{3 \times 10^{8}}{3} = 1 \times 10^{8}$ Hz = 100 MHz.
$f = v/\lambda = 12/0.4 = 30$ Hz. Period $T = 1/30 \approx 0.033$ s.
$\lambda = 1480/20 = 74$ m.
$f = 3 \times 10^{8} / (650 \times 10^{-9}) \approx 4.6 \times 10^{14}$ Hz.

Reasoning

Apply the ideas

Low frequency = long wavelength. Diffraction is strongest when wavelength is similar to or larger than the obstacle/gap. Bass has wavelengths of metres, comparable to door widths, so it diffracts around corners. Treble has wavelengths of centimetres and diffracts much less.
Microwaves: mobile phone signals, microwave ovens, radar. Infrared: TV remotes, thermal imaging. X-rays: medical imaging, airport security.
UV photons carry more energy per photon and can ionise atoms or damage DNA directly, causing sunburn and skin cancer. Visible photons have less energy and are mostly absorbed harmlessly.
Speed decreases (light is slower in water); wavelength decreases proportionally; frequency stays the same (set by the source).
Incorrect. Sound travels faster in steel (rigid bonds transmit vibration quickly) than in air; “heavier” is the wrong factor — stiffness matters more than density.

Problem solving

Wave scenarios

Distance $= 343 \times 6 \approx 2060$ m or about $2$ km.
$\lambda = 343 / (40\,000) = 0.00858$ m $\approx 8.6$ mm. Short wavelength lets bats resolve small objects (insects) and focus the beam more tightly.
$v = f\lambda = 2 \times 1.5 = 3$ m/s. Time $= 25 / 3 \approx 8.3$ s.
AM wavelength $= 3 \times 10^{8} / 10^{6} = 300$ m, comparable to hills — diffracts well. FM wavelength $= 3 \times 10^{8} / 10^{8} = 3$ m, much smaller than hills — little diffraction, so FM needs line of sight.

Reasoning

Challenge

Transverse waves cannot propagate through fluids because fluids have no shear rigidity. Observations show a “shadow zone” where S-waves do not arrive, meaning they pass through a liquid layer. P-waves still arrive (longitudinal, slower in liquid). This is evidence that the outer core is liquid.
$\lambda = 1540 / (3 \times 10^{6}) \approx 5.1 \times 10^{-4}$ m $= 0.51$ mm. Smallest resolvable feature is roughly the wavelength, so about $0.5$ mm — enough to see an unborn baby’s bones and organs.
Where waves from the two speakers arrive in phase (crests coincide), amplitudes add — loud (constructive interference). Where they arrive out of phase (crest meets trough), they cancel — quiet (destructive interference). The pattern depends on the path-length difference relative to wavelength.
Pulse travels to target and back, total distance $= c \times t = 3 \times 10^{8} \times 1.2 \times 10^{-4} = 3.6 \times 10^{4}$ m. Distance to target $= 1.8 \times 10^{4}$ m $= 18$ km. Ideas used: EM waves travel at $c$ ; distance = speed times time.

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