DP Physics HL C.2 Wave Model

DP Physics HL C.2 Wave Model - Full notes

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DP Physics HL | C.2 Wave Model

DP PHYSICS · THEME C: WAVE BEHAVIOUR

C.2

Wave Model

Full notes for HL · IB-style

Contents

1	Travelling waves
2	Wave graphs
3	Sound waves
4	Electromagnetic waves
5	Waves on strings and in water
6	Energy, intensity and the inverse-square law
7	Exam technique
8	Data-based questions and further practice
9	Common misconceptions and links
10	Exam-style question with markscheme
11	Check your understanding
12	Summary and topic map

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and do every Quick check before moving to the next section.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
describe transverse and longitudinal travelling waves
define wave quantities and use v = fλ
interpret displacement–distance and displacement–time graphs, including phase difference
describe the nature and speed of sound and measure it
describe electromagnetic waves and the spectrum
use intensity and the inverse-square law

There is no additional HL content in this subtopic. SL and HL students study the same material.

Guiding questions

What are the similarities and differences between different types of mechanical waves? How can the wave model describe the transfer of energy as a result of local disturbances in a medium? What effect does a change in the medium have on a wave?

How this topic is examined

C.2 appears in Papers 1 and 2: wave equation calculations, interpreting wave graphs, properties of sound and EM waves, and intensity calculations.

Always check whether a graph's horizontal axis is distance or time.

Command term	What the examiner wants
State	a brief answer with no explanation
Outline	a brief account or summary
Describe	a detailed account without reasons
Explain	a detailed account including causes, reasons or mechanisms
Calculate	a numerical answer with working and correct units
Determine	obtain the only possible answer, often from a graph or data
Show that	obtain the given result, showing every step (give one more significant figure)
Estimate	an approximate value using reasonable assumptions
Sketch	a graph or diagram showing the general shape and key features
Deduce / Suggest	reach a conclusion from given information; propose a reasonable explanation

How to use these notes

Each section teaches one part of the syllabus in full. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Key point	the one idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
In the real world	where the physics shows up outside the classroom
Try this	something quick you can do yourself to test the idea
Worked example	full solutions with IB-style marks (M1 method, A1 answer, R1 reasoning)
Quick check	short questions to test yourself at the end of each section

1 Travelling waves

Energy transfer without transfer of matter.

Think about it

When a Mexican wave goes round a stadium, nobody runs round the stadium. What actually travels?

A travelling wave transfers energy and information from one place to another without any net transfer of matter. The particles of the medium oscillate about fixed positions; the pattern moves.

1.1 What is a wave?

A mechanical wave is a disturbance that travels through a medium (string, water, air, solid) as each particle passes energy to its neighbour. Electromagnetic waves need no medium. In all waves, energy travels; the medium does not.

Quick check

1. Give one piece of evidence that water waves do not carry water with them.

Answers: 1. a floating cork bobs up and down but stays in roughly the same place

1.2 Transverse and longitudinal waves

In a transverse wave, the particles oscillate perpendicular to the direction of energy transfer (waves on strings, water ripples, electromagnetic waves, S-waves). In a longitudinal wave, the particles oscillate parallel to the direction of energy transfer, producing compressions and rarefactions (sound, P-waves, slinky pulses).

Feature	Transverse	Longitudinal
particle oscillation	perpendicular to travel	parallel to travel
examples	light, waves on strings, S-waves	sound, P-waves, ultrasound
features	crests and troughs	compressions and rarefactions
can be polarised?	yes	no

Figure 1. Transverse and longitudinal waves (C = compression, R = rarefaction).

1.3 Pulses

A single disturbance sent along a rope or slinky is a pulse. It moves at the wave speed while each part of the rope only moves up and down as the pulse passes.

Figure 2. A pulse on a rope at two instants.

WORKED EXAMPLE 1 Pulse speed (standard)

Use the graph to find the speed of the pulse.

Solution

Moves 1.5 m in 0.10 sM1

v = 15 m s⁻¹A1

1.4 Mechanical and electromagnetic waves compared

Feature	Mechanical waves	Electromagnetic waves
medium	needed	not needed
speed	depends on the medium (m s⁻¹ to km s⁻¹)	3.00 × 10⁸ m s⁻¹ in vacuum
types	transverse or longitudinal	transverse only
what oscillates	particles of the medium	electric and magnetic fields

1.5 Wave quantities

Displacement x is the distance of a particle from its equilibrium position. Amplitude A is the maximum displacement. Wavelength λ is the distance between successive points in phase (crest to crest). Period T is the time for one oscillation; frequency f = 1/T. Wave speed v is the speed at which the energy (wave pattern) travels.

WORKED EXAMPLE 2 Reading values (basic)

A wave has 6 complete wavelengths in 1.5 m and its source makes 40 oscillations in 2.0 s. Find λ, f and T.

Solution

λ = 1.5/6 = 0.25 mA1

f = 40/2.0 = 20 Hz; T = 0.050 sA1

1.6 The wave equation

In one period the wave travels one wavelength, so v = λ/T = fλ. The wave speed depends on the medium; the frequency is set by the source.

WORKED EXAMPLE 3 Sound (standard)

A 256 Hz tuning fork produces sound of wavelength 1.33 m. Find the speed of sound.

Solution

v = 256 × 1.33M1

= 340 m s⁻¹A1

WORKED EXAMPLE 4 Radio (standard)

Find the wavelength of a 98.5 MHz FM radio signal.

Solution

λ = 3.00 × 10⁸/9.85 × 10⁷M1

= 3.05 mA1

Common mistake

Thinking that increasing the frequency increases the wave speed. In a given medium the speed is fixed, so a higher frequency gives a shorter wavelength.

1.7 Wavefronts and rays

A wavefront joins neighbouring points that are in phase (for example, all crests). A ray shows the direction of energy transfer and is perpendicular to the wavefronts. A point source produces circular (spherical) wavefronts; far from the source they become nearly plane.

Figure 3. Circular wavefronts from a point source and plane wavefronts, with rays.

2 Wave graphs

Two kinds of graph, two different meanings.

2.1 Displacement–distance graphs

A displacement–distance graph is a snapshot of the whole wave at one instant. The wavelength and amplitude can be read from it.

Figure 4. Displacement–distance graph of a wave at one instant.

WORKED EXAMPLE 5 Reading (standard)

From the graph, find the wavelength and amplitude. If f = 50 Hz, find v.

Solution

λ = 0.80 m; A = 2.0 cmA1

v = 50 × 0.80 = 40 m s⁻¹A1

2.2 Displacement–time graphs

A displacement–time graph shows how one particle moves over time. The period and amplitude can be read from it.

Figure 5. Displacement–time graph for one point on the wave.

Key point

Check the horizontal axis first: distance gives λ; time gives T.

WORKED EXAMPLE 6 Reading (standard)

From the graph, find the period and frequency.

Solution

T = 20 msA1

f = 1/0.020 = 50 HzA1

2.3 Motion of particles from a snapshot

To find which way a particle is moving, compare two snapshots a short time apart, or imagine shifting the waveform slightly in its direction of travel. Particles ahead of a crest are moving up (for a wave moving right).

Figure 6. The same wave at two instants: the wave has moved 0.10 m to the right.

WORKED EXAMPLE 7 Particle direction (exam level)

For the wave moving right in the graph, state the direction of motion of the particle at x = 0.40 m at t = 0.

Solution

At x = 0.40 m the displacement is zero; the later curve is aboveM1

Particle is moving upwardsA1

WORKED EXAMPLE 8 Speed from shift (exam level)

Find the wave speed from the two snapshots.

Solution

v = 0.10/2.5 × 10⁻³M1

= 40 m s⁻¹A1

2.4 Phase difference along a wave

Two points a distance Δx apart along a wave have a phase difference φ = 2πΔx/λ. Points one wavelength apart are in phase; points half a wavelength apart are in antiphase.

WORKED EXAMPLE 9 Phase difference (standard)

Find the phase difference between two points 0.20 m apart on a wave of wavelength 0.80 m.

Solution

φ = 2π × 0.20/0.80M1

= π/2 radA1

Quick check

1. What is the phase difference between points 1.2 m apart on the wave above?

Answers: 1. 3π rad, equivalent to π (antiphase)

2.5 Longitudinal waves on graphs

Longitudinal waves are drawn on the same displacement–distance axes, with positive displacement meaning displacement in the direction of travel. Compressions occur where displacements converge (particles on either side move towards the point); for sound, the pressure variation is a quarter wavelength out of step with displacement.

Figure 7. Pressure variation and particle displacement along a sound wave.

Quick check

1. Where is the particle displacement zero in a sound wave?

Answers: 1. at the centres of compressions and rarefactions

2.6 Particle positions in a longitudinal wave

Plotting where particles actually are shows compressions and rarefactions directly. Particles with positive displacement (moved right) and those to their right with negative displacement crowd together into a compression.

Figure 8. Equilibrium and displaced positions of particles in a longitudinal wave (λ = 2.0 m).

Quick check

1. Where along the wave are the compressions in the figure?

Answers: 1. at x = 1.0 m and 3.0 m, where displaced particles bunch together

3 Sound waves

Longitudinal pressure waves.

Think about it

Why can you hear a train coming by putting your ear to the rail before you hear it through the air?

3.1 Nature of sound

Sound is a longitudinal mechanical wave consisting of compressions and rarefactions in a medium. It cannot travel through a vacuum. Humans hear frequencies from about 20 Hz to 20 kHz; higher frequencies are ultrasound; lower are infrasound.

Quick check

1. Why can sound not travel through space?

Answers: 1. there are no particles to pass the vibration on

3.2 Speed of sound in different media

Sound travels faster in solids and liquids than in gases because the particles are more strongly coupled. In air its speed increases with temperature.

Medium	Speed of sound / m s⁻¹
air (0 °C)	331
air (20 °C)	343
water	1480
steel	5900
granite	6000

WORKED EXAMPLE 10 Echo (standard)

An echo from a cliff returns 1.6 s after a shout. Find the distance to the cliff (v = 340 m s⁻¹).

Solution

Total distance = 340 × 1.6 = 544 mM1

Distance = 272 mA1

WORKED EXAMPLE 11 Rail and air (exam level)

A hammer strikes a steel rail 1.0 km away. Find the time difference between the sound arriving through the rail and through the air.

Solution

Air: 1000/340 = 2.94 s; steel: 1000/5900 = 0.17 sM1

Difference = 2.8 sA1

3.3 Measuring the speed of sound

Methods include timing an echo, using two microphones a known distance apart connected to a fast timer, or finding resonance in a tube (C.3). Two-microphone method: v = distance between microphones/time delay.

WORKED EXAMPLE 12 Two microphones (standard)

Two microphones 1.50 m apart detect a clap with a time delay of 4.4 ms. Find the speed of sound.

Solution

v = 1.50/0.0044M1

= 341 m s⁻¹A1

Try this

Clap in front of a large wall from 50 m away and adjust your clapping rate until each clap coincides with the echo of the last. Time 20 claps to find v.

3.4 Hearing range

The human ear detects roughly 20 Hz to 20 kHz; the upper limit falls with age. Many animals hear outside this range: dogs up to about 45 kHz, bats above 100 kHz, elephants communicate with infrasound.

Figure 9. Ranges of sound frequency.

WORKED EXAMPLE 13 Wavelength range (standard)

Find the range of wavelengths humans can hear in air (v = 340 m s⁻¹).

Solution

20 Hz: 17 m; 20 kHz: 1.7 cmM1

1.7 cm to 17 mA1

3.5 Measuring frequency with an oscilloscope

An oscilloscope displays pd against time. The time base (time per division) gives the period from the number of divisions per cycle; the y-gain gives the amplitude.

Figure 10. An oscilloscope trace of a sound signal.

WORKED EXAMPLE 14 Trace (exam level)

In the trace, one cycle spans 2.5 divisions and the peak is 2 divisions above the centre line. Find the frequency and peak voltage.

Solution

T = 2.5 × 0.5 ms = 1.25 ms: f = 800 HzM1 A1

Peak = 2 × 2 mV = 4 mVA1

3.6 Pitch and loudness

Pitch is determined by frequency: higher frequency, higher pitch. Loudness is related to amplitude (and the energy carried): larger amplitude, louder sound. Intensity is proportional to amplitude squared.

Figure 11. Oscilloscope-style traces comparing pitch and loudness.

WORKED EXAMPLE 15 Oscilloscope (standard)

A trace shows 4 cycles across 10 divisions at 0.5 ms per division. Find the frequency.

Solution

4 cycles in 5.0 ms: T = 1.25 msM1

f = 800 HzA1

3.7 Ultrasound

Ultrasound (above 20 kHz) has short wavelengths, so it reflects from small boundaries. It is used in medical imaging, sonar, cleaning and non-destructive testing. The depth of a boundary is found from the echo time: d = vt/2.

WORKED EXAMPLE 16 Medical scan (exam level)

An ultrasound pulse (v = 1540 m s⁻¹ in tissue) returns from an organ boundary after 65 μs. Find the depth of the boundary.

Solution

d = 1540 × 65 × 10⁻⁶/2M1

= 5.0 cmA1

In the real world

Bats and dolphins use echolocation with ultrasound to navigate and hunt in the dark or in murky water.

4 Electromagnetic waves

The electromagnetic spectrum.

4.1 Nature of electromagnetic waves

Electromagnetic (EM) waves are transverse oscillations of electric and magnetic fields, perpendicular to each other and to the direction of travel. They need no medium and all travel at c = 3.00 × 10⁸ m s⁻¹ in a vacuum.

Quick check

1. What oscillates in an electromagnetic wave?

Answers: 1. electric and magnetic fields

4.2 Electric and magnetic fields in an EM wave

In an EM wave, the electric and magnetic fields oscillate in phase, perpendicular to each other and to the direction of travel. Accelerating charges produce EM waves; for example, oscillating electrons in a radio aerial emit radio waves at the same frequency.

Figure 12. Electric and magnetic field oscillations in an electromagnetic wave.

Quick check

1. What produces electromagnetic waves?

Answers: 1. accelerating (oscillating) electric charges

4.3 The electromagnetic spectrum

EM waves are grouped by wavelength. All share the same speed in vacuum but differ in wavelength, frequency, photon energy and how they interact with matter.

Region	Typical source	Uses
radio	oscillating currents in aerials	broadcasting, communication
microwave	magnetrons	cooking, mobile phones, radar
infrared	warm objects	remote controls, thermal imaging
visible (400–700 nm)	hot objects, LEDs	seeing, optical fibres
ultraviolet	very hot objects, mercury lamps	sterilising, fluorescence
X-rays	fast electrons hitting metal	medical imaging
gamma rays	nuclear decay	cancer treatment, sterilising

Figure 13. The electromagnetic spectrum with typical wavelengths.

4.4 Photon energies across the spectrum

Each region of the EM spectrum carries photons of energy E = hf (E.1). The table shows why only the high-frequency end is ionising.

Region	Typical frequency / Hz	Photon energy / eV	Ionising?
radio	10⁸	4 × 10⁻⁷	no
microwave	10¹⁰	4 × 10⁻⁵	no
infrared	10¹³	0.04	no
visible	5 × 10¹⁴	2	no
ultraviolet	10¹⁶	40	partly
X-ray	10¹⁸	4000	yes
gamma	10²⁰	4 × 10⁵	yes

WORKED EXAMPLE 17 Photon energy (standard)

Find the energy in eV of a 10¹⁶ Hz UV photon (h = 4.14 × 10⁻¹⁵ eV s).

Solution

E = hf = 4.14 × 10⁻¹⁵ × 10¹⁶M1

= 41 eVA1

4.5 Calculations across the spectrum

Use c = fλ to convert between frequency and wavelength in a vacuum (or air, to a good approximation).

WORKED EXAMPLE 18 Microwave oven (standard)

A microwave oven operates at 2.45 GHz. Find the wavelength.

Solution

λ = 3.00 × 10⁸/2.45 × 10⁹M1

= 12.2 cmA1

WORKED EXAMPLE 19 Visible light (standard)

Find the frequency of red light of wavelength 700 nm.

Solution

f = 3.00 × 10⁸/7.00 × 10⁻⁷M1

= 4.3 × 10¹⁴ HzA1

4.6 Light in media

Light travels more slowly in materials than in a vacuum. Its frequency stays the same, so its wavelength decreases: λmedium = λvacuum × (v/c).

WORKED EXAMPLE 20 Light in glass (exam level)

Light of wavelength 600 nm in air enters glass where its speed is 2.0 × 10⁸ m s⁻¹. Find its frequency and wavelength in the glass.

Solution

f = 3.0 × 10⁸/6.0 × 10⁻⁷ = 5.0 × 10¹⁴ Hz (unchanged)A1

λ = 2.0 × 10⁸/5.0 × 10¹⁴ = 400 nmA1

4.7 Communication with waves

Information is carried by modulating waves: varying their amplitude (AM), frequency (FM) or using digital pulses. Radio and microwaves carry signals through the air; infrared and visible light carry signals through optical fibres with very little loss.

Signal	Wave used	Typical frequency
AM radio	radio	about 1 MHz
FM radio	radio	about 100 MHz
mobile phones	microwave	0.8–4 GHz
WiFi	microwave	2.4 or 5 GHz
fibre-optic internet	infrared	about 2 × 10¹⁴ Hz

WORKED EXAMPLE 21 Fibre wavelength (standard)

Find the wavelength in a vacuum of 1.94 × 10¹⁴ Hz fibre-optic signals.

Solution

λ = 3.0 × 10⁸/1.94 × 10¹⁴M1

= 1.55 μm (infrared)A1

4.8 Dangers and safety

Higher-frequency EM radiation (UV, X-rays, gamma) has enough photon energy to ionise atoms and damage DNA, increasing cancer risk. Microwaves and infrared heat tissue. Exposure is limited by shielding, distance and time.

Quick check

1. Why is UV more harmful to skin than visible light?

Answers: 1. UV photons have higher energy and can ionise or damage DNA

5 Waves on strings and in water

Waves you can see.

5.1 Waves on strings

The speed of a transverse wave on a string depends on the tension and the mass per unit length: it is faster on a tight, light string. Plucking a guitar string sends waves along it; their reflections set up standing waves (C.3).

WORKED EXAMPLE 22 String wave (standard)

A pulse travels along a 4.0 m rope in 0.25 s. The hand shakes the rope at 3.0 Hz. Find the wavelength.

Solution

v = 4.0/0.25 = 16 m s⁻¹M1

λ = 16/3.0 = 5.3 mA1

5.2 Wave speed on a string: data

The speed of waves on a stretched string increases with tension. Experiments show v ∝ √T, so a graph of v² against tension is a straight line through the origin.

Figure 14. Square of wave speed against tension for a string.

WORKED EXAMPLE 23 Mass per unit length (exam level)

For a string, v² = T/μ. Use the gradient to find μ, the mass per unit length.

Solution

Gradient = 200 m² s⁻² N⁻¹ = 1/μM1

μ = 5.0 × 10⁻³ kg m⁻¹A1

5.3 Wave speed and medium

Wave speed is set by the properties of the medium: tension and mass per unit length for strings, water depth for shallow water waves, stiffness and density for sound. When a wave passes into a new medium, its speed and wavelength change but its frequency does not.

Quick check

1. A wave passes from a light string into a heavier one. What happens to its speed, frequency and wavelength?

Answers: 1. speed decreases, frequency unchanged, wavelength decreases

5.4 The ripple tank

A ripple tank shows water waves clearly: a vibrating dipper makes waves, a light above projects their pattern onto a screen, and a stroboscope 'freezes' them so the wavelength can be measured.

Figure 15. A ripple tank with a straight dipper producing plane waves.

WORKED EXAMPLE 24 Ripple tank (exam level)

Ten wavelengths span 15 cm on the screen; the image is magnified 1.5 times. The dipper vibrates at 12 Hz. Find the wave speed.

Solution

Real λ = 1.5/1.5 = 1.0 cmM1

v = 12 × 0.010 = 0.12 m s⁻¹A1

Try this

Use a phone camera at a high frame rate to film ripples in a shallow tray. Measure the wavelength against a ruler and compare v = fλ with a timing measurement.

5.5 Water waves and depth

Water waves slow down as the water becomes shallower. The frequency stays the same, so the wavelength decreases. This is why waves bunch up and grow taller as they approach a beach, and why tsunamis, fast in deep ocean, rear up near the coast.

Figure 16. Water waves passing from deep to shallow water: same frequency, shorter wavelength.

WORKED EXAMPLE 25 Tsunami (exam level)

A tsunami travels at 200 m s⁻¹ in the open ocean with wavelength 150 km. Near the coast it slows to 15 m s⁻¹. Find its period and its wavelength near the coast.

Solution

T = 150 000/200 = 750 sA1

λ = 15 × 750 = 11 kmA1

5.6 Seismic waves

Earthquakes produce longitudinal P-waves and transverse S-waves. P-waves travel faster and can pass through liquids; S-waves cannot travel through liquids. The time delay between their arrivals at a seismometer gives the distance to the earthquake.

WORKED EXAMPLE 26 Seismometer (exam level)

P-waves travel at 8.0 km s⁻¹ and S-waves at 4.5 km s⁻¹. S-waves arrive 60 s after P-waves. Find the distance to the epicentre.

Solution

d/4.5 − d/8.0 = 60M1

d = 620 kmA1

In the real world

The absence of S-waves on the far side of Earth after earthquakes showed that Earth's outer core is liquid.

6 Energy, intensity and the inverse-square law

How waves spread out.

6.1 Intensity

Intensity I is the power per unit area perpendicular to the direction of travel (W m⁻²). Intensity is proportional to the square of the amplitude: I ∝ A².

WORKED EXAMPLE 27 Amplitude and intensity (standard)

The amplitude of a wave doubles. By what factor does its intensity change?

Solution

I ∝ A²M1

Increases 4 timesA1

6.2 The inverse-square law

From a point source radiating power P equally in all directions, the energy spreads over a sphere of area 4πr²: I = P/4πr². Doubling the distance reduces the intensity to a quarter (and the amplitude to a half).

Figure 17. Intensity from a 100 W point source against distance.

WORKED EXAMPLE 28 Lamp (standard)

A 60 W lamp radiates uniformly. Find the intensity 2.0 m away.

Solution

I = 60/(4π × 2.0²)M1

= 1.2 W m⁻²A1

WORKED EXAMPLE 29 Distance ratio (exam level)

The intensity of sound 3.0 m from a speaker is 0.020 W m⁻². Find the intensity at 12 m.

Solution

Distance × 4: intensity ÷ 16M1

= 1.25 × 10⁻³ W m⁻²A1

6.3 Amplitude and distance

Since I ∝ A² and I ∝ 1/r², the amplitude of a wave from a point source falls as 1/r.

Figure 18. Amplitude against distance from a point source.

WORKED EXAMPLE 30 Amplitude ratio (standard)

The amplitude of sound at 2.0 m is 0.60 μm. Find it at 6.0 m.

Solution

A ∝ 1/rM1

= 0.20 μmA1

6.4 Energy carried by waves

Waves carry energy at a rate given by their power. Ocean waves can deliver tens of kilowatts per metre of coastline, which wave-power devices aim to capture.

WORKED EXAMPLE 31 Solar intensity (standard)

Sunlight delivers 1000 W m⁻² at the ground. Find the power falling on a 1.6 m² solar panel.

Solution

P = IA = 1000 × 1.6M1

= 1.6 kWA1

7 Exam technique

Common question types.

7.1 Method checklist

•Check the graph axis (distance or time) before reading λ or T.
•Convert MHz, GHz, nm and ms to SI units.
•Use v = fλ with the speed for the correct medium.
•For particle motion, shift the waveform slightly in its direction of travel.
•For intensity, use I ∝ A² and I ∝ 1/r².

7.2 Multi-step problem: wave on a string

WORKED EXAMPLE 32 String (exam level)

A string vibrator at 60 Hz produces a wave with crests 0.25 m apart. A particle on the string has amplitude 4.0 mm. Find (a) the wave speed; (b) the time for a crest to travel 3.0 m; (c) the total distance moved by one particle in 1.0 s.

Solution

(a) v = 60 × 0.25 = 15 m s⁻¹A1

(b) t = 3.0/15 = 0.20 sA1

7.3 Multi-step problem: echo sounding

WORKED EXAMPLE 33 Depth survey (exam level)

A ship emits 40 kHz sonar pulses (v = 1500 m s⁻¹ in seawater). Echoes return after 0.12 s from a shoal and 0.40 s from the seabed. Find (a) the wavelength; (b) the depths of the shoal and seabed; (c) the minimum time between pulses so echoes from the seabed are not confused with the next pulse.

Solution

(a) λ = 1500/40 000 = 3.75 cmA1

(b) shoal 90 m; seabed 300 mA1

8 Data-based questions and further practice

Applying the wave model.

8.1 Experimental data: speed of sound

A student measures the time for sound to travel between two microphones at different separations.

d / m	0.50	1.00	1.50	2.00	2.50
t / ms	1.5	2.9	4.4	5.9	7.3

Figure 19. Distance against time delay for the microphone data.

WORKED EXAMPLE 34 Gradient (exam level)

Find the speed of sound from the graph.

Solution

Gradient = 2.5/7.3 × 10⁻³M1

= 342 m s⁻¹A1

8.2 Graded practice

WORKED EXAMPLE 35 Wave on a slinky (standard)

Compressions on a slinky are 0.30 m apart and pass a point at 4 per second. Find the wave speed.

Solution

λ = 0.30 m, f = 4.0 HzM1

v = 1.2 m s⁻¹A1

WORKED EXAMPLE 36 Sonar (exam level)

A ship's sonar pulse returns from the sea floor after 0.80 s (v = 1500 m s⁻¹). Find the depth.

Solution

d = 1500 × 0.80/2M1

= 600 mA1

WORKED EXAMPLE 37 Wavelength change (exam level)

A 500 Hz sound passes from air (340 m s⁻¹) into water (1480 m s⁻¹). Find the wavelengths in each.

Solution

Air: 340/500 = 0.68 mA1

Water: 1480/500 = 2.96 mA1

8.3 Graded practice: graphs

WORKED EXAMPLE 38 Two graphs (exam level)

A displacement–distance graph shows λ = 1.2 m; a displacement–time graph for one point shows T = 0.040 s. Find f and v.

Solution

f = 1/0.040 = 25 HzA1

v = 25 × 1.2 = 30 m s⁻¹A1

WORKED EXAMPLE 39 Phase from time (standard)

Two points on a wave oscillate with T = 0.20 s; one reaches its peak 0.05 s after the other. Find their phase difference.

Solution

φ = 2π × 0.05/0.20M1

= π/2 radA1

WORKED EXAMPLE 40 Distance between points (exam level)

Find the minimum separation of two points in antiphase on a 340 Hz sound wave in air.

Solution

λ = 1.0 mM1

λ/2 = 0.50 mA1

8.4 Graded practice: sound

WORKED EXAMPLE 41 Thunder (standard)

Thunder is heard 4.5 s after a lightning flash. Estimate the distance to the storm.

Solution

Light arrives almost instantly; d = 340 × 4.5M1

≈ 1.5 kmA1

WORKED EXAMPLE 42 Ultrasound frequency (exam level)

A medical scanner uses 3.5 MHz ultrasound in tissue (v = 1540 m s⁻¹). Find the wavelength and explain why high frequencies are used.

Solution

λ = 1540/3.5 × 10⁶ = 0.44 mmA1

Short wavelength resolves small detailsR1

WORKED EXAMPLE 43 Temperature and speed (exam level)

The speed of sound is proportional to √T (kelvin). If v = 331 m s⁻¹ at 273 K, find v at 303 K.

Solution

v = 331√(303/273)M1

= 349 m s⁻¹A1

8.5 Graded practice: EM waves and intensity

WORKED EXAMPLE 44 Radar (exam level)

A radar pulse reflects from an aircraft and returns after 0.20 ms. Find the distance to the aircraft.

Solution

d = 3.0 × 10⁸ × 0.20 × 10⁻³/2M1

= 30 kmA1

WORKED EXAMPLE 45 Laser intensity (standard)

A 5.0 mW laser beam has a cross-sectional area of 2.0 mm². Find its intensity.

Solution

I = 5.0 × 10⁻³/2.0 × 10⁻⁶M1

= 2.5 × 10³ W m⁻²A1

WORKED EXAMPLE 46 Light from Sun (exam level)

The Sun emits 3.8 × 10²⁶ W. Find the intensity at Mars, 2.3 × 10¹¹ m away.

Solution

I = 3.8 × 10²⁶/(4π(2.3 × 10¹¹)²)M1

= 570 W m⁻²A1

8.6 Key terms

Term	Meaning
transverse wave	oscillations perpendicular to the direction of energy transfer
longitudinal wave	oscillations parallel to the direction of energy transfer
wavelength	distance between successive points in phase
frequency	number of oscillations per second (Hz)
wavefront	line joining points in phase
ray	direction of energy transfer, perpendicular to wavefronts
intensity	power per unit area (W m⁻²)
compression/rarefaction	regions of higher/lower pressure in a longitudinal wave

8.7 Self-test

Try these without notes, then check the answers.

Quick check

1. Find v for f = 50 Hz and λ = 0.40 m.

2. Name two transverse and two longitudinal waves.

3. What can you read from a displacement–time graph?

4. Find the wavelength of 2.4 GHz WiFi.

5. How does intensity change if distance from a point source triples?

6. Why does the frequency of light not change when it enters glass?

Answers: 1. 20 m s⁻¹ 2. light, water ripples; sound, P-waves 3. period and amplitude 4. 12.5 cm 5. falls to one ninth 6. frequency is set by the source

8.8 Common exam errors

•Reading wavelength from a displacement–time graph (it gives the period).
•Saying the medium moves along with the wave.
•Forgetting to halve echo distances.
•Thinking wave speed depends on frequency in a given medium.
•Mixing up amplitude (from equilibrium) with peak-to-trough height.

9 Common misconceptions and links

Students often think…	Actually…
waves transport matter	they transport energy
higher frequency means higher speed	speed depends on the medium
sound can travel in a vacuum	it needs a medium
displacement–time graphs show wavelength	they show period
EM waves need a medium	they travel through a vacuum

How C.2 connects to the rest of the course

Topic	Connection
C.1 SHM	particles oscillate with SHM
C.3 Wave phenomena	reflection, refraction, interference, standing waves
E.1 Structure of the atom	photons and the EM spectrum
B.1 Thermal energy	infrared radiation

10 Exam-style question with markscheme

Try this under exam conditions (about 18 minutes) before reading the markscheme.

Figure 20. Displacement–distance graph for a wave on a string.

The graph shows a wave on a string at one instant. The source vibrates at 50 Hz.

(a) State the amplitude and wavelength. [2]

(b) Calculate the wave speed. [1]

(d) A sound wave of the same frequency travels in air at 340 m s⁻¹. Calculate its wavelength. [1]

(e) Describe how the motion of air particles in the sound wave differs from that of the string. [2]

Markscheme

Part	Answer	Marks
(a)	A = 2.0 cm	A1
	λ = 0.80 m	A1
(b)	v = 50 × 0.80 = 40 m s⁻¹	A1
(c)	Δx = 0.40 m = λ/2	M1
	π rad	A1
(d)	λ = 340/50 = 6.8 m	A1
(e)	air particles oscillate parallel to the direction of travel (longitudinal)	R1
	string particles oscillate perpendicular (transverse)	R1

Examiner's comments. In (a), several gave the peak-to-trough distance (4.0 cm) as the amplitude.

11 Check your understanding

Section A: multiple choice

1. Sound waves are

A. longitudinal B. transverse C. electromagnetic D. stationary

2. Wave speed equals

A. f/λ B. λ/f C. fλ D. 1/fλ

3. All EM waves in a vacuum have the same

A. frequency B. speed C. wavelength D. energy

4. A displacement–time graph shows the

A. wavelength B. wave speed C. phase velocity D. period

5. Two points half a wavelength apart are

A. in phase B. in antiphase C. π/2 apart D. 2π apart

6. If the amplitude triples, intensity

A. increases 9 times B. triples C. is unchanged D. increases 6 times

7. Which cannot travel through a vacuum?

A. light B. X-rays C. radio waves D. sound

8. Visible light wavelengths are about

A. 400–700 mm B. 1–1000 m C. 400–700 nm D. 1–10 pm

Section B: structured

9. Distinguish between transverse and longitudinal waves, with an example of each. [4]

10. A wave has frequency 25 Hz and speed 5.0 m s⁻¹. Find the phase difference between points 5.0 cm apart. [2]

11. Explain why sound travels faster in steel than in air. [2]

12. A point source emits 2.0 W of sound. Find the intensity 5.0 m away. [2]

Answers

1. A 2. C 3. B 4. D 5. B 6. A 7. D 8. C

9. Transverse: perpendicular (light); longitudinal: parallel (sound).

10. λ = 0.20 m; φ = 2π(0.05/0.20) = π/2.

11. Particles strongly coupled; vibrations passed on faster.

12. I = 2.0/(4π × 25) = 6.4 × 10⁻³ W m⁻².

12 Summary and topic map

Idea	Remember
Wave types	transverse and longitudinal
Quantities	A, λ, T, f, v
Wave equation	v = fλ
Graphs	distance gives λ; time gives T
Sound	longitudinal; needs a medium
EM waves	transverse; c in vacuum
Intensity	I ∝ A²; I = P/4πr²

Before the exam, make sure you…

•check graph axes before reading values
•use v = fλ in the correct medium
•halve echo distances
•use I ∝ A² and inverse-square law

Wave Model on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 21. Topic map for Wave Model.

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