IB DP Physics · Theme C · Wave behavior

Wave model.

C.2 · Traveling waves, the wave equation, and the nature of sound and of light
Standard level and higher level

SL + HL · 3 hours C.1 Simple harmonic motion ↗

The journey

Three parts.

Part 01

Traveling waves

Transverse and longitudinal waves, and \(v = f \lambda\)

Part 02

Two graphs of one wave

Displacement against position, and against time

Part 03

Sound and light

Mechanical waves and electromagnetic waves

C.1 described one oscillator. A wave is a line of oscillators, each a little behind the one before.

C.2 WAVE MODEL · SL + HL

Part 01

Traveling waves.

Energy on the move, with nothing carried along

C.2 WAVE MODEL · SL + HL

Energy without matter

What a traveling wave is.

Each particle oscillates about its own place
It passes its motion to the next, a little later
The pattern travels. The medium does not

A wave transfers energy, with no net transfer of matter.

A TRANSVERSE WAVE

C.2 WAVE MODEL · SL + HL

Follow the dark particle

It goes up and down in simple harmonic motion and ends each cycle exactly where it began. Nothing moves to the right except the shape. A stadium wave works the same way: each person stands and sits, nobody changes seats, and the wave runs round the ground.

Why energy must be traveling

A particle far along the line was at rest before the wave arrived and is oscillating afterwards, so it has gained energy. That energy came from whatever shook the first particle, passed along from neighbor to neighbor. A cork on a pond bobs when ripples reach it but is not swept to the shore.

“No resultant displacement of the medium.”

Averaged over a cycle, each particle’s displacement is zero. The guide stresses that energy is transferred all the same.

Each particle copies its neighbor with a delay. That delay, as a phase difference, is what C.3 and C.4 build on.

Two ways to oscillate

Transverse and longitudinal.

Transverse: oscillations at right angles to the direction of travel
Longitudinal: oscillations along the direction of travel
Crowded together: a compression. Spread out: a rarefaction

A LONGITUDINAL WAVE

C.2 WAVE MODEL · SL + HL

Examples

Transverse: waves on a string, ripples on water (nearly), the S-waves of an earthquake and all electromagnetic waves. Longitudinal: sound in air, the pressure pulses you can send along a stretched spring by pushing its end, and the P-waves of an earthquake.

Why a gas carries only longitudinal waves

A transverse wave needs each layer of the medium to drag the next one sideways, which takes a force along the surfaces between layers. Solids can supply it. Gases and liquids cannot: their layers slide freely over each other. They can, however, be squashed and spring back, and that is all a longitudinal wave needs. This is how the Earth’s outer core was shown to be liquid: S-waves do not pass through it.

In the animation the compressions travel to the right, but the dark particle only moves to and fro. Nothing travels with the wave.

Only transverse waves can be polarized, since only they have a direction of oscillation to choose. Sound cannot be polarized.

Measuring a wave

Wavelength, frequency and period.

Wavelength \(\lambda\): the distance between neighboring points that are in step
Period \(T\): the time for one oscillation of a particle
Frequency \(f\): oscillations per second
Amplitude: the greatest displacement

A WAVE AT ONE INSTANT

C.2 WAVE MODEL · SL + HL

“In step”

Two particles one wavelength apart are doing exactly the same thing at every moment: both at a crest together, both passing through the center together, and moving the same way. Crest to crest is the easiest pair to see, but any point to the next matching point will do. For a longitudinal wave, it is the distance from one compression to the next.

What sets what

The frequency is decided by the source: every particle is forced to oscillate at the rate of the thing that is shaking the medium. The speed is decided by the medium. The wavelength then has no choice: it is whatever distance the wave covers in one period. This matters at boundaries in C.3, where the speed changes and the frequency does not.

Measuring the amplitude from trough to crest. It is from the undisturbed level to a crest: half the full height.

Speed, frequency and wavelength

The wave equation.

\(v = f \lambda = \dfrac{\lambda}{T}\)

In one period the wave moves forward one wavelength
Speed \(= \dfrac{\text{distance}}{\text{time}} = \dfrac{\lambda}{T}\)
True for every kind of wave

A 440 Hz note in air: \(v = 340 \,\mathrm{m} \,\mathrm{s}^{-1}\) \(\lambda = \dfrac{v}{f} = \dfrac{340}{440} = \class{res}{0.77 \,\mathrm{m}}\)

Red light of wavelength 650 nm \(f = \dfrac{c}{\lambda} = \dfrac{3.00 \times 10^{8}}{650 \times 10^{-9}} = \class{res}{4.6 \times 10^{14} \,\mathrm{Hz}}\)

C.2 WAVE MODEL · SL + HL

Seeing why

Watch one crest in the animation of the transverse wave. During the time the dark particle takes to go once up and down, that crest moves along to exactly where the next crest was. One wavelength traveled in one period: \(v = \frac{\lambda}{T}\). And since \(f = \frac{1}{T}\), \(v = f \lambda\).

Two different speeds

\(v\) is the speed of the pattern, constant in a given medium. Each particle has its own speed, which changes all the time as it oscillates, from zero at the extremes to \(\omega x_{0}\) at the center. The two have nothing to do with each other: a loud note and a quiet one travel equally fast, though the particles move faster in the loud one.

Sizes

Audible sound runs from 20 Hz to 20 kHz, which in air is from 17 m down to 17 mm: comparable with rooms, doors and people. Visible light has wavelengths below a thousandth of a millimeter. That difference decides how each behaves at an obstacle, in C.3.

Leaving a wavelength in nanometers. 650 nm is \(650 \times 10^{-9}\) m.

Quick check

Traveling waves.

01What does a traveling wave transfer?

Energy, not matter

02How do the particles move in a longitudinal wave?

Along the direction of travel

03Is sound in air transverse or longitudinal?

Longitudinal

04A wave has \(f\) = 50 Hz and \(\lambda\) = 0.30 m. Speed?

15 m s⁻¹

05A wave of period 0.20 s travels at 4.0 m s⁻¹. Wavelength?

0.80 m

06What decides the frequency of a wave?

The source

PRESS → TO REVEAL EACH ANSWER

Part 02

Two graphs of one wave.

They look the same and mean different things

C.2 WAVE MODEL · SL + HL

A photograph of the wave

Displacement against position.

All the particles, at one instant
Peak to peak: the wavelength
Height of a peak: the amplitude
It cannot tell you the period

A SNAPSHOT

C.2 WAVE MODEL · SL + HL

A graph, not a picture

For a transverse wave on a string the graph happens to look like the string itself. For any other wave it does not. For sound, the displacements are along the direction of travel, yet the graph is still drawn with displacement up the page. The curve shows how far each particle is from its undisturbed position, not where the particles are in space.

What changes with time

A moment later the whole curve has shifted along in the direction of travel, keeping its shape. Exam questions often give one snapshot and ask you to draw the wave a quarter of a period later: slide the curve forward by a quarter of a wavelength.

Read the label on the horizontal axis before anything else. If it is a distance, the repeat length is the wavelength.

A film of one particle

Displacement against time.

One particle, followed through time
Peak to peak: the period
Height of a peak: the amplitude
It cannot tell you the wavelength

ONE PARTICLE

C.2 WAVE MODEL · SL + HL

This is the graph from C.1

Each particle of the medium is a simple harmonic oscillator, and this is its displacement–time graph. Its gradient at any moment is the velocity of the particle (not of the wave).

Using the two together

One graph alone gives either the wavelength or the period. The speed needs both: \(v = \frac{\lambda}{T}\). A typical question shows a snapshot with a repeat distance of 4.0 m and a displacement–time graph with a repeat time of 0.50 s. The speed is \(\frac{4.0}{0.50} = 8.0\) m s⁻¹.

A neighboring particle has the same graph, shifted along the time axis: it does the same thing a little later. Half a wavelength away, the graph is upside down.

Reading a wavelength off a displacement–time graph. The horizontal axis is time, so the repeat length is a period, in seconds.

From a snapshot

The direction each particle is moving.

Draw the wave a moment later, shifted in its direction of travel
Each particle moves straight up or down to the new curve
At a crest or trough: momentarily at rest
At zero displacement: moving fastest

NOW, AND A MOMENT LATER

C.2 WAVE MODEL · SL + HL

Why the trick works

A particle cannot move sideways with the wave. All it can do is go up or down. A moment from now it must lie on the new, shifted curve, directly above or below where it is now. So the direction from the solid curve to the dashed curve, at that position, is the direction the particle is moving.

A pattern worth noticing

For a wave traveling to the right, the particles on the leading side of each crest (just to its right) are rising, because the crest is about to arrive. The particles behind it are falling, because it has just passed. Reverse the direction of travel and every arrow reverses.

Speed of a particle

Where the curve is steepest, at zero displacement, the gap between the two curves is greatest: those particles are moving fastest. At a crest the two curves are level with each other: that particle is at rest. This is C.1 again, read off a wave.

Always start by marking the direction of travel on the diagram. Without it the question cannot be answered.

The same graph for sound

Graphs of a longitudinal wave.

Displacement is along the wave: say, right is positive
Compression: particles on both sides have moved towards it
Rarefaction: they have moved away

At both, the displacement itself is zero.

PARTICLES ABOVE, GRAPH BELOW

C.2 WAVE MODEL · SL + HL

Finding the compressions

Look at a point where the graph crosses the axis going downwards. Just to its left the displacement is positive: those particles have moved right, towards the point. Just to its right the displacement is negative: those particles have moved left, also towards the point. Particles arriving from both sides make a compression. Where the graph crosses going upwards, the particles on each side have moved away, leaving a rarefaction.

The surprise

The particle at the center of a compression has not moved at all. The largest displacements are found half way between a compression and a rarefaction, where the spacing of the particles is normal.

If positive is taken to the left, the two swap. Check which direction the question calls positive.

Placing compressions at the peaks of a displacement graph. A peak is where a particle is furthest from home, not where the particles are closest together.

Quick check

Two graphs of one wave.

01Which graph gives the wavelength?

Displacement against position

02Which graph gives the period?

Displacement against time

03A snapshot repeats every 2.0 m; a particle repeats every 0.40 s. Wave speed?

5.0 m s⁻¹

04How fast is a particle moving when it is at a crest?

It is at rest

05What is the displacement at the center of a compression?

Zero

06Two particles are half a wavelength apart. How do their motions compare?

Exactly opposite

PRESS → TO REVEAL EACH ANSWER

Part 03

Sound and light.

One model, two very different kinds of wave

C.2 WAVE MODEL · SL + HL

A mechanical wave

Sound waves.

Longitudinal: compressions and rarefactions of the medium
Needs a medium: no sound in a vacuum
Higher frequency: higher pitch
Larger amplitude: louder
Fastest in solids, slowest in gases

Medium (approximate values)	speed of sound / m s⁻¹
Air at 20 °C	340
Helium at 20 °C	1000
Water	1500
Steel	5900

C.2 WAVE MODEL · SL + HL

What a loudspeaker does

Its cone moves forwards and squashes the air in front of it, then moves back and leaves the air thinned. The squashed air pushes on the air beyond it, and so the compression moves outwards at the speed of sound, followed by the rarefaction. Your eardrum is pushed in and out by the tiny changes of pressure as they arrive. For ordinary speech those changes are about a millionth of atmospheric pressure.

What decides the speed

How stiff the medium is and how dense. A stiff medium springs back quickly, which helps; a dense one is sluggish, which hinders. Steel is far denser than air but so much stiffer that sound travels seventeen times faster in it. In a gas, the speed is tied to the speed of the molecules themselves: from B.3, about 500 m s⁻¹ for air. Warm the gas and sound travels faster.

The speed does not depend on frequency or loudness

If it did, the high and low notes of a distant band would arrive at different times.

A bell ringing in a jar becomes inaudible as the air is pumped out, while it can still be seen. Light needs no medium; sound does.

A wave of fields

Electromagnetic waves.

Oscillating electric and magnetic fields
At right angles to each other and to the direction of travel: transverse
Need no medium
All travel at \(c = 3.00 \times 10^{8} \,\mathrm{m} \,\mathrm{s}^{-1}\) in a vacuum

FIELDS, NOT PARTICLES

C.2 WAVE MODEL · SL + HL

What is oscillating

Nothing material. At each point in space the electric field strength and the magnetic field strength (D.2) rise, fall and reverse, in step with each other. The “displacement” on a graph of a light wave is the value of the field, not the position of anything.

Why it can cross empty space

A changing magnetic field creates an electric field: that is induction (D.4). A changing electric field, in turn, creates a magnetic field. Each keeps regenerating the other, so the pair can travel on with nothing to carry it. Maxwell worked this out in the 1860s and found that such a wave must travel at one particular speed, which turned out to be the measured speed of light.

Where they come from

Accelerating charge: electrons driven up and down an aerial, or changing energy level inside an atom.

In a transparent material electromagnetic waves travel more slowly than \(c\), and at slightly different speeds for different frequencies. In a vacuum they all travel at exactly \(c\).

The story · Karlsruhe, 1886

Background story · not assessed

Hertz makes Maxwell’s waves.

Photograph of Heinrich Hertz, a young man with a beard and mustache in a dark suit — **Heinrich Hertz**, about 1890

Engraving of two curved metal reflectors facing each other, one with an induction coil and spark gap, the other with a receiver — A **spark transmitter and receiver** of the Hertz type

In 1864 James Clerk Maxwell’s theory had predicted electromagnetic waves. Nobody had made one
From 1886 to 1889 Hertz made them with a spark and detected them with a loop of wire. They reflected, refracted and traveled at the speed of light
Asked what use they were: “Nothing, I guess.” He died at 36; the unit of frequency took his name in 1930

Portrait: Robert Krewaldt, about 1890, public domain. Engraving: unknown author, before 1912, public domain.

C.2 WAVE MODEL · SL + HL

One kind of wave, many names

The electromagnetic spectrum.

WAVELENGTH IN A VACUUMORDERS OF MAGNITUDE, AS IN THE DATA BOOKLET

The same wave throughout. Only the wavelength, and so the frequency, differs: \(c = f \lambda\).

C.2 WAVE MODEL · SL + HL

How wide it is

The scale is logarithmic: each mark is a thousand times shorter than the one to its left. The whole chart covers a factor of \(10^{16}\) in wavelength, and visible light is a sliver of it, from about 400 nm (violet) to 700 nm (red), less than a factor of two.

Wavelengths you can picture

Radio waves: from the size of a person to the size of a mountain. Microwaves: centimeters, the size of a finger; a microwave oven uses 12 cm. Infrared: the width of a hair down to the size of a cell. Visible: half a thousandth of a millimeter. Ultraviolet and X-rays: down to the size of an atom, \(10^{-10}\) m. Gamma rays: towards the size of a nucleus.

The boundaries are conventions

Nothing changes abruptly at any of them, and the regions overlap. X-rays and gamma rays are told apart by their origin, electrons or nuclei, not strictly by wavelength.

Know the order and the approximate order of magnitude of each region. The data booklet has the chart.

Compare

Mechanical and electromagnetic waves.

	Mechanical waves	Electromagnetic waves
What oscillates	Particles of a medium	Electric and magnetic fields
Medium	Essential	Not needed: can cross a vacuum
Type	Transverse or longitudinal	Always transverse
Speed	Set by the medium: 340 m s⁻¹ for sound in air	\(3.00 \times 10^{8}\) m s⁻¹ in a vacuum; slower in matter
Examples	Sound, waves on strings and water, seismic waves	Radio, microwaves, infrared, light, ultraviolet, X-rays, gamma rays

C.2 WAVE MODEL · SL + HL

One model for both

Wavelength, frequency, period, amplitude and the wave equation apply to a sound wave and to a light wave without any change. So do the two graphs of Part 2, as long as “displacement” is read as the field strength for light. That is the sense of the topic’s title: a single model of what a wave is.

Thunder and lightning

Both leave the same place at the same moment. The light arrives almost at once; the sound follows at 340 m s⁻¹. If the thunder comes 6.0 s after the flash, the storm is \(340 \times 6.0 = 2.0\) km away. The time the light took, \(\frac{2000}{3.00 \times 10^{8}} = 7\) μs, can be ignored.

Light also behaves as a stream of photons (E.1, E.2). The wave model is the one that explains everything in theme C.

Quick check

Sound and light.

01Why can sound not cross a vacuum?

It needs particles to oscillate

02What oscillates in an electromagnetic wave?

Electric and magnetic fields

03Which has the longer wavelength: infrared or ultraviolet?

Infrared

04Frequency of a radio wave of wavelength 3.0 m?

\(\frac{3.00 \times 10^{8}}{3.0} = 1.0 \times 10^{8} \,\mathrm{Hz}\)

05An echo from a cliff returns after 1.0 s. Distance to the cliff? (340 m s⁻¹)

170 m: there and back is 340 m

06Are electromagnetic waves transverse or longitudinal?

Transverse

PRESS → TO REVEAL EACH ANSWER

Summary

The wave model on one slide.

A traveling wave

Energy, not matter

Each particle oscillates about its own place

Two types

Transverse · longitudinal

Across, or along, the direction of travel

Wave equation

\(v = f \lambda = \frac{\lambda}{T}\)

One wavelength in one period

Two graphs

Position: \(\lambda\) · time: \(T\)

A snapshot of all particles, or the history of one

Sound

Longitudinal, needs a medium

Compressions and rarefactions

Electromagnetic waves

Transverse, no medium

Fields oscillate; speed \(c\) in a vacuum

C.2 WAVE MODEL · SL + HL

Summary

Key words.

Traveling waveTransfers energy without transferring matter

TransverseOscillations at right angles to the direction of travel

LongitudinalOscillations along the direction of travel

WavelengthDistance between neighboring points in step

Wave speedSpeed of the pattern: \(f \lambda\)

CompressionWhere particles are crowded together

RarefactionWhere particles are spread out

Mechanical waveNeeds a medium

Electromagnetic waveOscillating electric and magnetic fields

C.2 WAVE MODEL · SL + HL