C.2 Wave model

A wave is a disturbance that carries energy from one place to another. Light from the Sun, the sound of a voice, ripples on a pond and the shaking of the ground in an earthquake are all waves. In every case energy moves outwards from the source, while the material the wave passes through only oscillates about where it already was. The wave model, built on the oscillations of C.1, describes all of these with the same few quantities and one equation.

Most of the information people receive also travels as waves: vision, hearing, radio, mobile phones, Wi-Fi and medical ultrasound scans all depend on them. The rest of theme C (wave phenomena, standing waves and the Doppler effect) builds on the description set out here.

Travelling waves and the transfer of energy

When one end of a long rope is moved up and down, each small piece of the rope pulls on the next, which pulls on the next, and the disturbance travels along the rope. The pieces of rope themselves do not travel along; each one moves up and down about its own equilibrium position and then comes to rest. What travels is the pattern of the disturbance and the energy it carries. A single disturbance is called a pulse; a source that oscillates continuously produces a continuous wave.

A travelling wave on a rope or a slinky spring is a mechanical wave: it needs a material, the medium, whose particles are linked by forces so that one can pass the disturbance to the next. Electromagnetic waves are different, because they are oscillations of electric and magnetic fields and can travel through a vacuum. Both families obey the same wave model.

Each particle of the medium in a mechanical wave oscillates, often in simple harmonic motion, about its equilibrium position. Neighbouring particles start their oscillations a little later than the ones nearer the source, so they are at different stages of their cycle at any instant. This steady delay from one particle to the next is what makes the shape of the wave move forwards.

Energy and amplitude

A wave of larger amplitude carries more energy, because the particles of the medium oscillate with larger displacements and higher maximum speeds. The rate at which a wave delivers energy to each square metre of a surface facing it is its intensity, measured in $\text{W m}^{-2}$. For a wave of a given frequency, the intensity is proportional to the square of the amplitude, so doubling the amplitude of a sound wave increases its intensity by a factor of four.

Transverse and longitudinal waves

Waves are classified by the direction in which the particles of the medium oscillate, compared with the direction in which the wave transfers energy.

Transverse waves

A wave on a rope or a guitar string is a transverse wave. The points of maximum positive displacement are crests and the points of maximum negative displacement are troughs. All electromagnetic waves are transverse, as are S waves in the solid rock of the Earth.

The rope diagram shows that particles at a crest or a trough are momentarily at rest, exactly like an oscillator at its extreme, while those passing through the equilibrium line move fastest. Each particle moves only up and down; none moves along the rope.

Longitudinal waves

Pushing and pulling one end of a slinky spring along its length produces a longitudinal wave. The coils move back and forth along the spring, so in some places they are squeezed together and in others spread apart. A region where the particles are closer together than normal is a compression; a region where they are further apart is a rarefaction. Sound waves and P waves in the Earth are longitudinal.

In the particle diagram, every particle oscillates about its own equilibrium position with the same amplitude. The compressions and rarefactions are where neighbouring particles happen to be displaced towards or away from each other, and these regions travel along the medium at the wave speed.

Describing a wave

The quantities that describe a wave combine those of an oscillation (C.1) with one that describes the wave in space.

• Displacement $y$: the distance of a particle of the medium from its equilibrium position, in a stated direction.

• Amplitude $x_0$: the maximum displacement of a particle from its equilibrium position.

• Wavelength $\lambda$: the distance between two neighbouring points on the wave that are oscillating in phase, for example two adjacent crests or two adjacent compressions.

• Period $T$: the time for one particle to make one complete oscillation; equivalently, the time for the wave to move forward one wavelength.

• Frequency $f$: the number of oscillations of a particle per second, $f = 1/T$, equal to the frequency of the source.

• Wave speed $v$: the speed at which the wave (the energy, or a crest) travels through the medium.

Displacement–distance and displacement–time graphs

A wave can be drawn in two ways that look alike but show different things. A displacement–distance graph plots the displacement of every particle against its position along the wave at one instant, like a photograph. The wavelength is read along its horizontal axis.

A displacement–time graph plots the displacement of a single particle against time, like a video of one point. It is the same graph as the $x$–$t$ graph of an oscillator in C.1, and the period is read along its horizontal axis.

The two graphs have the same sinusoidal shape and the same amplitude, so the label on the horizontal axis is the only way to tell them apart. The wavelength cannot be found from a displacement–time graph, and the period cannot be found from a displacement–distance graph.

The wave equation

In one period $T$, each particle completes one oscillation and the wave moves forward by exactly one wavelength $\lambda$. Speed is distance divided by time, so

The equation holds for every kind of wave, mechanical or electromagnetic. It also explains a pattern seen in any one medium: waves of different frequency travel at the same speed, so a higher frequency means a shorter wavelength. A note one octave higher (double the frequency) has half the wavelength in air.

The speed of sound in air at 20 °C is about $343\ \text{m s}^{-1}$. Human hearing covers frequencies from about 20 Hz to 20 kHz, so audible sound in air has wavelengths from about 17 m down to 1.7 cm.

Phase along a wave

Points on a wave are compared using phase, as for two oscillators. Points one whole wavelength apart (or any whole number of wavelengths) oscillate in phase: they move in the same direction with the same displacement at every instant. Points half a wavelength apart are in antiphase: when one is at a crest the other is at a trough. In general, two points separated by a distance $\Delta x$ along the wave differ in phase by the fraction $\Delta x/\lambda$ of a cycle, which is $2\pi\Delta x/\lambda$ radians.

The motion of particles in a travelling wave

A snapshot of a wave shows where the particles are, but not which way each one is moving. That depends on the direction in which the wave travels. A reliable method is to sketch the wave a moment later, shifted slightly in the direction of travel, and see where each particle must move to lie on the new curve.

For a wave travelling to the right, the shifted-wave diagram shows that particles just ahead of a crest (on its front, right-hand side) are moving up, because the crest is about to arrive; particles just behind a crest are moving down. Particles at crests and troughs are momentarily at rest, and particles at the equilibrium position move fastest.

Particle velocity and wave velocity

The speed of the particles and the speed of the wave are different quantities. The wave speed $v = f\lambda$ is set by the medium and is the same everywhere along the wave. The particle velocity changes continuously throughout each cycle, from zero at the extremes to a maximum at equilibrium, exactly as for simple harmonic motion. A particle on a rope can move up and down at $1\ \text{m s}^{-1}$ while the wave travels along the rope at $20\ \text{m s}^{-1}$.

Longitudinal waves on a displacement–distance graph

A longitudinal wave can also be drawn as a displacement–distance graph, although the particles move along the axis rather than across it. A positive direction is chosen (usually to the right), and the displacement of each particle along the wave is plotted upwards if it is positive and downwards if it is negative. The graph is then a sinusoid, but its crests are not compressions.

The aligned diagram shows how to find the compressions. At a compression the particles on the left are displaced to the right (positive) and those on the right are displaced to the left (negative), so they crowd together. This happens where the graph crosses the axis going from positive to negative. Where the graph crosses the axis going from negative to positive, the particles move away from each other, giving a rarefaction. At the crests and troughs of the graph the particles have their largest displacements, but their neighbours have nearly the same displacement, so the spacing is normal.

When a wave changes medium

The speed of a wave depends on the medium: how strongly its particles are linked and how much inertia they have. Sound travels faster in water than in air, and faster still in steel; light travels more slowly in glass than in a vacuum. The frequency, by contrast, is set by the source. Every oscillation that arrives at a boundary passes on one oscillation to the far side, so the frequency cannot change when a wave crosses from one medium into another.

Since $v = f\lambda$ and $f$ is fixed, the wavelength changes in proportion to the speed. A wave that slows down on entering a new medium has its wavelength shortened; one that speeds up has its wavelength lengthened.

Sound waves

Sound is produced by a vibrating object, such as a loudspeaker cone, a guitar string or the vocal folds. As the cone moves forward it pushes the neighbouring air molecules together, creating a compression; as it moves back it leaves a region of lower pressure, a rarefaction. These pressure variations travel outwards as a longitudinal wave. The air molecules oscillate back and forth along the direction of travel; they do not flow from the loudspeaker to the listener.

Because sound needs particles to pass on the disturbance, it cannot travel through a vacuum. A ringing electric bell inside a sealed glass jar becomes quieter and quieter as the air is pumped out, although the bell can still be seen vibrating, because light crosses the near-vacuum in the jar while sound cannot.

Sound is fastest in solids, where the particles are close together and strongly bonded, and slowest in gases. Sound with frequencies above the range of human hearing, above about 20 kHz, is called ultrasound; bats and dolphins use it for echolocation and hospitals use it for imaging.

Electromagnetic waves

An electromagnetic wave is a travelling oscillation of an electric field and a magnetic field. The two fields oscillate in phase, at right angles to each other and at right angles to the direction of travel, so electromagnetic waves are transverse. They are produced by accelerating charges, such as electrons oscillating in a radio aerial, and by energy changes inside atoms and nuclei, such as electrons moving between energy levels or nuclei emitting gamma rays.

Because the fields themselves carry the disturbance, electromagnetic waves need no medium. In a vacuum all of them travel at the same speed, the speed of light $c = 3.00\times10^{8}\ \text{m s}^{-1}$, whatever their frequency. In materials they travel more slowly; in glass, for example, light travels at about $2\times10^{8}\ \text{m s}^{-1}$.

The electromagnetic spectrum

Electromagnetic waves form a continuous electromagnetic spectrum of wavelengths, from gamma rays shorter than the size of a nucleus to radio waves many kilometres long. The spectrum is divided into named regions according to how the waves are produced, detected and used, but the physics is the same throughout, and the boundaries between regions are not sharp.

Since $c = f\lambda$ is the same for all of them in a vacuum, the shortest wavelengths have the highest frequencies. Gamma rays and X-rays overlap in wavelength and are distinguished by their source: gamma rays come from nuclei, X-rays from electrons.

The same method works in reverse: the order of magnitude of a wavelength places the radiation in its region, and $c = f\lambda$ then gives its frequency.

Mechanical and electromagnetic waves compared

The wave model applies equally to mechanical and electromagnetic waves: both transfer energy, both have a wavelength, frequency and speed linked by $v = f\lambda$, and both show the wave phenomena of C.3. They differ in what oscillates and in whether a medium is needed.

Seismic waves: P and S waves

An earthquake releases energy that travels through the Earth as seismic waves. Two kinds travel through the body of the Earth. P waves (primary waves) are longitudinal: the rock is alternately compressed and stretched along the direction of travel. S waves (secondary waves) are transverse: the rock is sheared from side to side. In the crust, P waves travel at roughly $6\ \text{km s}^{-1}$ and S waves at roughly $4\ \text{km s}^{-1}$, so P waves always arrive first.

Earthquake early-warning systems use this delay. During the magnitude 9.0 Tōhoku earthquake off the north-east coast of Japan on 11 March 2011, the Japan Meteorological Agency's network detected the first P waves at the nearest coastal station and issued a public warning about 8 seconds later. Because the damaging S waves travel more slowly, places further from the epicentre received the warning several seconds or more before the strongest shaking arrived. Coastal seismometers on the Tōhoku Shinkansen line cut the power to the high-speed trains in the same way, and every passenger train stopped without derailing.

The difference between the two kinds of wave also provides evidence about the Earth's interior. S waves are transverse and cannot travel through a liquid, which does not resist shearing. Seismometers on the far side of the Earth from an earthquake detect P waves but no S waves, which shows that the outer core of the Earth is liquid.

Applying the ideas in exam questions

Many wave questions start from a graph. Read the axis first, then take measurements over several wavelengths or periods where the graph allows it, to reduce the reading uncertainty.

Paper 1A: speed from a wave graph

Paper 1A: sound and light

Paper 2: a sound wave

A loudspeaker produces a sound wave of frequency 850 Hz in air. The graph shows the variation of the displacement of the air molecules with position along the wave at one instant.

Paper 1B: waves in a ripple tank

A student measures the wavelength $\lambda$ of water waves in a ripple tank for several frequencies $f$ of the vibrating bar. The uncertainty in each wavelength is $\pm 0.1\ \text{cm}$.

Self-check

• To do:

  I can explain how a travelling wave transfers energy without a net transfer of the medium.

• To do:

  I can distinguish transverse and longitudinal waves by the direction of particle oscillation relative to energy transfer, and give examples of each.

• To do:

  I can define displacement, amplitude, wavelength, period, frequency and wave speed for a wave.

• To do:

  I can read the wavelength from a displacement–distance graph and the period from a displacement–time graph.

• To do:

  I can derive and use v = fλ = λ/T.

• To do:

  I can state the phase relationship between points a given distance apart along a wave.

• To do:

  I can find the direction of motion of a particle in a travelling wave, and distinguish particle velocity from wave velocity.

• To do:

  I can locate compressions and rarefactions on a displacement–distance graph of a longitudinal wave.

• To do:

  I can explain why the frequency stays the same and the wavelength changes when a wave enters a new medium.

• To do:

  I can describe the nature of sound waves and explain why sound cannot travel through a vacuum.

• To do:

  I can describe the nature of electromagnetic waves and identify the region of the spectrum from a wavelength.

• To do:

  I can compare mechanical and electromagnetic waves, using seismic waves and light as examples.

The atmosphere of Mars is very thin. Ignore air resistance unless a part says otherwise.

1. (a)

   Show that the gravitational field strength at the surface of Mars is about $3.7\ \text{N kg}^{-1}$.

   [2]2 marks

   Markscheme (a)

   $g = \dfrac{6.67 \times 10^{-11} \times 6.42 \times 10^{23}}{(3.39 \times 10^{6})^2}$ ✓
   $3.73\ \text{N kg}^{-1}$ ✓
   Must see the substitution OR an answer to at least 3 s.f.

2. (b)(i)

   Calculate the time of flight of the capsule.

   [2]2 marks

   Markscheme (b)(i)

   vertical component = «$42 \sin 50^\circ =$» $32.2\ \text{m s}^{-1}$ ✓
   «$\dfrac{2 \times 32.2}{3.7} =$

3. (b)(ii)

   Calculate the horizontal distance travelled by the capsule.

   [2]2 marks

   Markscheme (b)(ii)

   horizontal component = «$42 \cos 50^\circ =$» $27.0\ \text{m s}^{-1}$ ✓
   «$27.0 \times 17.3 =$

4. (b)(iii)

   Calculate the maximum height reached by the capsule.

   [2]2 marks

   Markscheme (b)(iii)

   $h = \dfrac{32.2^2}{2 \times 3.7}$ ✓
   140 m ✓
   Accept 138–140 m.

5. (c)

   The same launcher is used on Earth, with the same launch speed and angle. Determine the ratio $\dfrac{\text{horizontal distance on Mars}}{\text{horizontal distance on Earth}}$. Ignore air resistance on Earth.

   [2]2 marks

   Markscheme (c)

   time of flight and so horizontal distance are proportional to $\dfrac{1}{g}$

6. (d)

   Explain, without calculation, why the capsule lands with the same speed as it was launched.

   [2]2 marks

   Markscheme (d)

   there is no air resistance, so mechanical energy is conserved ✓
   the capsule lands at the same height, so its gravitational potential energy and hence its kinetic energy «speed» are the same as at launch ✓
   Accept an argument using components: the horizontal component is unchanged and the vertical component has the same magnitude at landing.

7. (e)(i)

   A later mission must send a sample capsule away from Mars completely. Show that the escape speed from the surface of Mars is about $5\ \text{km s}^{-1}$.

   [2]2 marks

   Markscheme (e)(i)

   $v_\text{esc} = \sqrt{\dfrac{2GM}{r}} = \sqrt{\dfrac{2 \times 6.67 \times 10^{-11} \times 6.42 \times 10^{23}}{3.39 \times 10^{6}}}$

8. (e)(ii)

   Calculate the minimum energy needed to move the 0.50 kg capsule from the surface of Mars to infinity.

   [2]2 marks

   Markscheme (e)(ii)

   $\Delta E_\text{p} = \dfrac{GMm}{r}$ OR $\tfrac{1}{2}mv_\text{esc}^{\,2}$

9. (e)(iii)

   Outline why the escape speed does not depend on the direction in which the capsule is launched.

   [1]1 mark

   Markscheme (e)(iii)

   escape depends only on the kinetic energy being at least equal to the gain in gravitational potential energy, and energy is a scalar ✓
   Ignore references to the atmosphere.

10. (f)

    Radio signals from the rover travel to Earth at the speed of light. On one day the distance between Mars and Earth is $2.25 \times 10^{11}\ \text{m}$. Calculate the time taken for a signal from the rover to reach Earth.

    [1]1 mark

    Markscheme (f)

    «$\dfrac{2.25 \times 10^{11}}{3.00 \times 10^{8}} =$

11. (g)

    The thin atmosphere of Mars does exert a small drag force on the capsule. Predict how the horizontal distance and the speed on landing compare with your answers to (b)(ii) and (d).

    [2]2 marks

    Markscheme (g)

    horizontal distance is «slightly» less ✓
    landing speed is «slightly» less than the launch speed, as some energy is transferred to the atmosphere ✓

(b) $0.30$ «m s⁻¹» ✓

(c) $\dfrac{0.1}{1.2} \times 100 = 8\%$ ✓

Accept 8.3%.

(d) measure the distance across several «e.g. 10» wavelengths and divide ✓

Do not accept "use better equipment".