Themes › Theme C Wave behaviour
C.2Wave model
A wave carries energy from one place to another without carrying matter with it. Ripples on a pond, sound, light, radio and earthquakes all follow the same simple model. Each part of the medium oscillates, often in SHM (C.1), and the pattern travels.
Knowledge and science
Nature of science
Models. One wave model, with wavelength, frequency, speed and amplitude, describes waves as different as sound in air and light in space. A single model that works across so many situations is a sign of a powerful idea.
Theories. In the 1860s James Clerk Maxwell's equations for electricity and magnetism predicted waves travelling at about $3 \times 10^8\ \text{m s}^{-1}$, the measured speed of light. He concluded that light is an electromagnetic wave. Heinrich Hertz produced and detected radio waves in the 1880s, confirming the prediction.
Observations. In 1895 Wilhelm Röntgen noticed a screen glowing near a covered discharge tube, and discovered X-rays by accident. Within months they were used in hospitals. He chose not to patent them.
ToK: questions to think about
- Can something wave if nothing is waving? Sound needs a medium, but light crosses empty space. For decades physicists assumed an invisible "ether" carried light, until experiments found no sign of it. What does a model mean when part of it turns out not to exist?
- How do instruments extend what we can know? Our eyes detect only a tiny band of the electromagnetic spectrum. Is knowledge gained only through instruments as reliable as what we see directly?
- Similar maths, same thing? Sound and light obey the same wave equation $v = f\lambda$. Does a shared mathematical description tell us they are fundamentally alike, or is it just a convenient analogy?
- Who should benefit from a discovery? Röntgen gave X-rays to the world for free. Should scientific discoveries be patented and owned, or shared openly?
1. What is a wave?
A travelling wave transfers energy from one place to another without any overall transfer of matter. When a wave passes along a rope, each piece of rope moves up and down about its own equilibrium position and ends up where it started. Only the disturbance, and the energy it carries, moves along.
- A single disturbance is a pulse. A repeated one, often a source moving in SHM, gives a continuous wave train.
- The material the wave travels through is the medium. Each particle of the medium oscillates with the same frequency as the source, but slightly later than the one before it.
- The wave speed depends on the medium, not on the source. Waves travel faster on a tighter rope, and sound travels faster in water than in air.
2. Transverse and longitudinal waves
- Transverse wave: the particles oscillate perpendicular to the direction the wave travels. Examples: waves on a string, and all electromagnetic waves. They have crests and troughs.
- Longitudinal wave: the particles oscillate parallel to the direction the wave travels. Examples: sound, and a push-pull pulse on a slinky. They have compressions (particles closer together, higher pressure) and rarefactions (particles further apart, lower pressure).
3. Wavelength, frequency and wave speed
- Displacement: how far a particle is from its equilibrium position.
- Amplitude: the maximum displacement.
- Wavelength $\lambda$: the shortest distance between two points moving in phase, for example crest to crest, or compression to compression.
- Time period $T$: the time for one complete oscillation of a particle. Frequency $f = 1/T$: oscillations per second.
- Wave speed $v$: how fast the wave pattern (and the energy) moves.
In one period, each particle completes one oscillation and the wave moves forward exactly one wavelength. So:
The frequency is set by the source and doesn't change when a wave enters a new medium. The speed is set by the medium. So if the speed changes, the wavelength must change too.
Worked example: the same note in air and water
A tuning fork produces sound of frequency 440 Hz. Find the wavelength in air (speed 340 $\text{m s}^{-1}$) and in water (speed 1500 $\text{m s}^{-1}$).
In air: $\lambda = \dfrac{v}{f} = \dfrac{340}{440} = 0.77$ m. In water: $\lambda = \dfrac{1500}{440} = 3.4$ m.
The frequency (and so the pitch) is the same in both. Only the speed and wavelength change.
4. Two kinds of wave graph
Wave graphs look identical, so always read the horizontal axis first:
Which way is a particle moving? On a displacement–distance graph, imagine the whole wave shifted slightly in its direction of travel. A particle's new displacement shows whether it is moving up or down. Particles at crests and troughs are momentarily at rest. Particles at the equilibrium position move fastest.
For a longitudinal wave, the same graphs are used, but "displacement" means displacement along the direction of travel (positive meaning, say, to the right). A compression is centred where particles on both sides are displaced towards it.
Worked example: reading both graphs
For a wave on a string, a displacement–distance graph shows a wavelength of 0.80 m. A displacement–time graph for one point on the string shows a period of 0.20 s. Find the frequency and the wave speed.
$f = \dfrac{1}{T} = \dfrac{1}{0.20} = 5.0$ Hz. $v = f\lambda = 5.0 \times 0.80 = 4.0\ \text{m s}^{-1}$.
5. Sound waves
- Sound is a longitudinal mechanical wave. A vibrating source, such as a loudspeaker cone, pushes on the air to make compressions, and pulls back to make rarefactions.
- It needs a medium. A ringing bell in a jar becomes silent as the air is pumped out, though you can still see it, because light needs no medium.
- Its speed is about 340 $\text{m s}^{-1}$ in air at room temperature, faster in liquids (about 1500 $\text{m s}^{-1}$ in water), and faster still in solids.
- Humans hear roughly 20 Hz to 20 kHz. Higher frequencies (ultrasound) are used in medical scanning and sonar.
- Pitch depends on frequency, and loudness on amplitude.
6. Electromagnetic waves
- An electromagnetic (EM) wave is made of oscillating electric and magnetic fields, at right angles to each other and to the direction of travel. So EM waves are transverse.
- They are produced by accelerating charges, and they need no medium. In a vacuum, all EM waves travel at the same speed, $c = 3.00 \times 10^{8}\ \text{m s}^{-1}$, so $c = f\lambda$.
- The family of EM waves is the electromagnetic spectrum. The data booklet shows the approximate order of magnitude of each wavelength range:
| Region | Approximate wavelength | Example use |
|---|---|---|
| Gamma rays | below about $10^{-11}$ m | cancer treatment, sterilising equipment |
| X-rays | $10^{-11}$ to $10^{-8}$ m | medical imaging, airport scanners |
| Ultraviolet | $10^{-8}$ to $4 \times 10^{-7}$ m | sterilising water, detecting forged banknotes |
| Visible light | 400 nm (violet) to 700 nm (red) | vision, optical fibres |
| Infrared | $7 \times 10^{-7}$ to $10^{-3}$ m | thermal cameras, remote controls |
| Microwaves | $10^{-3}$ to about $10^{-1}$ m | mobile phones, wifi, microwave ovens |
| Radio waves | above about $10^{-1}$ m | broadcasting, communication |
Going from radio to gamma, the wavelength decreases, the frequency increases, and so does the energy each photon carries (E.2). The boundaries between regions aren't sharp.
Worked example: a radio station
An FM radio station broadcasts at 100 MHz. What is the wavelength?
$\lambda = \dfrac{c}{f} = \dfrac{3.00\times10^{8}}{100\times10^{6}} = 3.0$ m. That's in the radio region, as expected.
7. Mechanical and electromagnetic waves compared
| Mechanical waves (sound, water, string) | Electromagnetic waves | |
|---|---|---|
| What oscillates | particles of the medium | electric and magnetic fields |
| Medium needed? | yes | no, they travel through a vacuum |
| Type | transverse or longitudinal | always transverse |
| Speed | depends on the medium; far slower than light | $c = 3.00\times10^8\ \text{m s}^{-1}$ in a vacuum (slower in materials) |
| What they share | transfer energy without net transfer of matter; obey $v = f\lambda$; reflect, refract, diffract and interfere (C.3) | |
Link to B.1: as a wave spreads out, its energy is spread over a larger area. The intensity (power per unit area, W m$^{-2}$) from a point source falls with the square of the distance: $I = \dfrac{P}{4\pi r^2}$. Intensity is also proportional to the square of the amplitude, $I \propto A^2$.
8. Common mistakes
- Thinking the medium travels with the wave. Particles only oscillate about fixed positions. Energy is transferred, matter isn't.
- Reading a period off a displacement–distance graph, or a wavelength off a displacement–time graph. Check the axis.
- Thinking frequency changes when a wave enters a new medium. Frequency stays the same; speed and wavelength change.
- Saying a higher frequency makes a wave travel faster. In a given medium, a higher frequency means a shorter wavelength, at the same speed.
- Calling sound transverse, or saying that EM waves need a medium.
- Measuring the wavelength from a crest to the next trough. That's only half a wavelength.
9. Check your understanding
A duck floats on a pond as waves pass. Why doesn't it move along with the waves?
Water waves transfer energy, not water. The duck, like the water around it, moves up and down (actually in small circles) about the same position.
Light passes from air into glass, where it travels more slowly. What happens to its frequency and wavelength?
The frequency stays the same, because it's set by the source. Since $v = f\lambda$ and $v$ falls, the wavelength gets shorter.
During a storm you see lightning 3.0 s before you hear the thunder. Roughly how far away is the storm?
Light arrives almost instantly, so the delay is the travel time of the sound: $340 \times 3.0 \approx 1000$ m, about 1 km.
What is the frequency of green light of wavelength 530 nm?
$f = \dfrac{c}{\lambda} = \dfrac{3.00\times10^{8}}{530\times10^{-9}} = 5.7 \times 10^{14}$ Hz.