From ripples on a pond to radio and earthquakes: types of waves, wave speed, reflection, refraction, diffraction, interference and standing waves, with diagrams, animations and worked examples.
Drop a pebble into a still pond and a ring of ripples spreads outwards. A floating leaf bobs up and down as the ripples pass, but it does not travel with them. That simple picture holds the key idea of this whole article: a wave carries energy from place to place without carrying the material along with it. Waves bring us sound, light, radio and mobile signals, X-rays, earthquakes and tsunamis. The same few rules describe all of them, from ripples on a pond to the light of distant stars. This article explains the types of wave, how to describe and calculate with them, and how waves reflect, refract, diffract, interfere and form standing waves.
A wave is a disturbance that transfers energy from one place to another without transferring matter. A source (a vibrating object, an oscillating charge, a sudden movement of the ground) sets the first particles or fields into motion, and the disturbance is passed on from point to point.
There are two broad families:
The particles of a medium do not travel with a mechanical wave. They only vibrate about their equilibrium (rest) positions, passing the energy to their neighbours, as the bobbing leaf shows. A stadium "Mexican wave" is a good picture: each person only stands up and sits down, but the wave runs round the stadium.
Waves are classified by the direction in which the particles (or fields) vibrate compared with the direction in which the wave travels.
| Term | Meaning | Unit |
|---|---|---|
| Amplitude (A) | Maximum displacement from the rest position. It is related to the energy carried: bigger amplitude, more energy (loudness for sound, brightness for light). | m (for a rope or water wave) |
| Wavelength (λ) | Distance between two consecutive identical points (crest to crest, or compression to compression). | metre (m) |
| Frequency (f) | Number of complete waves passing a point each second (or the number of vibrations per second at the source). | hertz (Hz) |
| Period (T) | Time for one complete vibration: T = 1 ÷ f. | second (s) |
| Wave speed (v) | Distance travelled by a wave pattern each second. | m/s |
| Wavefront | A line joining points of the wave that are in the same state (for example all on one crest). The wave travels at right angles to its wavefronts. | — |
Two different graphs are used and they should not be confused. A graph of displacement against distance is a "snapshot" of the whole wave at one instant, from which you can read the wavelength. A graph of displacement against time shows one point on the wave as it vibrates, from which you read the period.
A signal generator produces a wave of frequency 50 Hz. Find its period. A trace on an oscilloscope shows one complete cycle across 4.0 divisions with the time base set to 2.0 ms per division. Find the period and the frequency of this wave.
(a) T = 1 ÷ f = 1 ÷ 50 = 0.020 s (20 ms).
(b) Period = 4.0 × 2.0 ms = 8.0 ms = 0.0080 s. Frequency f = 1 ÷ 0.0080 = 125 Hz.
In one period T the wave moves forward by one wavelength λ. The speed is distance ÷ time, so v = λ ÷ T. Since 1 ÷ T = f:
v = f × λ (speed = frequency × wavelength)
Two facts help in every problem: (1) the frequency is set by the source and does not change when the wave moves from one medium to another; (2) the speed is set by the medium, so a change of medium changes the speed and therefore the wavelength.
A ripple tank vibrator makes 5.0 waves per second, and the distance between successive crests is 2.0 cm. Find the speed of the ripples.
v = fλ = 5.0 × 2.0 cm = 10 cm/s = 0.10 m/s.
An FM radio station broadcasts at 100 MHz. Find the wavelength of the waves (c = 3.00 × 10⁸ m/s).
λ = v ÷ f = 3.00 × 10⁸ ÷ 1.00 × 10⁸ = 3.0 m.
What decides the speed in different media?
When a wave meets a barrier it can be reflected, bouncing back into the medium it came from. Reflection obeys the same law of reflection for every kind of wave: the angle of incidence equals the angle of reflection, both measured from the normal (the line at right angles to the surface). The speed, wavelength and frequency of the wave are unchanged by reflection. Examples are echoes of sound (see the article on sound), mirrors for light, and the reflection of radio waves from the ionosphere. A curved reflector, such as a satellite dish, focuses plane waves to a point, which is how dishes and telescopes collect weak signals. In a ripple tank, plane waves hitting a straight barrier bounce back as plane waves, and those hitting a curved barrier are focused or spread.
Refraction is the change of direction of a wave when its speed changes as it enters a different medium (or a region of different depth). Because the frequency stays constant while the speed changes, the wavelength changes too: a wave that slows down has a shorter wavelength. If the wave hits the boundary at an angle, the wavefronts pivot and the wave changes direction.
sin i ÷ sin r = v₁ ÷ v₂ = λ₁ ÷ λ₂
For light this gives the refractive index n = c ÷ v and Snell's law, described in the article on light.
Water waves of frequency 10 Hz travel from deep water, where the wavelength is 2.0 cm, into shallow water, where it is 1.2 cm. They strike the boundary at an angle of incidence of 40°. Find both speeds and the angle of refraction.
Speeds: v₁ = fλ₁ = 10 × 2.0 = 20 cm/s; v₂ = 10 × 1.2 = 12 cm/s.
sin r = sin i × (v₂ ÷ v₁) = sin 40° × (12 ÷ 20) = 0.6428 × 0.60 = 0.3857.
r = sin⁻¹(0.3857) ≈ 23°. The wave bends towards the normal as it slows down.
Refraction explains why a straw looks bent in water, why sound carries farther over a calm lake at dusk (the cooler air near the water slows the sound, bending it back down), and why ocean waves arrive almost parallel to a beach, however they were travelling out at sea.
Diffraction is the spreading of waves when they pass through a gap or round the edge of an obstacle. The Dutch scientist Christiaan Huygens explained it in 1678 by imagining that every point on a wavefront acts as a source of tiny circular wavelets; the new wavefront is the envelope of all of them. The amount of diffraction depends on the size of the gap compared with the wavelength:
Diffraction explains many everyday observations:
When two waves meet in the same place, the resulting displacement is the sum of the individual displacements. This is the principle of superposition. Afterwards each wave carries on as if the other had never been there.
For two coherent sources (sources with the same frequency and a constant phase difference), the result at any point depends on the path difference, the extra distance one wave has travelled compared with the other:
In 1801 the English scientist Thomas Young shone light through two narrow slits and saw a pattern of bright and dark fringes on a screen. This was strong evidence that light behaves as a wave. For slits a distance a apart, a screen at distance D, and light of wavelength λ, the fringe spacing is x = λD ÷ a (when a is much smaller than D).
Light of wavelength 600 nm passes through two slits 0.50 mm apart. The screen is 2.0 m away. Find the fringe spacing.
x = λD ÷ a = (600 × 10⁻⁹ × 2.0) ÷ (0.50 × 10⁻³) = 1.2 × 10⁻⁶ ÷ 5.0 × 10⁻⁴ = 2.4 × 10⁻³ m = 2.4 mm.
Interference is used in noise-cancelling headphones, which produce a sound wave in antiphase with the outside noise, and in anti-reflection coatings on lenses. It also explains the colours in soap films and oil slicks on water.
When a wave reflects from a fixed end and meets the next incoming wave, the two superpose and can form a standing (stationary) wave. A standing wave does not travel. It has points of zero amplitude called nodes and points of maximum amplitude called antinodes. Every point between two nodes vibrates in phase, and the distance between neighbouring nodes is λ ÷ 2.
A string of length L fixed at both ends can only vibrate in patterns that fit a whole number of half-wavelengths: L = n × λ ÷ 2. These patterns are the harmonics:
λₙ = 2L ÷ n and fₙ = n × v ÷ 2L = n × f₁ (n = 1, 2, 3, …)
The lowest frequency f₁ is the fundamental. Shorter, tighter or lighter strings give higher notes.
A guitar string 0.65 m long carries waves at 260 m/s. Find the frequency of the fundamental and of the second harmonic.
Fundamental: λ₁ = 2L = 1.30 m, so f₁ = v ÷ λ₁ = 260 ÷ 1.30 = 200 Hz.
Second harmonic: f₂ = 2f₁ = 400 Hz (λ₂ = 0.65 m).
Standing waves explain how musical instruments work: guitar and violin strings, air columns in flutes and organ pipes, and the vibrating skin of drums. They are also the reason for "hot spots" and "cold spots" in a microwave oven (which is why it has a turntable) and for the sound resonating in a bathroom or a bottle when you blow across the top.
A transverse wave can vibrate in many directions at right angles to its direction of travel. If the vibrations are limited to one plane, the wave is plane-polarised. Longitudinal waves cannot be polarised because they vibrate along the direction of travel, so the fact that light can be polarised proved that light is a transverse wave. A polarising filter, such as the material in polarised sunglasses, lets through only light vibrating in one direction. Because light reflected from a road or water surface is partly polarised horizontally, polarised lenses cut glare. Radio and TV aerials must be lined up with the polarisation of the transmitted signal.
EM waves are transverse waves that travel at c in a vacuum. They differ in frequency and wavelength, which determines how they interact with matter. All obey v = fλ with v = c.
| Region | Wavelength (approx.) | Frequency (approx.) | Examples of use | Hazard |
|---|---|---|---|---|
| Radio | > 1 m | < 3 × 10⁸ Hz | Broadcasting, communications | Very low at normal levels |
| Microwave | 1 mm–1 m | 3 × 10⁸–3 × 10¹¹ Hz | Cooking, mobile phones, radar, satellite | Internal heating at high intensity |
| Infrared | 0.7 µm–1 mm | 3 × 10¹¹–4 × 10¹⁴ Hz | Heating, remote controls, night vision | Skin burns |
| Visible light | 400–700 nm | 4.3 × 10¹⁴–7.5 × 10¹⁴ Hz | Seeing, photography, fibre optics | Eye damage from very bright sources |
| Ultraviolet | 10–400 nm | 7.5 × 10¹⁴–3 × 10¹⁶ Hz | Sterilising, security marks, vitamin D | Sunburn, skin cancer, eye damage |
| X-rays | 0.01–10 nm | 3 × 10¹⁶–3 × 10¹⁹ Hz | Medical imaging, security scanners | Ionising: can damage cells |
| Gamma rays | < 0.01 nm | > 3 × 10¹⁹ Hz | Radiotherapy, sterilising, tracers | Ionising: can damage cells |
Higher-frequency EM waves carry more energy per photon, which is why UV, X-rays and gamma rays can ionise atoms and harm living tissue, while radio waves cannot. The spectrum and its uses are described further in the articles on light and on atomic and nuclear physics.
The Doppler effect applies to all waves: if a source and observer move towards each other, the observed frequency is higher; if they move apart, it is lower. Radar speed guns, medical blood-flow scanners and the "red shift" of light from distant galaxies all use it (see the articles on sound and on gravity and space).
An earthquake sends out several types of wave through and over the Earth. P waves (primary) are longitudinal and travel fastest (about 6 km/s in the crust), so they arrive first. S waves (secondary) are transverse and slower, and they cannot travel through liquids. Because S waves from an earthquake do not reach the opposite side of the Earth, seismologists concluded that the Earth has a liquid outer core. Surface waves travel along the ground and cause most of the damage. Seismometers record all of these, and comparing the arrival times at several stations locates the earthquake.
A tsunami is a series of very long water waves, usually caused by an undersea earthquake, landslide or volcanic eruption. In the deep ocean its wavelength may be hundreds of kilometres, so it is only a small bump in the open sea, but it travels extremely fast because the speed of such long waves depends on the depth: v = √(g × d).
Find the speed of a tsunami (a) in ocean 4000 m deep and (b) in water 10 m deep near the coast. (g = 9.8 m/s²)
(a) v = √(9.8 × 4000) = √39 200 ≈ 198 m/s, about 710 km/h, similar to a jet airliner.
(b) v = √(9.8 × 10) = √98 ≈ 9.9 m/s, about 36 km/h.
As the wave slows near the coast, the energy it carries is squeezed into a shorter wavelength and the wave grows much taller, a process called shoaling. Many countries operate tsunami warning systems, and people near the coast should move inland or to high ground immediately after a strong earthquake or an official warning.
Try each question before opening the answer.
v = fλ = 4.0 × 3.0 = 12 m/s. T = 1 ÷ f = 1 ÷ 4.0 = 0.25 s.
λ = v ÷ f = 3.0 × 10⁸ ÷ 9.0 × 10⁷ ≈ 3.3 m.
Transverse: light (or a wave on a rope). Longitudinal: sound (or a compression wave in a spring).
f = v ÷ λ = 24 ÷ 8.0 = 3.0 Hz (unchanged). λ = 16 ÷ 3.0 ≈ 5.3 cm.
Sound waves have wavelengths similar to the size of a doorway, so they diffract strongly. Light has a wavelength thousands of times smaller than a doorway, so it hardly diffracts.
Path difference ÷ λ = 0.50 ÷ 0.20 = 2.5 = (2 + ½), so it is destructive.
f₁ = v ÷ 2L = 200 ÷ 1.0 = 200 Hz. f₃ = 3 × 200 = 600 Hz.
S waves are transverse and cannot travel through a liquid. The fact that they do not pass through shows that the outer core is liquid.
Easy demonstrations: a long slinky (or a long spring) to show transverse and longitudinal pulses and reflection; a rope held by two students to find how the wavelength changes with frequency; a ripple tank (or a shallow tray of water with an overhead lamp and white paper underneath) to show reflection, refraction over a submerged glass plate, and diffraction through gaps made from wooden blocks; a free oscilloscope or tone-generator app on a phone to measure frequency and period; a laser pointer (never directed at eyes) through a hair or fine slit to see diffraction. Standing waves can be shown with a vibrating string, a Kundt's tube or a simple water-filled tube. Numerical values are rounded; please check that they match the data booklet used by your examination board.
This article is original EDUSAMBAM writing. Definitions, constants and facts were checked against the sources below. Diagrams are original and were not copied from any source.
Note on the numbers: typical speeds and wavelength ranges are rounded values; boundaries between EM regions overlap. Please check numerical values against your own syllabus or exam board specification before relying on them.
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