Waves

Describing waves

A wave is a travelling oscillation. It carries energy forward while the matter it passes through simply vibrates in place.

Energy, not matter

A wave is described by its wavefront, wavelength, amplitude, frequency, the crest and trough, and its wave speed. As a wave passes, each point in the material oscillates about a fixed position; the energy moves on, but the matter stays put.

Transverse and longitudinal

Waves come in two kinds. In a transverse wave the vibration is at right angles to the direction of travel — light and other electromagnetic waves, water surface ripples and seismic S-waves. In a longitudinal wave the vibration is along the direction of travel, producing compressions and rarefactions — sound and seismic P-waves.

The wave equation

Wave speed, frequency and wavelength are tied together by one equation that holds for every wave, transverse or longitudinal — from radio to a ripple in a tank.

Speed, frequency and wavelength

A wave travels one wavelength forward in the time it takes one full oscillation. Multiplying how many waves pass each second (frequency) by the length of each (wavelength) gives the speed: v = fλ.

Worked example: a sound wave of frequency 1700 Hz travelling at 340 m/s has wavelength λ = v / f = 340 / 1700 = 0.20 m.

Worked example: ripples of wavelength 0.30 m passing a post at 0.50 waves per second travel at v = fλ = 0.50 × 0.30 = 0.15 m/s.

Wave behaviour: reflection, refraction, diffraction

Every wave can bounce, bend and spread. These three behaviours — reflection, refraction and diffraction — are seen most clearly with water waves in a ripple tank, and they govern light and sound just the same.

Reflect, refract, diffract

A ripple tank shows all three. Waves reflect off a barrier with the angle of incidence equal to the angle of reflection. They refract — change speed and direction — when the water depth changes, slowing and bending in shallower water while the frequency stays fixed. And they diffract, spreading out after passing through a gap or around an edge.

ExtendedHow much spreading?

The amount of diffraction through a gap depends on the gap size compared with the wavelength: the spreading is most pronounced when the gap is roughly equal to the wavelength, and becomes slight when the gap is much larger. At an edge, longer wavelengths diffract — bend around the edge — more than shorter ones.

Reflection of light

Light is a wave, so it obeys the law of reflection exactly. A plane mirror turns that simple rule into the familiar image staring back at you.

The law of reflection

When light strikes a mirror, the angle of incidence equals the angle of reflection, both measured from the normal. A plane mirror forms an image that is the same size as the object, as far behind the mirror as the object is in front, virtual, and laterally inverted (left and right swapped).

The image in a plane mirror

The image is virtual because no light actually passes through it — the rays only appear to come from behind the mirror. It cannot be projected onto a screen, unlike a real image.

Refraction and total internal reflection

Light changes speed when it enters glass or water, and that change of speed bends its path. The same effect makes a straw look broken at the surface of a drink.

Bending at a boundary

As light passes from air into glass it slows down and bends towards the normal; passing back into air it speeds up and bends away. Through a parallel-sided block the emerging ray is parallel to the original, just shifted sideways.

ExtendedRefractive index

The refractive index n measures how strongly a material bends light, and equals the ratio of the wave’s speed in the two regions: n = sin i / sin r, where i is the angle in air and r the angle in the medium.

Worked example: light entering glass at an angle of incidence of 40° refracts to 25°, so n = sin 40° / sin 25° = 0.643 / 0.423 = 1.5 — typical for glass.

When light cannot escape

As the angle of incidence inside glass increases, the refracted ray bends further from the normal until, at the critical angle, it travels along the surface. Beyond that angle no light escapes — it all reflects back into the glass. This is total internal reflection.

The critical angle is linked to the refractive index by n = 1 / sin c. In an optical fibre, light strikes the walls beyond the critical angle and is totally internally reflected over and over, travelling along the fibre with almost no loss — the basis of modern telecommunications.

Lenses and images

A lens bends light to gather it or spread it. A converging lens can collect rays from an object into a sharp, real image — the principle behind a camera and a projector.

Converging and diverging lenses

A converging (convex) lens brings parallel rays together at the principal focus; a diverging (concave) lens spreads them so they appear to come from a focus. The distance from lens to focus is the focal length. To locate the image, two rays are enough: one parallel to the axis that then passes through the far focus, and one straight through the centre of the lens.

Describing the image

As the object moves closer to a converging lens the image changes: far away it is real, inverted and diminished; closer it grows; and once the object is inside the focal length, the image becomes virtual, upright and enlarged.

This last case is the magnifying glass. The rays leaving the lens diverge, so they never meet; tracing them back gives a magnified virtual image behind the object.

ExtendedCorrecting sight

A converging lens corrects long-sightedness, bringing the focus forward onto the retina; a diverging lens corrects short-sightedness, pushing the focus back. The same lens types appear in cameras, projectors and magnifying glasses.

Dispersion of light

White light is a mixture of every colour. A glass prism bends each colour by a slightly different amount, fanning them out into the familiar spectrum of the rainbow.

The visible spectrum

When white light passes through a prism, each colour slows by a slightly different amount in the glass and so refracts by a different angle. Red bends least and violet most, spreading the light into seven colours: red, orange, yellow, green, blue, indigo, violet — in order of decreasing wavelength and increasing frequency.

The electromagnetic spectrum

Visible light is one thin slice of a far wider family of waves. From radio to gamma, all are electromagnetic, all are transverse, and all travel through a vacuum at the same enormous speed.

Seven regions in order

In order of increasing frequency and decreasing wavelength, the regions are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays.

Uses across the spectrum

Radio for broadcasts and communications; microwaves for satellite and mobile-phone links and cooking; infrared for thermal imaging, remote controls and grills; visible light for sight and photography; ultraviolet for sterilising and security marking; X-rays for medical and security imaging; and gamma rays for sterilising equipment and treating cancer.

Harmful effects

Excessive exposure carries risks that rise with frequency: microwaves heat the water in body tissue; infrared causes skin burns; ultraviolet damages skin cells (skin cancer) and eyes; and X-rays and gamma rays, being ionising, can mutate cells and cause cancer.

ExtendedCommunication and digital signals

Communication with orbiting satellites uses microwaves, because they pass easily through the atmosphere; radio waves carry broadcasting over long distances around the Earth.

A signal can be analogue (continuously varying) or digital (a series of on/off pulses). Digital transmission can be cleaned up and regenerated along the way, so quality is preserved over long distances; more information can be sent in a given time; and errors can be detected and corrected. This is why modern telephone, internet and broadcast systems are digital.

Sound

Sound is a longitudinal wave made by something vibrating. It needs particles to travel through, which is why — unlike light — sound cannot cross the vacuum of space.

A longitudinal wave

A vibrating source — a loudspeaker cone, a string, vocal cords — pushes and pulls the air, sending out a longitudinal wave of compressions (particles squeezed together) and rarefactions (particles spread apart). Because it relies on particles, sound travels through solids, liquids and gases but not through a vacuum.

Sound travels faster in solids than in liquids, and faster in liquids than in gases. The more closely packed the particles, the more quickly each one passes the vibration to the next.

Pitch, loudness and echoes

A higher frequency gives a higher pitch; a larger amplitude gives a greater loudness. The healthy human audible range runs from 20 Hz to 20 000 Hz. A reflected sound is an echo, which gives a way to measure the speed of sound — time an echo over a known distance and use distance ÷ time, remembering the sound travels to the surface and back.

ExtendedUltrasound and its uses

Ultrasound is used for medical scanning (imaging an unborn baby), for sonar — measuring the depth of water or locating shoals of fish by timing an echo — and for non-destructive testing of materials for hidden cracks. A pulse is sent out and the returning echo timed; the distance is found from the speed of sound, halved because the pulse travels there and back.