Topic 3: Waves

Waves transfer energy from place to place without transferring matter. This topic looks at how we describe waves, how light bends and reflects, how lenses form images, and how the same wave ideas explain everything from radio signals to ultrasound scans.

General wave properties and the wave equation

Every wave can be described by a few quantities. The wavelength is the distance between two neighbouring points that are in step, such as crest to crest, measured in metres (m). The frequency is the number of complete waves passing a point each second, measured in hertz (Hz). The amplitude is the largest distance a point moves from its rest position, and the period is the time for one complete wave. Wave speed (in m/s) links frequency and wavelength through the wave equation: speed = frequency x wavelength, often written v = f x wavelength. Worked example: a water wave has a frequency of 5 Hz and a wavelength of 0.4 m. Its speed is v = f x wavelength = 5 x 0.4 = 2 m/s. You can rearrange the same equation to find frequency (f = v divided by wavelength) or wavelength (wavelength = v divided by f).

Transverse and longitudinal waves

Waves come in two main kinds. In a transverse wave the vibrations are at right angles (perpendicular) to the direction the wave travels. Light, all electromagnetic waves, and ripples on water are transverse. These waves have crests and troughs. In a longitudinal wave the vibrations are along the same line as the direction of travel. Sound is the key example. Instead of crests and troughs, a longitudinal wave has compressions, where particles are squeezed close together, and rarefactions, where they are spread apart. Both types transfer energy without moving the medium permanently; the particles only vibrate about fixed positions.

Reflection of light

When light hits a flat mirror it bounces off, or reflects. We measure angles from the normal, an imaginary line drawn at 90 degrees to the surface where the ray strikes. The law of reflection states that the angle of incidence equals the angle of reflection, both measured from the normal. A plane (flat) mirror forms an image that is the same size as the object, upright, and the same distance behind the mirror as the object is in front. The image is virtual, meaning the rays only appear to come from it and cannot be caught on a screen. The image is also laterally inverted, so left and right appear swapped.

Refraction and refractive index

Refraction is the bending of light as it passes from one transparent material into another, caused by a change in the speed of light. When light slows down, for example going from air into glass, it bends towards the normal. When it speeds up, going from glass into air, it bends away from the normal. The refractive index n of a material tells us how much it slows light: n = sin(i) divided by sin(r), where i is the angle in air and r is the angle in the material. This relationship is Snell's law. A larger refractive index means more bending. For glass n is roughly 1.5, and for water about 1.33. Worked idea: if a ray enters glass at 30 degrees and refracts to 19 degrees, n = sin(30) divided by sin(19), which is about 1.5.

Total internal reflection and the critical angle

When light travels from a denser material (like glass) towards a less dense one (like air), it bends away from the normal. As the angle of incidence increases, the refracted ray bends more until it travels along the surface. The angle of incidence that causes this is called the critical angle, c. Beyond the critical angle no light escapes and it is all reflected back inside; this is total internal reflection. Two conditions are needed: the light must be going from the denser to the less dense medium, and the angle of incidence must be greater than the critical angle. The critical angle is linked to refractive index by n = 1 divided by sin(c). Total internal reflection is used in optical fibres, which carry telephone and internet signals as pulses of light, and in prisms inside binoculars and periscopes.

Thin converging lenses and ray diagrams

A converging (convex) lens is thicker in the middle and bends parallel rays of light so they meet at a point called the principal focus. The distance from the lens to this point is the focal length. To draw a ray diagram, use two simple rays from the top of the object: one ray travelling parallel to the axis that bends to pass through the principal focus, and one ray passing straight through the centre of the lens without bending. Where these rays cross is the top of the image. If the object is beyond the focal length, the image is real (can be caught on a screen) and inverted; this is how cameras and the eye work. If the object is closer than the focal length, the image is virtual, upright and magnified, which is how a magnifying glass works.

Dispersion of light

White light is a mixture of all the colours. When it passes through a triangular glass prism, the light refracts and the different colours bend by different amounts because each colour travels at a slightly different speed in glass. This spreading out is called dispersion, and it produces a spectrum. Red light bends the least and violet light bends the most, giving the familiar order red, orange, yellow, green, blue, indigo, violet. The same effect in raindrops creates a rainbow. Dispersion shows that the refractive index of a material depends slightly on the colour, or wavelength, of the light.

The electromagnetic spectrum and its uses

The electromagnetic (EM) spectrum is a family of transverse waves that all travel at the same speed in a vacuum, about 3 x 10^8 m/s. In order of increasing frequency (and decreasing wavelength) the regions are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. Radio waves are used for broadcasting and communications. Microwaves are used in cooking and mobile phone and satellite links. Infrared is used in remote controls, thermal imaging and heating. Visible light lets us see and is used in photography and fibre optics. Ultraviolet is used in security marking and sterilising water. X-rays produce medical images of bones, and gamma rays are used to sterilise equipment and treat cancer. High frequency waves like ultraviolet, X-rays and gamma rays carry more energy and can damage living cells.

Sound waves and ultrasound

Sound is a longitudinal wave produced by a vibrating object that makes the surrounding air compress and stretch. Because it needs particles to travel, sound cannot pass through a vacuum. The speed of sound in air is roughly 330 to 340 m/s, much slower than light, which is why thunder is heard after lightning is seen. The pitch of a sound depends on its frequency, and the loudness depends on its amplitude. Humans can typically hear frequencies from about 20 Hz to 20000 Hz. Sound above 20000 Hz is called ultrasound. Ultrasound is reflected at boundaries between materials, so by sending pulses and timing the echoes we can measure depth or build images. It is widely used in prenatal scans, cleaning delicate objects, and detecting cracks inside metals.