Waves

A wave 波 carries energy 能量 from place to place without carrying matter. For example, a water wave makes a floating cork bob up and down, but the cork does not travel along with the wave.

Describing a wave

• wavefront 波前: a line joining points on a wave that move together (for example, the top of one ripple)
• wavelength 波长 ($\lambda$): the distance between two neighbouring wavefronts (one full wave)
• frequency 频率 ($f$): the number of waves passing a point each second, measured in hertz (Hz)
• crest 波峰 (top) and trough 波谷 (bottom)
• amplitude 振幅: the largest distance a point moves from its rest position
• wave speed 波速 ($v$): how fast a wavefront travels

The parts of a wave: wavelength, amplitude, crest and trough

These are linked by the wave equation:

The wavelength is the distance between two crests; the amplitude is the height from the rest position to a crest

Worked example. A sound wave travels at $340\ \text{m/s}$ and has a frequency of $170\ \text{Hz}$. Find its wavelength.

Rearranging $v = f\lambda$ gives $\lambda = \dfrac{v}{f}$:

Two types of wave

• In a transverse wave 横波 the particles vibrate at right angles (90°) to the direction the wave travels (its propagation 传播). Light and water waves are transverse.
• In a longitudinal wave 纵波 the particles vibrate along the same direction as the wave travels. Sound is longitudinal.

In a transverse wave the particles vibrate across the travel direction; in a longitudinal wave they vibrate along it, making compressions and rarefactions

Wave behaviour

All waves can show three behaviours, which you can see in a ripple tank 波纹水槽:

• reflection 反射: the wave bounces off a surface.
• refraction 折射: the wave changes speed (and usually direction) when it enters a different material or depth.
• diffraction 衍射: the wave spreads out after passing through a gap or around an edge. The spreading is greatest when the gap is about the same size as the wavelength.

Plane waves passing through a narrow gap spread out into curved waves — diffraction is strongest when the gap is about one wavelength wide

Two dippers in a ripple tank make circular water waves that overlap — a quick way to study how waves behave

Reflection

When light hits a mirror, it reflects. We measure angles from the normal 法线 (a line at 90° to the surface).

The law of reflection: the angle of incidence 入射角 equals the angle of reflection 反射角.

A plane mirror 平面镜 forms an image that is the same size as the object, the same distance behind the mirror, and virtual (it cannot be caught on a screen).

At a plane mirror the angle of incidence $i$ equals the angle of reflection $r$, both measured from the normal

Refraction

When light passes from one material into another, it changes speed and bends. The angle in the second material is the angle of refraction 折射角.

Going from air into glass the light slows down and bends towards the normal, so $i > r$

The refractive index 折射率 $n$ compares the speed of light in the two materials:

Worked example. A ray of light passes from air into glass. The angle of incidence is 60° and the angle of refraction is 35°. Find the refractive index of the glass. (Take $\sin i = 0.87$ and $\sin r = 0.57$.)

When light tries to leave glass or water and the angle is too large, it cannot get out and instead reflects completely. This is total internal reflection 全反射. It happens when the angle inside is bigger than the critical angle 临界角 $c$:

Below the critical angle the light refracts out; above it the light is totally internally reflected

This effect is used in optical fibres 光纤, thin glass threads that carry light signals for telephones and the internet.

Total internal reflection keeps light bouncing along each fibre until it shines out at the tip

Lenses

A converging lens 凸透镜 (fat in the middle) bends parallel rays inwards to a point called the principal focus 焦点. The distance from the lens to this point is the focal length 焦距. A diverging lens 凹透镜 (thin in the middle) spreads rays out.

A converging lens bends parallel rays to meet at the principal focus; the focal length $f$ is the lens-to-focus distance

A converging lens can form a real image 实像 (rays really meet; can be shown on a screen) or, when the object is very close, a virtual image 虚像 (rays only seem to come from it). Used close to the eye, a converging lens is a magnifying glass 放大镜.

Held away from the eye, a converging lens forms a real image — here the houses behind it appear upside down

Dispersion

A glass prism 棱镜 splits white light into the colours of the spectrum 光谱. This splitting is called dispersion 色散.

A prism refracts violet light most and red light least, so white light spreads into a spectrum

The order of colours is red, orange, yellow, green, blue, indigo, violet. Red light has the longest wavelength and the lowest frequency; violet is the opposite. Light of a single frequency (one pure colour) is monochromatic 单色.

White light entering a real glass prism spreads into the full spectrum of colours

The electromagnetic spectrum is a family of waves that all travel at the same high speed in a vacuum 真空, $3.0 \times 10^8\ \text{m/s}$. In order from longest wavelength (lowest frequency) to shortest:

As you go from radio waves to gamma rays, the frequency rises, the wavelength falls, and the energy (and danger) rises.

From radio waves to gamma rays the wavelength falls while the frequency and energy rise

Digital and analogue signals

Radio waves and microwaves carry information as signals, and a signal is either analogue 模拟 or digital 数字. An analogue signal varies continuously and can take any value; a digital signal is a stream of just two values, on or off (1 or 0).

Modern communication prefers digital signalling for two reasons the exam asks for:

• a higher rate of data transmission;
• a greater range: a weak digital signal can be regenerated back to a clean 1/0 exactly, because only two levels have to be told apart, so noise picked up along the way is removed. An analogue signal cannot be cleaned up this way, so its noise builds up.

Sound is made by a vibrating object. It is a longitudinal wave: the air is squeezed into a compression 压缩 (particles close together) and stretched into a rarefaction 稀疏 (particles far apart).

• Sound needs a medium 介质 (a material) to travel through, so it cannot travel through a vacuum.
• Humans can hear frequencies from about $20\ \text{Hz}$ to $20\,000\ \text{Hz}$.
• Sound travels much slower than light (about $340\ \text{m/s}$ in air), and faster in liquids and solids than in gases.
• A larger amplitude makes a louder sound (greater loudness 响度); a higher frequency makes a higher pitch 音调. So a dolphin's rapid clicks are high-pitched, and a hard drum-hit (big vibration) is loud.

A reflected sound is an echo 回声. You can find the speed of sound by timing an echo over a known distance.

Worked example. A student stands 85 m from a large wall and claps once. The echo returns 0.50 s later. Find the speed of sound.

The sound travels to the wall and back, a distance of $2 \times 85 = 170\ \text{m}$, in $0.50\ \text{s}$:

Ultrasound 超声波 is sound above $20\,000\ \text{Hz}$ – too high for humans to hear. Its uses rely on reflection: sonar measures water depth, medical scanning images a baby or soft tissue, and non-destructive testing finds cracks inside metal without cutting it open. Like an echo, an ultrasound pulse travels there and back, so halve the total distance to reach the object.

Worked example. A sonar pulse returns from the seabed $0.10\ \text{s}$ after it is sent; sound travels at $1500\ \text{m/s}$ in water. The pulse covers $s = v\times t = 1500\times0.10 = 150\ \text{m}$ there and back, so the sea is $150/2 = 75\ \text{m}$ deep.

• Learn $v = f\lambda$ and be ready to rearrange it. Measure all angles of incidence, reflection and refraction from the normal (the line at 90° to the surface), never from the surface itself.
• Transverse waves vibrate across the direction of travel (light, water); longitudinal waves vibrate along it (sound). Sound needs a medium, so it cannot cross a vacuum; light can.
• Total internal reflection happens only when light travels from a denser material to a less dense one and the angle inside is bigger than the critical angle.
• Know the electromagnetic spectrum in order — radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. All travel at the same speed in a vacuum; frequency and energy rise from radio to gamma.
• Going into a denser material (air → glass) light slows down and bends towards the normal; leaving it (glass → air) light speeds up and bends away.