波

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. You can make a wave on a rope by shaking one end up and down, or on a long spring by pushing one end back and forth. In a ripple tank 波纹水槽 (a shallow tray of water with a light above it) you can watch water waves on a screen below.

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 (also called the peak 峰, the 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

These are linked by the wave equation:

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

Worked example (ripple tank). In a ripple tank, $12$ wavefronts pass a point in $4.0\ \text{s}$. On the screen, the distance from the 1st wavefront to the 6th wavefront is $10\ \text{cm}$. Find the frequency, the wavelength and the wave speed.

From the 1st to the 6th wavefront there are $5$ gaps, not $6$, so

Two types of wave

• In a transverse wave 横波 the oscillations are at right angles (90°) to the direction the wave travels (its propagation 传播). Light, water waves and seismic S-waves (the sideways shaking in an earthquake) are transverse.
• In a longitudinal wave 纵波 the particles vibrate along the same direction as the wave travels. Sound and seismic P-waves are longitudinal.

Wave behaviour

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

• 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.

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.

The incident ray, reflected ray and normal lie in one plane.

Worked example. A ray makes $30^{\circ}$ with the mirror surface. Draw the normal at the point of incidence. The normal makes $90^{\circ}$ with the surface, so the angle of incidence is the remaining angle:

Apply the law of reflection:

Draw the reflected ray away from the mirror on the opposite side of the normal. It makes $60^{\circ}$ with the normal and $30^{\circ}$ with the mirror surface. Do not use the surface angle as the angle of incidence.

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).

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.

Seeing refraction. Shine a narrow ray from a ray box 光线盒 into a rectangular glass block standing on paper. Mark the ray going in and the ray coming out, remove the block, and join the marks to draw the ray inside the glass. Draw the normal where the ray enters and measure the angle of incidence and the angle of refraction with a protractor. Repeat with a semicircular block or a prism: the ray always bends towards the normal on entering the glass and away from the normal on leaving.

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 $\sin r = 0.57$.)

The refractive index is also the speed of light in air divided by its speed in the glass. Light travels at $3.0 \times 10^8\ \text{m/s}$ in air, so in this glass it travels at $\dfrac{3.0 \times 10^8}{1.5} = 2.0 \times 10^8\ \text{m/s}$.

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 $c$:

Worked example. Glass has a refractive index of $1.5$. (a) Find the critical angle. (b) A ray inside the glass meets the glass–air surface at $50^{\circ}$ to the normal. What happens? (c) Another ray meets the surface at $30^{\circ}$ to the normal. Find the angle at which it leaves.

(a) $\sin c = \dfrac{1}{n} = \dfrac{1}{1.5} = 0.667$, so $c = 42^{\circ}$.

(b) $50^{\circ}$ is bigger than $42^{\circ}$, so the ray is totally internally reflected. It reflects back into the glass at $50^{\circ}$ to the normal.

(c) $30^{\circ}$ is smaller than $42^{\circ}$, so the ray refracts out into the air. The angle in the air is the larger angle, so use $n = \dfrac{\sin(\text{angle in air})}{\sin(\text{angle in glass})}$:

The ray bends away from the normal ($49^{\circ} > 30^{\circ}$), as it should when light leaves glass.

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

Lenses

A converging lens 凸透镜 (fat in the middle) brings rays parallel to its principal axis to a point called the principal focus 焦点. The principal axis 主光轴 is the straight line through the centre of the lens at right angles to it; the principal focus lies on this axis. The distance from the centre of the lens to the principal focus is the focal length 焦距. A diverging lens 凹透镜 (thin in the middle) spreads parallel rays out, as if they came from a focus on the incoming side.

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.

Describe an image with three pairs of words: enlarged 放大的 or diminished 缩小的 (or the same size), upright 正立 or inverted 倒立 (upside down), and real or virtual. An object far beyond the focus gives a diminished, inverted, real image (as in a camera). An object between the focus and the lens gives an enlarged, upright, virtual image (a magnifying glass).

Drawing a ray diagram. Draw the lens as a vertical line on the principal axis and mark a principal focus $F$ on each side. From the top of the object draw two rays:

1. a ray parallel to the principal axis, which leaves the lens through the principal focus on the far side;
2. a ray through the centre of the lens, which goes straight on without bending.

The top of the image is where these two rays meet. If the rays spread apart after the lens, extend them backwards with dashed lines: they meet at a virtual image on the same side as the object.

Worked example (ray diagrams). A converging lens has a focal length of $3.0\ \text{cm}$.

(a) An object $2.0\ \text{cm}$ tall stands $9.0\ \text{cm}$ from the lens (left diagram). The two rays meet $4.5\ \text{cm}$ behind the lens, $1.0\ \text{cm}$ below the axis. The image is real (the rays really meet, so it can be caught on a screen), inverted and diminished ($1.0\ \text{cm}$ tall compared with $2.0\ \text{cm}$).

(b) An object $1.0\ \text{cm}$ tall stands $2.0\ \text{cm}$ from the lens, inside the focal length (right diagram). After the lens the rays spread apart and never meet. Extended backwards, they meet $6.0\ \text{cm}$ from the lens on the same side as the object, $3.0\ \text{cm}$ above the axis. The image is virtual, upright and enlarged (three times as tall). This is how a magnifying glass works.

Draw the diagram to scale on squared paper, then read the image distance and height from the grid. A sharp pencil and a ruler matter: a small error in the angle of a ray moves the image a long way.

Correcting sight. A person with short-sightedness 近视 sees near things clearly but distant things are blurred: the eye focuses the light in front of the retina 视网膜 (the back of the eye). A diverging lens in front of the eye spreads the rays out first, so they focus on the retina. A person with long-sightedness 远视 sees distant things clearly but near things are blurred: the eye focuses the light behind the retina. A converging lens brings the rays together a little before they enter the eye.

Dispersion

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

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.

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. The speed of electromagnetic waves in air is almost the same as in a vacuum.

Worked example. A radio station broadcasts at $100\ \text{MHz}$ ($1.0 \times 10^8\ \text{Hz}$). Find the wavelength of its radio waves.

Satellites. Communication with artificial satellites 人造卫星 mainly uses microwaves, because they pass through the atmosphere. Some satellite phones use satellites in low orbit 低轨道, close to the Earth. Other satellite phones, and direct broadcast satellite television, use satellites in geostationary orbit 地球静止轨道, which stay above the same point on the Earth.

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). Sound can be sent as either kind of signal: a microphone gives an analogue signal, which can be turned into a digital one.

Digital communication can support a higher rate of data transmission and a greater range due to accurate signal regeneration. These are separate benefits; the actual rate depends on the equipment and channel.

Worked example. A simplified receiver reads levels below 0.5 as 0 and above 0.5 as 1. It receives 0.12, 0.86, 0.73 and 0.19. The recovered message is 0110. A repeater sends fresh signals at clean levels 0, 1, 1 and 0, removing the small changes in level. Repeating this along a link supports increased range while the two levels can still be distinguished.

Regeneration is not guaranteed to repair every damaged message. If the third level becomes 0.41, this receiver reads 0 instead of the original 1. Excessive noise can cause a wrong digit. For an analogue signal, a received value such as 0.73 could be the true value or a different value changed by noise. There are no two fixed levels to restore; simply amplifying the signal also amplifies its noise.

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.

Measuring the speed of sound. Two students stand a known distance apart, for example 500 m, measured with a tape or a trundle wheel. One bangs two blocks of wood together. The other starts a stopwatch when they see the blocks hit and stops it when they hear the bang. Light arrives almost at once, so the time measured is the time the sound takes, and speed = distance ÷ time. Repeat and take an average to reduce the reaction-time error. The echo method needs one person: clap at a known distance from a large wall and time the reflected sound, remembering that the sound travels there and back.

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 material with a higher refractive index to one with a lower refractive index (for example glass → air) 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 an optically denser material (higher refractive index, air → glass) light slows down and bends towards the normal; leaving it (glass → air) light speeds up and bends away.

Common mistakes

• Counting wavefronts instead of gaps. From the 1st to the 6th wavefront is 5 wavelengths.
• Measuring angles from the surface. Angles of incidence, reflection and refraction are measured from the normal.
• Saying total internal reflection happens when light goes from air into glass. It only happens inside the denser material (glass or water), at an angle bigger than the critical angle.
• Drawing the ray through the centre of a lens with a bend in it. That ray goes straight on.
• Calling an image "virtual" because it is small or upside down. Real means the rays really meet (a screen can show it); virtual means they only seem to come from it.
• Forgetting to halve the distance in an echo or sonar question. The sound travels there and back.
• Leaving MHz or kHz in $v = f\lambda$. Change to Hz first: $1\ \text{MHz} = 1\,000\,000\ \text{Hz}$.