Waves

Ripples spreading on water are progressive waves that carry energy outward.

Two waves of the same frequency, shifted by a phase difference

A wave 波 carries energy 能量 from one place to another without moving matter overall. The particles of the medium 介质 oscillate 振动 about fixed rest positions; only the disturbance (and its energy) propagates 传播. Examples: a transverse wave 横波 on a rope, a longitudinal wave 纵波 on a slinky spring, ripples on water, and sound in air. A wave that travels and carries energy is a progressive wave 行波.

Key terms

• displacement 位移 $y$ — how far a particle has moved from its rest position at a moment. A vector 矢量.
• amplitude 振幅 $A$ — the largest displacement from the rest position.
• wavelength 波长 $\lambda$ — the shortest distance along the wave between two points that move in phase 同相 (for example, two next-door crests 波峰).
• period 周期 $T$ — the time for one full oscillation of a particle.
• frequency 频率 $f$ — the number of full oscillations per second; $f = 1/T$. Unit: hertz 赫兹, $\text{Hz}$.
• speed 速率 $v$ — how fast a crest travels along the medium.
• phase difference 相位差 — the fraction of a cycle by which one oscillation leads or lags another. Given in radians 弧度 (a full cycle is $2\pi$) or degrees (a full cycle is $360°$).

Two points one wavelength apart are in phase (phase difference 0 or $2\pi$). Two points half a wavelength apart are exactly out of phase (phase difference $\pi$).

The wave equation

In one period $T$, the wave moves forward by one wavelength $\lambda$. So speed $= \text{distance} / \text{time} = \lambda / T = \lambda f$:

This comes straight from the definitions of speed, frequency and wavelength, and works for every progressive wave.

Reading a CRO trace

A cathode-ray oscilloscope 示波器 (CRO) draws a voltage signal — for sound, the output of a microphone — against time. Two controls matter:

• time-base 时基 (seconds per division across): turns horizontal distance on the screen into time. Read the period $T$ as the distance between two next-door peaks, then $f = 1/T$.
• y-gain 垂直增益 (volts per division up): turns vertical distance into voltage. The amplitude in volts is the peak height from the centre line.

If the time-base is $5\ \text{ms}/\text{div}$ and one full cycle takes $4$ divisions, then $T = 4 \times 5\ \text{ms} = 20\ \text{ms}$ and $f = 50\ \text{Hz}$.

Reading the period T from a CRO trace using the grid and time-base

A real oscilloscope: the grid lets you read off the period and the amplitude

Intensity of a wave

A wave carries energy. The intensity 强度 at a point is the power 功率 passing through unit area at right angles to the direction of travel:

Unit: $\text{W m}^{-2}$.

Intensity is proportional to the square of the amplitude:

For a point source 点源 sending out energy equally in all directions, the wavefronts 波前 are spheres; the surface area at distance $r$ is $4\pi r^{2}$, so

Doubling the distance cuts the intensity to a quarter, which means the amplitude is halved (since $I \propto A^{2}$).

Worked example. A lamp emits $60\ \text{W}$ of light equally in all directions. Find the intensity of the light $2.0\ \text{m}$ away.

Transverse waves

The particles oscillate perpendicular 垂直 to the direction the energy travels. A wave on a rope, all electromagnetic waves, and S-waves in the Earth are transverse.

Transverse wave on a rope

Longitudinal waves

The particles oscillate parallel to the direction the energy travels. Sound in any medium, P-waves in the Earth, and the squashes on a slinky are longitudinal. The wave is made of compressions 压缩 (higher pressure, particles close together) and rarefactions 稀疏 (lower pressure, particles spread out).

Longitudinal wave on a slinky spring

Graphs of waves

A graph of particle displacement against position at one moment looks like a sine curve 正弦曲线 for both kinds of wave. The difference: for a transverse wave the displacement axis is the real sideways displacement; for a longitudinal wave it is the small back-and-forth displacement along the direction of travel (positive one way, negative the other).

A displacement–distance graph shows the wave's amplitude and wavelength

A graph of particle displacement against time at one point in space is also a sine curve for both kinds. Read the period $T$ from this graph.

A displacement–time graph shows the wave's amplitude and period

When the source 波源 of a sound moves relative to a stationary 静止 observer 观察者, the heard frequency is different from the source frequency. This is the Doppler effect 多普勒效应.

• source moving towards the observer: the wavefronts in front are squashed, so the wavelength is shorter and the heard frequency is higher.
• source moving away from the observer: the wavefronts behind are spread out, so the wavelength is longer and the heard frequency is lower.

A moving source squashes the wavefronts ahead of it, raising the observed frequency

The formula (source moving at speed $v_{\text{s}}$ along the line to the observer; wave speed $v$, source frequency $f_{\text{s}}$, heard frequency $f_{\text{o}}$):

Choose the sign to match the physics:

• minus sign on the bottom when the source moves towards the observer ($f_{\text{o}} > f_{\text{s}}$),
• plus sign when the source moves away ($f_{\text{o}} < f_{\text{s}}$).

You only need the case of a stationary observer.

Worked example. A car horn at $f_{\text{s}} = 800\ \text{Hz}$ moves at $30\ \text{m s}^{-1}$ towards a still listener. Speed of sound $v = 340\ \text{m s}^{-1}$:

All electromagnetic waves 电磁波 (EM waves) are transverse and travel in a vacuum 真空 at the same speed:

The electromagnetic spectrum 电磁波谱 includes radio waves, microwaves 微波, infrared 红外线, visible light, ultraviolet 紫外线, X-rays X射线 and $\gamma$-rays γ射线.

The electromagnetic spectrum

Approximate wavelength ranges in free space (learn the orders of magnitude):

• radio waves: $> 10^{-1}\ \text{m}$ (up to many km).
• microwaves: $10^{-3}\ \text{m}$ to $10^{-1}\ \text{m}$.
• infrared: $\sim 7 \times 10^{-7}\ \text{m}$ to $10^{-3}\ \text{m}$.
• visible light: $400\ \text{nm}$ (violet) to $700\ \text{nm}$ (red), i.e. $4 \times 10^{-7}\ \text{m}$ to $7 \times 10^{-7}\ \text{m}$.
• ultraviolet: $\sim 10^{-8}\ \text{m}$ to $4 \times 10^{-7}\ \text{m}$.
• X-rays: $\sim 10^{-11}\ \text{m}$ to $10^{-8}\ \text{m}$.
• $\gamma$-rays: $< 10^{-11}\ \text{m}$.

The boundaries between regions are not sharp. Use $c = f\lambda$ to change between wavelength and frequency. Only light with wavelengths $400$–$700\ \text{nm}$ can be seen.

Polarisation 偏振 means making a transverse wave oscillate in one plane only.

• Only transverse waves can be polarised — the oscillation is perpendicular to the direction of travel, so different perpendicular planes are real choices.
• Longitudinal waves (sound) cannot be polarised — the oscillation is along the direction of travel, so there is no other plane.

So polarisation is a test: if a wave can be polarised, it must be transverse.

Unpolarised waves vibrate in many planes; a polarised wave vibrates in one plane

A see-through plastic protractor placed between two crossed polarising filters. Light only reaches your eye because the stressed plastic rotates its plane of polarisation — the colours map where the plastic is squeezed most. With ordinary light it would just look clear

Malus's law

Plane-polarised 平面偏振 light of intensity $I_{0}$ passes through a polarising filter 偏振片 whose transmission axis 透光轴 is at angle $\theta$ to the plane of polarisation. The transmitted intensity is given by Malus's law 马吕斯定律:

• $\theta = 0°$: filter lined up with the polarisation, $I = I_{0}$, all passes through.
• $\theta = 90°$: filter at right angles, $I = 0$, all blocked.
• $\theta = 60°$: $I = I_{0} \cos^{2} 60° = I_{0} \cdot 0.25 = I_{0}/4$.

Worked example. Plane-polarised light of intensity $12\ \text{W m}^{-2}$ meets a polarising filter whose axis is at $30°$ to the plane of polarisation. Find the transmitted intensity.

Crossed filters (a) block the light; parallel filters (b) let it pass

For two filters in a row, use Malus's law twice with the angle between each pair. Be careful with the angle each time — after the first filter the polarisation is along that filter's axis, and the second filter's angle is measured from there.

(You do not need to work out the effect of a polarising filter on an unpolarised wave.)

• Define terms precisely (displacement, amplitude, wavelength, period, frequency) and use $v = f\lambda$.
• Distinguish transverse (vibration perpendicular to travel) from longitudinal (parallel); only transverse waves can be polarised.
• For the Doppler effect, the observed frequency rises as the source approaches and falls as it recedes.
• Learn the electromagnetic spectrum order; all its waves travel at $c$ in a vacuum.