IB WJEC Science: Waves Key Points | IB WJEC 科学：波考点精讲

📚 IB WJEC Science: Waves Key Points | IB WJEC 科学：波考点精讲

Waves are one of the most pervasive phenomena in nature, carrying energy and information without transporting matter. From water ripples to seismic tremors and from audible sound to invisible X-rays, a unified set of principles describes their behaviour. For students following IB and WJEC science courses, the waves topic demands both a qualitative understanding of phenomena such as interference, diffraction and polarisation, and the quantitative skill to apply equations confidently. This revision guide highlights essential definitions, formulas and common exam pitfalls to deepen your understanding of wave physics.

波是自然界中最普遍的现象之一，在传递能量和信息的同时并不搬运物质。从水面涟漪到地震，从可听声波到不可见的X射线，一套统一的原理解释了它们的行为。对于学习IB和WJEC科学课程的同学而言，“波”这一专题既要求定性地理解干涉、衍射、偏振等现象，也需要能熟练应用公式进行计算。本复习指南聚焦核心定义、关键公式和常见考试陷阱，帮助你加深对波动物理的理解。

1. Types of Waves | 波的分类

Waves are broadly divided into mechanical waves and electromagnetic waves. Mechanical waves require a material medium to propagate; examples include sound waves in air, water waves on a surface and seismic waves through the Earth. In contrast, electromagnetic waves consist of oscillating electric and magnetic fields and can travel through a vacuum at the speed of light (c ≈ 3.00 × 10⁸ m s⁻¹). All waves can also be classified as either transverse or longitudinal depending on the direction of particle oscillation relative to the direction of energy transfer.

波大致分为机械波和电磁波。机械波需要介质才能传播，例如空气中的声波、水面上的水波以及穿透地球的地震波。电磁波则由振荡的电场和磁场构成，可以在真空中以光速（c ≈ 3.00 × 10⁸ m s⁻¹）传播。所有波还可以根据介质粒子振动方向相对于能量传播方向，分为横波或纵波。

Property	Transverse	Longitudinal
Particle oscillation	Perpendicular to wave direction	Parallel to wave direction
Examples	Light, water ripples, S-waves	Sound, P-waves
Can be polarised?	Yes	No

Understanding these categories is fundamental: in transverse waves we can describe crests and troughs, whereas longitudinal waves feature compressions and rarefactions. Electromagnetic waves are transverse, making polarisation a key property unique to them.

理解这些分类是基础：横波中我们可以描述波峰与波谷，而纵波则表现出压缩区与稀疏区。电磁波都是横波，因此偏振成为它们独有的重要特性。

2. Describing Waves Quantitatively | 波的定量描述

Several parameters allow us to describe waves precisely. Displacement (y) measures the distance a particle moves from its equilibrium position; the maximum displacement is the amplitude (A). The wavelength (λ) is the distance between two consecutive points that are in phase, for instance crest to crest. The period (T) is the time taken for one complete oscillation, and the frequency (f) is the number of complete waves passing a point per second. Frequency and period are reciprocals: f = 1/T. The unit of frequency is the hertz (Hz).

几个参数可以用来精确描述波。位移（y）是介质粒子离开平衡位置的距离；最大位移称为振幅（A）。波长（λ）是两个相邻同相点之间的距离，比如波峰到波峰。周期（T）是完成一次全振动所需的时间，频率（f）是每秒通过某点的完整波的数量。频率和周期互为倒数：f = 1/T。频率的单位是赫兹（Hz）。

Phase difference describes how much one wave is ‘ahead’ or ‘behind’ another. It is usually measured in radians or degrees: a full cycle corresponds to 2π rad or 360°. Two points half a wavelength apart are in antiphase (π rad, or 180°). In-phase points have a phase difference of 0, 2π, 4π, etc.

相位差描述了一个波超前或滞后另一个波的程度，通常以弧度或度为单位：一个完整周期对应 2π 弧度或 360°。相距半个波长的两点反相（π 弧度或 180°）。同相点的相位差为 0、2π、4π 等。

3. The Wave Equation | 波动方程

The relationship between wave speed (v), frequency (f) and wavelength (λ) is given by the wave equation:

v = f × λ

This equation holds for all wave types. If a wave moves from one medium to another, its frequency stays constant because it is determined by the source, but its speed and wavelength change. For electromagnetic waves in a vacuum, v = c and the equation becomes c = f λ, linking frequency and wavelength directly.

该方程对所有波的类型都成立。波浪形从一个介质进入另一个介质时，由于频率由波源决定，因此保持不变，但波速和波长都会改变。对于真空中的电磁波，v = c，方程变为 c = f λ，从而直接联系起频率和波长。

Common exam tasks involve calculating one quantity when the other two are given, or using the constancy of frequency to find the new wavelength in a different medium. Always check units: v in m s⁻¹, λ in m, f in Hz.

常见考题是已知两个量求第三个量，或者利用频率不变性求出在新介质中的波长。务必检查单位：v 的单位是 m s⁻¹，λ 是 m，f 是 Hz。

4. Reflection and Refraction | 反射与折射

When a wave meets a boundary between two media, some of its energy may be reflected and some transmitted. The law of reflection states that the angle of incidence equals the angle of reflection, both measured relative to the normal. Refraction is the change in direction of a wave as it crosses a boundary where its speed changes. This is governed by Snell’s law:

n₁ sin θ₁ = n₂ sin θ₂

where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles to the normal. The refractive index n is defined as the ratio of the speed of light in a vacuum (c) to the speed in the medium (v): n = c / v. A larger refractive index indicates a slower wave speed and greater optical density.

当波遇到两个介质的界面时，一部分能量被反射，一部分能量透射。反射定律指出入射角等于反射角，两者均相对于法线测量。折射是波在穿过速度发生变化的边界时方向发生改变的现象，它遵循斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂，其中 n₁ 和 n₂ 是两介质的折射率，θ₁ 和 θ₂ 是相对于法线的角度。折射率 n 定义为真空中光速（c）与介质中光速（v）的比值：n = c / v。折射率越大，光速越慢，光密性越强。

During refraction, frequency remains constant but wavelength and speed change: λ_medium = λ_vacuum / n. A useful rule: when light enters a denser medium, it bends toward the normal; when it enters a less dense medium, it bends away from the normal.

折射过程中频率不变，但波长和波速改变：λ_介质 = λ_真空 / n。一个有用的规则：当光进入光密介质时，光线向法线方向偏折；当光进入光疏介质时，光线偏离法线。

5. Total Internal Reflection | 全内反射

When light travels from a denser medium to a less dense one (n₁ > n₂), the refracted ray bends away from the normal. At a particular angle of incidence called the critical angle (θ_c), the refracted ray grazes the boundary (θ₂ = 90°). If the angle of incidence exceeds the critical angle, all light is reflected back into the denser medium. This is total internal reflection (TIR).

当光从光密介质射向光疏介质（n₁ > n₂）时，折射光线偏离法线。当入射角达到特定值——临界角（θ_c）时，折射光线沿界面掠射（θ₂ = 90°）。如果入射角超过临界角，所有光线都会反射回光密介质，这就是全内反射（TIR）。

The critical angle is found from Snell’s law by setting θ₂ = 90°:

sin θ_c = n₂ / n₁ (with n₁ > n₂)

TIR is exploited in optical fibres for communication and in endoscopes. No light energy is lost at the interface, making TIR highly efficient.

令 θ₂ = 90°，可由斯涅尔定律求得临界角：sin θ_c = n₂ / n₁（n₁ > n₂）。全内反射应用于光纤通信和内窥镜中。由于界面上没有光能损失，全内反射非常高效。

6. Diffraction | 衍射

Diffraction is the spreading out of waves when they pass through a narrow gap or around an obstacle. The amount of diffraction increases when the wavelength is comparable to the size of the gap or obstacle. If the gap is much wider than the wavelength, the wave passes through with only slight spreading; if the gap is similar to or smaller than the wavelength, the wave fans out in a semicircular pattern.

衍射是波在穿过窄缝或绕过障碍物时发生的扩散现象。当波长与缝隙或障碍物的尺寸相当时，衍射效果最明显。如果缝隙远大于波长，波通过时仅有轻微扩散；如果缝隙与波长相近或更小，波将以半圆形向外展开。

For a single slit of width a, the central maximum of the diffraction pattern extends to an angle θ given by a sin θ ≈ λ. The diffraction envelope is fundamental to understanding two-slit interference results. Diffraction explains why we can hear sound around corners but cannot see around them: sound wavelengths are long (metres), while light wavelengths are extremely short (hundreds of nanometres).

对于宽度为 a 的单缝，衍射图样的中央极大延伸到满足 a sin θ ≈ λ 的角度。这一衍射包络是理解双缝干涉结果的基础。衍射也解释了为什么我们能听到拐角处的声音却看不到拐角后面的景象——声波波长较长（米级），而光波波长极短（几百纳米）。

7. Two-Source Interference | 双源干涉

When two coherent waves overlap, they superpose to produce a pattern of alternating constructive and destructive interference. For two coherent sources separated by a distance d and viewed on a screen a distance D away (D >> d), bright fringes occur where the path difference is an integer multiple of λ, and dark fringes where it is an odd multiple of λ/2.

当两列相干波叠加时，会产生交替的相长干涉和相消干涉图案。对于间距为 d 的两个相干源，在距离 D 处的屏幕上（D >> d）观察，亮纹出现在光程差为波长整数倍的位置，暗纹则出现在光程差为半波长奇数倍的位置。

The fringe separation (Δx) in Young’s double-slit experiment is given by:

Δx = λ D / d

This formula is central to measuring the wavelength of light. Coherence – having a constant phase relationship – is essential; this is usually achieved by using the same source to illuminate both slits. The intensity distribution shows a series of equally spaced bright fringes modulated by the single-slit diffraction envelope.

杨氏双缝实验中的条纹间距（Δx）由下式给出：Δx = λ D / d。这一公式是测量光波波长的核心。相干性——即保持恒定的相位关系——至关重要，通常通过用同一光源照亮双缝来实现。强度分布表现为一系列等间距的亮纹，且受到了单缝衍射包络的调制。

8. Standing Waves | 驻波

A standing wave is formed when two identical waves travelling in opposite directions superpose. It is characterised by nodes (points of zero displacement) and antinodes (points of maximum amplitude). Standing waves do not transfer energy; instead, energy is stored in the oscillation of the medium. They are observed on stretched strings, in air columns inside pipes, and on the surface of a cup of coffee vibrated at a resonant frequency.

驻波由两列相同、沿相反方向传播的波叠加而成。它的特征是存在波节（位移为零的点）和波腹（振幅最大的点）。驻波不传递能量，能量储存在介质的振荡中。常见的驻波出现在拉紧的弦上、管内的空气柱中，以及在共振频率下振动的咖啡杯表面。

For a string fixed at both ends, the resonant frequencies (harmonics) are:

fₙ = n v / (2L), n = 1, 2, 3, …

where L is the length of the string and v is the wave speed. For a tube open at both ends, the same formula applies for all harmonics. For a tube closed at one end, only odd harmonics exist:

fₙ = n v / (4L), n = 1, 3, 5, …

Understanding these boundary conditions allows you to predict the fundamental frequency and overtones, a favourite topic in both IB and WJEC examinations.

对于两端固定的弦，共振频率（谐频）为：fₙ = n v / (2L)，n = 1, 2, 3, …，其中 L 是弦长，v 是波速。对于两端开口的管，所有谐频均适用同一公式。对于一端封闭的管，只存在奇数谐频：fₙ = n v / (4L)，n = 1, 3, 5, …。理解这些边界条件，你就能预测基频和泛音，这在IB和WJEC的考试中是常见考点。

9. Doppler Effect | 多普勒效应

The Doppler effect describes the change in observed frequency when there is relative motion between a wave source and an observer. For sound, the perceived frequency is higher when the source and observer move toward each other and lower when they move apart. The relationship is:

f’ = f (v ± vₒ) / (v ∓ vₛ)

where f’ is the observed frequency, f is the source frequency, v is the speed of sound in the medium, vₒ is the speed of the observer (positive if moving toward the source), and vₛ is the speed of the source (positive if moving toward the observer). The sign convention ensures that numerator increases and denominator decreases when the motion reduces the separation, raising f’.

多普勒效应描述了波源与观察者之间存在相对运动时观测频率的变化。对于声波，当波源和观察者相互靠近时，感知到的频率升高；彼此远离时频率降低。关系式为：f’ = f (v ± vₒ) / (v ∓ vₛ)。其中 f’ 是观测频率，f 是波源频率，v 是介质中的声速，vₒ 是观察者的速度（迎向波源为正），vₛ 是波源的速度（迎向观察者为正）。符号规则确保距离缩小时分子增大、分母减小，从而使 f’ 升高。

For electromagnetic waves, the Doppler shift produces redshift (when a light source recedes) and blueshift (when it approaches). At low speeds, the shifted wavelength can be approximated by Δλ/λ₀ ≈ v/c, where v is the relative recession velocity and c is the speed of light. This principle underpins the evidence for an expanding universe.

对于电磁波，多普勒频移产生红移（光源远离时）和蓝移（光源靠近时）。在低速情况下，波长的变化可近似为 Δλ/λ₀ ≈ v/c，其中 v 是相对退行速度，c 是光速。这一原理为宇宙膨胀提供了证据。

10. Electromagnetic Spectrum & Polarisation | 电磁波谱与偏振

The electromagnetic spectrum orders all EM waves by frequency or wavelength, from radio waves (longest λ, lowest f) to gamma rays (shortest λ, highest f). All travel at speed c in a vacuum and obey the wave equation c = f λ. The main regions are: radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma. Each region finds distinct applications: radio for broadcasting, microwaves for cooking and communication, IR for thermal imaging, visible for sight, UV for sterilisation, X-rays for medical imaging, and gamma rays for cancer treatment.

电磁波谱按照频率或波长排列所有电磁波，从无线电波（最大 λ，最低 f）到伽马射线（最小 λ，最高 f）。所有电磁波在真空中以光速 c 传播，且满足波动方程 c = f λ。主要区域有：无线电波、微波、红外线、可见光、紫外线、X 射线和伽马射线。每个区域都有独特应用：无线电用于广播，微波用于烹饪和通信，红外线用于热成像，可见光用于视觉，紫外线用于消毒，X 射线用于医学成像，伽马射线用于癌症治疗。

Polarisation is a property unique to transverse waves. An unpolarised wave has oscillations in many planes perpendicular to the direction of energy transfer. A polarising filter selects only those oscillations aligned with its transmission axis, producing a plane-polarised wave. If a second polariser (analyser) is placed after the first, the transmitted intensity varies with the angle θ between their axes according to Malus’s law:

I = I₀ cos² θ

where I₀ is the intensity incident on the analyser. This effect is used in polarising sunglasses, liquid crystal displays (LCDs), and stress analysis of materials. Polarisation provides conclusive evidence that light is a transverse wave, and it cannot occur with longitudinal waves such as sound.

偏振是横波独有的性质。非偏振波在与传播方向垂直的多个平面内振动。偏振滤光片只选择与自身透振轴方向一致的振动，从而产生平面偏振波。如果在第一个偏振片之后再放一个检偏器，透过光的强度随两轴夹角 θ 变化，遵循马吕斯定律：I = I₀ cos² θ，其中 I₀ 是入射到检偏器的光强。这一效应被用于偏光太阳镜、液晶显示器（LCD）以及材料应力分析。偏振为光是横波提供了确凿证据，而对于声波等纵波则无法发生偏振。

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IB WJEC Science: Waves Key Points | IB WJEC 科学：波 考点精讲