IB & CIE Physics: Waves – Key Concepts & Exam Tips | IB 与 CIE 物理：波

📚 IB & CIE Physics: Waves – Key Concepts & Exam Tips | IB 与 CIE 物理：波 – 核心考点精讲

Waves form a central pillar in both IB and CIE A-Level Physics, linking oscillations to optics, sound and even quantum ideas. This revision guide distills the essential definitions, formulas and exam strategies you need to master wave behaviour, interference, standing waves, the Doppler effect and polarisation.

波动是 IB 与 CIE A-Level 物理的核心内容之一，它将振动同光学、声学乃至量子观念紧密相连。这篇复习指南提炼了最重要的定义、公式和应试策略，帮助你掌握波的传播特性、干涉、驻波、多普勒效应以及偏振。

1. Types of Waves and Basic Quantities | 波的种类与基本物理量

Waves transfer energy without any net transfer of matter. A transverse wave has particle oscillations perpendicular to the direction of energy propagation; examples include light, water ripples and seismic S-waves. A longitudinal wave has oscillations parallel to the propagation direction, as in sound and seismic P-waves.

波传递能量而并不迁移物质。横波的质点振动方向垂直于能量传播方向，例如光、水波和地震 S 波；纵波的质点振动方向平行于传播方向，例如声波和地震 P 波。

The key kinematic quantities are displacement y (m), amplitude A (m), wavelength λ (m), frequency f (Hz), period T (s) and wave speed v (m s⁻¹). They are linked by T = 1/f and v = f λ.

关键运动学量包括位移 y（m）、振幅 A（m）、波长 λ（m）、频率 f（Hz）、周期 T（s）和波速 v（m s⁻¹）。它们满足 T = 1/f 以及 v = f λ。

Always identify whether you are dealing with a transverse or longitudinal wave from the context, as this determines how the particle motion is drawn and how polarisation can be applied.

做题时首先要根据情景判断是横波还是纵波，因为这决定了画粒子振动方向的方式，也决定了偏振是否适用。

2. The Wave Equation and Graphical Representations | 波速公式与图像表示

The fundamental wave equation is used constantly in both CIE and IB papers:

v = f λ

波的基本公式在 CIE 和 IB 考试中反复出现：

v = f λ

A displacement–distance graph for a fixed instant shows the wavelength λ as the distance between two successive in‑phase points. A displacement–time graph for a single particle shows the period T as the time between successive identical states. The shape of the wave on a displacement–distance graph is a snapshot of the whole wave profile, while the displacement–time graph resembles a simple harmonic oscillation.

固定时刻的位移–距离图显示波形，两个相邻同相点之间的距离就是波长 λ。单质点的位移–时间图则显示该点振动情况，两次相同振动状态之间的时间间隔为周期 T。位移–距离图是一幅波形“快照”，而位移–时间图则类似于简谐振动图像。

Calculating frequency from the period or vice‑versa is a common first step. For electromagnetic waves in a vacuum, c = 3.00×10⁸ m s⁻¹, and you can use c = f λ to find unknown quantities.

由周期求频率（或反求）是常见的解题第一步。对于真空中的电磁波，c = 3.00×10⁸ m s⁻¹，利用 c = f λ 即可求出未知量。

3. Phase and Phase Difference | 相位与相位差

Phase describes the fraction of a wave cycle that has been completed relative to a reference. Two points on a wave are in phase if their phase difference is an integer multiple of 2π (or 360°). They are in anti‑phase if the difference is π, 3π, 5π …

相位描述波在一个周期中所处的阶段。若两点相位差为 2π（或 360°）的整数倍，则它们同相；若相位差为 π, 3π, 5π …，则它们反相。

Phase difference Δφ is linked to path difference Δx by

Δφ = (2π / λ) × Δx

相位差 Δφ 与路程差 Δx 的关系为

Δφ = (2π / λ) × Δx

Understanding phase is crucial for interference and standing waves. For instance, a path difference of λ/2 corresponds to a phase difference of π, giving destructive interference.

理解相位是掌握干涉和驻波的关键。例如，路程差为 λ/2 对应相位差 π，产生相消干涉。

4. Superposition and Interference | 叠加原理与干涉

The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. For coherent sources (same frequency, same waveform and a constant phase difference), this leads to stable interference patterns.

叠加原理指出，当两列（或多列）波在空间某点相遇时，该点的合位移等于各列波单独引起的位移的矢量和。对于相干波源（频率相同、波形相同且具有恒定相位差），叠加会形成稳定的干涉图样。

Constructive interference occurs where the waves arrive in phase, producing a maximum amplitude (A₁ + A₂). Destructive interference occurs where they arrive in anti‑phase, giving a minimum amplitude (|A₁ – A₂|). For light and sound, constructive interference produces bright fringes or loud spots, while destructive interference yields dark fringes or quiet spots.

当两列波同相到达时，产生相长干涉，合成振幅最大（A₁ + A₂）；当反相到达时，产生相消干涉，合成振幅最小（|A₁ – A₂|）。对于光或声波，相长干涉对应亮条纹或响点，相消干涉对应暗条纹或静点。

In order to obtain clear two‑source interference patterns, the sources must be coherent. A laser or a single slit placed before double slits can provide the necessary coherence.

要获得清晰的双源干涉图样，波源必须相干。激光，或在双缝前放置单缝，都可以提供所需的相干性。

5. Young’s Double‑Slit Experiment | 杨氏双缝实验

Young’s double‑slit experiment provides direct evidence for the wave nature of light. Coherent light passing through two narrow slits produces an interference pattern of equally spaced bright and dark fringes on a distant screen.

杨氏双缝实验为光的波动性提供了直接证据。相干光通过两条狭缝后在远处屏幕上形成等间距的明暗相间的干涉条纹。

The fringe spacing Δx (or s) is given by

Δx = λD / d

条纹间距 Δx（或记作 s）由下式决定：

Δx = λD / d

where λ is the wavelength, D is the slit‑to‑screen distance and d is the separation between the two slits. All quantities must be in consistent SI units. If white light is used, the central fringe is white, and higher‑order fringes are coloured spectra because λ differs for different colours.

式中 λ 为波长，D 为缝屏间距，d 为双缝间距。计算时所有量必须使用一致的国际单位。若使用白光，中央条纹为白色，高级次条纹则会因不同颜色的波长差异而呈现彩色光谱。

6. Diffraction Gratings | 衍射光栅

A diffraction grating consists of many equally spaced parallel slits. It produces much sharper and more widely spaced maxima than a double slit. The condition for constructive interference (principal maxima) is the grating equation:

d sin θ = nλ (n = 0, 1, 2, …)

衍射光栅由大量等间距的平行刻线构成，产生的明纹比双缝更细更亮，且间距更大。相长干涉（主极大）的条件由光栅方程给出：

d sin θ = nλ (n = 0, 1, 2, …)

Here d is the grating spacing (e.g. 1/number of lines per metre), θ is the angle of the nth order maximum measured from the central axis, and n is the order number. The maximum possible order is limited because sin θ ≤ 1, so n ≤ d/λ.

式中 d 为光栅常数（例如每毫米刻线数的倒数），θ 为第 n 级明纹相对中央零级的衍射角，n 为级数。由于 sin θ ≤ 1，最大可能级数受限于 n ≤ d/λ。

Using a grating to measure an unknown wavelength is a classic practical; you normally measure the angles for several orders and average. Remember that zero order (n = 0) always lies on the central axis and is un‑deviated.

利用光栅测量未知波长是经典的实验题；通常测量多个级次的衍射角并求平均值。注意零级（n = 0）始终在中央轴线上，不发生偏折。

7. Standing Waves | 驻波

A standing (stationary) wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. The waveform does not propagate; instead, nodes (points of zero displacement) and antinodes (points of maximum displacement) are fixed in space.

当两列频率、振幅相同且相向传播的行波叠加时，会形成驻波（定波）。波形并不向前传播，波节（位移始终为零）和波腹（振幅最大）在空间固定不动。

For a string fixed at both ends, the allowed wavelengths and frequencies are given by

L = n (λₙ/2) and fₙ = n (v/2L), n = 1, 2, 3…

对于两端固定的弦，允许的波长与频率为

L = n (λₙ/2) 且 fₙ = n (v/2L), n = 1, 2, 3…

For a pipe open at both ends, the same harmonic series holds (displacement antinodes at both ends). For a pipe closed at one end, only odd harmonics exist:

L = (2n – 1) λₙ/4 fₙ = (2n – 1) v/4L, n = 1, 2, 3…

对于两端开口的管，谐频序列与两端固定弦相同（两端为位移波腹）。对于一端封闭的管，仅存在奇数倍谐频：

L = (2n – 1) λₙ/4 fₙ = (2n – 1) v/4L, n = 1, 2, 3…

Nodes are separated by λ/2; an antinode lies midway between two adjacent nodes. In exam sketches, clearly label nodes (N) and antinodes (A) and indicate the amplitude envelope.

相邻波节相距 λ/2，波腹位于两相邻波节的中点。在考试画图题中，务必标出波节 (N) 和波腹 (A)，并画出振幅包络线。

8. Doppler Effect | 多普勒效应

The Doppler effect is the change in observed frequency when a wave source moves relative to an observer. For sound and other mechanical waves, the observed frequency f′ is given by

f′ = f × (v ± vₒ) / (v ∓ vₛ)

多普勒效应是指当波源与观察者相对运动时，观察者接收到的频率发生变化的现象。对于声波等机械波，观测频率 f′ 由下式给出：

f′ = f × (v ± vₒ) / (v ∓ vₛ)

where f is the source frequency, v is the wave speed in the medium, vₒ is the observer’s speed and vₛ is the source’s speed. The sign convention: use ‘+’ in the numerator if the observer moves toward the source, ‘–’ if away; in the denominator use ‘–’ if the source moves toward the observer, ‘+’ if away. For electromagnetic waves, the relativistic Doppler formula is used, but at low speeds the approximation f′ ≈ f (1 ± v/c) can be applied.

式中 f 为波源频率，v 为介质中的波速，vₒ 为观察者速率，vₛ 为波源速率。符号约定：观察者靠近波源时分子取“+”，远离取“–”；波源靠近观察者时分母取“–”，远离取“+”。对于电磁波需用相对论多普勒公式，但在低速情况下可用近似 f′ ≈ f (1 ± v/c)。

Typical applications include police radar speed traps, redshift of galaxies, and medical ultrasound flow meters. Pay attention to which object is moving and choose the sign carefully.

常见应用包括警用雷达测速、星系红移和医用超声流量计。解题时要分清哪个物体在运动，符号选择务必小心。

9. Polarisation | 偏振

Polarisation is a phenomenon exclusive to transverse waves. It describes the restriction of the oscillations of a transverse wave to a single plane. Longitudinal waves (such as sound) cannot be polarised.

偏振是横波特有的现象，指将横波的振动限制在单一平面内。纵波（如声波）无法偏振。

Unpolarised light can be plane‑polarised by passing it through a Polaroid filter. The transmitted intensity I after passing through an ideal polariser is given by Malus’s law:

I = I₀ cos²θ

非偏振光通过偏振片后可变成平面偏振光。通过理想偏振片后的透射光强遵循马吕斯定律：

I = I₀ cos²θ

where I₀ is the intensity of the beam incident on the analyser and θ is the angle between the transmission axes of the polariser and the analyser. Note that when unpolarised light passes through a single polariser, its intensity is immediately halved (because the average of cos²θ over all orientations is ½).

式中 I₀ 为入射到检偏器的光强，θ 为起偏器与检偏器透振轴的夹角。需要注意：非偏振光通过一个偏振片后强度立即减半（因为 cos²θ 对随机取向的平均值为 ½）。

Polarisation provides clear evidence that light is a transverse wave. It is used in Liquid Crystal Displays (LCDs), glare‑reducing sunglasses, and stress analysis of materials.

偏振为光的横波特性提供了有力证据，并广泛应用于液晶显示器（LCD）、防眩光太阳镜以及材料应力分析中。

10. Exam Tips for Waves | 波的考试技巧

Always convert quantities to base SI units before substituting into formulas. For fringe spacing calculations, make sure slit separation d and fringe spacing Δx are in metres, and D is in metres. Check that your answer for wavelength is reasonable (visible light ≈ 4×10⁻⁷ to 7×10⁻⁷ m).

代入公式前务必将所有量换算为基本国际单位。计算条纹间距时，确认双缝间距 d、条纹间距 Δx 和缝屏距 D 均以米为单位。波长结果应落在合理范围（可见光约为 4×10⁻⁷ 至 7×10⁻⁷ m）。

When sketching standing waves, draw a smooth envelope and clearly mark nodes and antinodes. For two‑source interference, label the central maximum and the first‑order maxima. In phase‑difference questions, draw a reference circle or use Δφ = (2π/λ) Δx to find the phase.

画驻波图时，要画出平滑的包络线并明确标注波节与波腹。对于双源干涉图，标出中央极大和一级极大。涉及相位差的题目可画参考圆或直接使用 Δφ = (2π/λ) Δx 进行计算。

For the Doppler effect, underline the direction of motion before choosing signs; a quick rule – frequency rises when source and observer approach each other. In Malus’s law, remember that the first polariser halves the intensity of unpolarised light before you apply the cos²θ factor.

处理多普勒效应时，先在草稿纸上标出运动方向再选择符号；一个快速判断法则是：当波源与观察者相互靠近时频率升高。应用马吕斯定律时，非偏振光经过第一个偏振片强度减半，此后才能代入 cos²θ 因子。

During practical assessments or data‑analysis questions, repeat measurements, calculate a mean wavelength, and determine the uncertainty using half the range. Use a protractor diagram when measuring angles for a diffraction grating, and always read the scale precisely.

在实验评估或数据分析题中，要重复测量，计算平均波长，并用半极差法估算不确定度。用分光计测量光栅衍射角时，务必精确读刻度，并在记录纸上画出光路示意图。

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IB & CIE Physics: Waves – Key Concepts & Exam Tips | IB 与 CIE 物理：波 – 核心考点精讲