A-Level Physics: Waves Revision Notes | A-Level 物理：波考点精讲

📚 A-Level Physics: Waves Revision Notes | A-Level 物理：波考点精讲

Waves are a core topic in A-Level Physics, bridging mechanical oscillations with light, sound, and the electromagnetic spectrum. A thorough understanding of wave behaviour, from the simplest harmonic descriptions to interference and diffraction, is essential for tackling both theoretical and practical exam questions. This revision guide breaks down the key ideas, formulas, and common pitfalls into twelve focused sections, pairing English explanations with Chinese translations so you can master the content bilingually.

波是 A-Level 物理中的核心课题，将机械振动与光、声音及电磁波谱连接起来。深入理解波的行为——从最简单的简谐描述到干涉和衍射——对于应对理论和实验考题至关重要。本复习指南把关键概念、公式和常见易错点分成十二个专题小节，并将英文解释与中文翻译配对，帮助你双语掌握考点。

1. Types of Waves | 波的种类

Waves can be classified as transverse or longitudinal. In a transverse wave, particle displacement is perpendicular to the direction of energy propagation (e.g. electromagnetic waves, water ripples, waves on a string). In a longitudinal wave, particle displacement is parallel to the direction of energy propagation (e.g. sound waves, seismic P-waves). Both types transfer energy without net transfer of matter.

波可以分为横波和纵波。在横波中，质点的位移与能量传播方向垂直（例如电磁波、水波、弦上的波）。在纵波中，质点的位移与能量传播方向平行（例如声波、地震 P 波）。两种波都传递能量而不发生物质的净转移。

2. Wave Parameters | 波的基本参数

Key quantities describe any periodic wave. The displacement y of a particle from its equilibrium position is measured in metres. The amplitude A is the maximum displacement. The wavelength λ is the distance between two adjacent points in phase (e.g. crest to crest). The period T is the time for one full oscillation, and the frequency f is the number of cycles per second, measured in hertz (Hz). The relationship f = 1/T always holds. Phase is measured in radians or degrees and determines the state of oscillation at a point.

下列关键量描述任何周期性波。质点偏离平衡位置的位移 y 以米为单位。振幅 A 是最大位移。波长 λ 是两个相邻同相点之间的距离（例如波峰到波峰）。周期 T 是一次完整振动所需的时间，频率 f 是每秒钟的循环次数，单位为赫兹 (Hz)。关系式 f = 1/T 始终成立。相位以弧度或度为单位，决定某一点的振动状态。

3. The Wave Equation | 波动方程

The wave speed v, frequency f and wavelength λ are linked by the wave equation. This equation applies to all waves, provided the medium (or vacuum) is uniform.

波速 v、频率 f 和波长 λ 通过波动方程联系起来。该方程适用于所有波，只要介质（或真空）是均匀的。

v = f λ

Since v is constant for a given medium, increasing the frequency shortens the wavelength. In electromagnetic waves, v = c = 3.00 × 10⁸ m s⁻¹ in a vacuum.

由于在给定介质中 v 是常量，频率增大时波长会变短。对于电磁波，真空中的波速为 v = c = 3.00 × 10⁸ m s⁻¹。

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

The phase of a wave describes its stage in a cycle, often expressed in radians. Two points on a wave have a phase difference Δφ = (2π/λ) × path difference. When the path difference is a whole number of wavelengths, the points are in phase (Δφ = 0, 2π, 4π …). When the path difference is an odd multiple of half a wavelength, they are in antiphase (Δφ = π, 3π, 5π …). Phase difference is crucial for understanding interference.

波的相位描述它在周期中的阶段，通常以弧度表示。波上两点之间的相位差 Δφ = (2π/λ) × 路程差。当路程差为波长的整数倍时，两点同相 (Δφ = 0, 2π, 4π …)；当路程差为半波长的奇数倍时，两点反相 (Δφ = π, 3π, 5π …)。相位差对于理解干涉至关重要。

5. Superposition and Interference | 波的叠加与干涉

When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements (principle of superposition). For sustained interference, the sources must be coherent – they must have the same frequency and a constant phase difference. Constructive interference occurs when the waves are in phase, giving maximum amplitude. Destructive interference occurs when they are in antiphase, giving minimum amplitude. The path difference condition for constructive interference is nλ (n = 0, 1, 2 …); for destructive interference it is (n + ½)λ.

当两个或多个波在一点相遇时，合位移是各个位移的矢量和（叠加原理）。要产生稳定的干涉，波源必须相干——即具有相同的频率和恒定的相位差。当波同相时发生相长干涉，振幅最大；当波反相时发生相消干涉，振幅最小。相长干涉的路程差条件为 nλ (n = 0, 1, 2 …)；相消干涉的条件为 (n + ½)λ。

6. Young’s Double-Slit Experiment | 杨氏双缝干涉实验

Young’s experiment demonstrates the wave nature of light. Monochromatic light passes through two narrow slits, acting as coherent sources. An interference pattern of equally spaced bright and dark fringes is observed on a screen. The fringe separation Δx is given by:

杨氏实验证明了光的波动性。单色光通过两条狭缝，成为相干光源。在屏幕上观察到等间距的明暗条纹。条纹间距 Δx 由下式给出：

Δx = λD / d

where d is the slit separation, D is the distance from slits to screen, and λ is the wavelength. The formula relies on the small-angle approximation. Bright fringes occur where d sinθ = nλ, with n = 0, ±1, ±2 …

其中 d 为双缝间距，D 为缝到屏幕的距离，λ 为波长。该公式基于小角度近似。亮纹出现在满足 d sinθ = nλ 的位置，n = 0, ±1, ±2 …

7. Diffraction and Diffraction Grating | 衍射与衍射光栅

Diffraction is the spreading of waves when they pass through an aperture or around an obstacle. Significant diffraction occurs when the aperture size is comparable to the wavelength. A diffraction grating consists of many equally spaced slits. For a grating with slit spacing d, constructive interference maxima are observed at angles θ given by:

衍射是波通过小孔或绕过障碍物时发生的展宽现象。当小孔尺寸与波长可比时，衍射显著。衍射光栅由许多等间距的狭缝组成。对于缝间距为 d 的光栅，相长干涉的极大出现在满足下式的角度 θ 上：

d sinθ = nλ

where n = 0, ±1, ±2 … (order number). The grating produces sharper, brighter fringes than a double slit and is used to measure wavelengths precisely. The maximum order n is limited by sinθ ≤ 1.

其中 n = 0, ±1, ±2 …（级次）。光栅产生的条纹比双缝更锐利、更亮，且常用于精确测量波长。最大级次 n 受 sinθ ≤ 1 的限制。

8. Stationary Waves | 驻波

A stationary (standing) wave forms when two identical progressive waves travel in opposite directions and superpose. Unlike progressive waves, stationary waves do not transfer energy; they store it in a pattern of nodes (zero displacement) and antinodes (maximum displacement). For a string fixed at both ends, the resonant frequencies are:

当两列相同的行波沿相反方向传播并叠加时，会形成驻波（驻立波）。与行波不同，驻波不传递能量，而是将能量储存为节点（位移为零）和腹点（位移最大）的图样。对于两端固定的弦，共振频率为：

fₙ = n v / (2L) n = 1, 2, 3 …

Here L is the length of the string and v is the wave speed. The fundamental frequency f₁ occurs when n = 1. The distance between adjacent nodes (or antinodes) is λ/2. In pipes (open or closed), similar standing waves form with different harmonic series.

其中 L 是弦长，v 是波速。基频 f₁ 在 n = 1 时出现。相邻节点（或相邻腹点）之间的距离为 λ/2。在管乐器（开管或闭管）中，也会形成类似的驻波，但谐波序列有所不同。

9. The Doppler Effect | 多普勒效应

The Doppler effect is the change in observed frequency when a wave source and observer move relative to each other. For sound, the observed frequency fₒ is related to the source frequency fₛ by:

多普勒效应是指当波源与观察者相对运动时，观测频率发生改变的现象。对于声波，观测频率 fₒ 与源频率 fₛ 的关系为：

fₒ = fₛ (v ± vₒ) / (v ∓ vₛ)

where v is the wave speed, vₒ is the observer’s speed, and vₛ is the source speed. Choose the top signs (+) for relative approach (higher frequency) and the bottom signs (−) for relative recession (lower frequency). For light, the Doppler shift produces redshift (receding) and blueshift (approaching) and is used in astrophysics.

其中 v 是波速，vₒ 是观察者速度，vₛ 是波源速度。分子取加号、分母取减号表示相互靠近（频率变高）；分子取减号、分母取加号表示相互远离（频率变低）。对于光，多普勒频移产生红移（远离）和蓝移（靠近），在天体物理学中得到应用。

10. Polarisation | 偏振

Polarisation is a property unique to transverse waves. An unpolarised wave vibrates in all planes perpendicular to the direction of propagation. After passing through a polarising filter (e.g. a polaroid film), the wave oscillates in only one plane. For light, Malus’s law describes the transmitted intensity through a second polariser (analyser) at angle θ:

偏振是横波独有的特性。非偏振波在与传播方向垂直的所有平面上振动。通过偏振片（如偏振膜）后，波只在一个平面内振动。对于光，马吕斯定律描述了透过第二偏振片（检偏器）的强度，该偏振片与第一偏振片透射轴夹角为 θ：

I = I₀ cos²θ

where I₀ is the intensity after the first polariser. Polarisation provides evidence that light is a transverse wave. Applications include glare-reducing sunglasses, LCD screens, and stress analysis in materials.

其中 I₀ 是通过第一个偏振片后的强度。偏振为光是横波提供了证据。应用包括减少眩光的太阳镜、液晶显示屏和材料的应力分析。

11. The Electromagnetic Spectrum | 电磁波谱

Electromagnetic (EM) waves are transverse waves consisting of oscillating electric and magnetic fields. All EM waves travel at the speed of light c in vacuum and differ only in frequency and wavelength. The spectrum, in order of increasing frequency (decreasing wavelength), is: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each region has distinct properties and uses.

电磁波是由振荡的电场和磁场组成的横波。所有电磁波在真空中都以光速 c 传播，仅在频率和波长上存在差异。电磁波谱按频率递增（波长递减）的顺序依次为：无线电波、微波、红外线、可见光、紫外线、X 射线和伽马射线。每一区域都具有不同的特性和用途。

Region / 区域	Wavelength range / 波长范围	Typical application / 典型应用
Radio / 无线电波	> 0.1 m	Communication, MRI
Microwave / 微波	1 mm – 0.1 m	Cooking, radar, Wi-Fi
Infrared / 红外线	700 nm – 1 mm	Thermal imaging, remote controls
Visible / 可见光	400 – 700 nm	Human vision, photography
Ultraviolet / 紫外线	10 – 400 nm	Sterilisation, fluorescent lamps
X-rays / X 射线	0.01 – 10 nm	Medical imaging, security
Gamma rays / 伽马射线	< 0.01 nm	Cancer therapy, sterilisation

The energy of an EM photon is E = hf, where h is Planck’s constant. Higher frequency radiation carries more energy per photon.

电磁波光子的能量为 E = hf，其中 h 为普朗克常量。频率越高的辐射，每个光子携带的能量越大。

12. Refraction and Total Internal Reflection | 折射与全内反射

When a wave passes from one medium to another, its speed changes, causing a change in direction – refraction. Frequency remains constant, but wavelength and speed alter. Snell’s law relates the angles of incidence θ₁ and refraction θ₂ to the refractive indices n₁ and n₂:

当波从一种介质进入另一种介质时，其速度发生变化，从而引起方向的改变——折射。频率保持不变，但波长和速度会改变。斯涅尔定律将入射角 θ₁ 和折射角 θ₂ 与折射率 n₁、n₂ 联系起来：

n₁ sinθ₁ = n₂ sinθ₂

The refractive index n of a medium is the ratio of the speed of light in vacuum c to its speed in the medium v: n = c / v. When light travels from a denser to a rarer medium (n₁ > n₂), total internal reflection can occur if the angle of incidence exceeds the critical angle θc, given by:

介质的折射率 n 定义为光在真空中的速度 c 与在该介质中的速度 v 之比：n = c / v。当光从光密介质射向光疏介质 (n₁ > n₂) 时，若入射角超过临界角 θc，就会发生全内反射，临界角满足：

sinθc = n₂ / n₁

This principle is exploited in optical fibres, where light is trapped inside the core, enabling high-speed data transmission.

这一原理被应用于光纤中，光被限制在纤芯内传输，从而实现高速数据通信。

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A-Level Physics: Waves Revision Notes | A-Level 物理：波 考点精讲