📚 Waves: CIE A-Level Physics Exam Essentials | 波：CIE A-Level 物理考点精讲
Waves form a cornerstone of CIE A-Level Physics, linking together topics as diverse as oscillations, optics, sound, and the electromagnetic spectrum. A thorough grasp of wave terminology, mathematical relationships, and experimental methods is essential for both Paper 1 and Paper 2, as well as for the practical examination. This guide distils the most tested concepts into clear explanations, equations, and diagrams, ensuring you can confidently tackle multiple-choice questions, structured problems, and data analysis tasks.
波是 CIE A-Level 物理的核心内容之一，将振动、光学、声学和电磁波谱等不同专题紧密联系在一起。透彻理解波的术语、数学关系和实验方法，对于选择题、计算题以及实验考核都至关重要。本篇考点精讲将常考概念提炼为清晰的解释、方程与图示，帮助你从容应对选择题、结构化问题与数据分析。
1. Wave Basics and Types | 波的基础与类型
A wave is a disturbance that transfers energy from one point to another without net transfer of matter. Progressive waves carry energy away from a source; stationary waves store energy in a confined region. All waves are either mechanical (requiring a medium, e.g. sound, water waves) or electromagnetic (able to travel through a vacuum).
波是一种扰动，能在不产生物质净转移的情况下，将能量从一点传递到另一点。行波将能量从波源带走；驻波则将能量约束在一个区域内。所有波要么是机械波（需要介质，如声波、水波），要么是电磁波（可在真空中传播）。
Waves are further classified as transverse or longitudinal. In transverse waves, particle displacement is perpendicular to the direction of energy propagation (e.g. light, waves on a string). In longitudinal waves, particles oscillate parallel to the direction of propagation (e.g. sound). Polarisation is a property unique to transverse waves and is used to distinguish between the two types.
波可进一步分类为横波与纵波。横波中，质元位移垂直于能量传播方向（例如光、绳上的波）。纵波中，质元沿传播方向振动（例如声波）。偏振是横波独有的性质，常被用来区分两类波。
| Property | Transverse | Longitudinal |
| Displacement direction | Perpendicular to propagation | Parallel to propagation |
| Can be polarised? | Yes | No |
| Examples | EM waves, S-waves, string | Sound, P-waves |
2. Key Quantities: Amplitude, Wavelength, Frequency, Speed | 关键物理量：振幅、波长、频率、波速
Displacement (y) describes the distance of a particle from its equilibrium position. Amplitude (A) is the maximum displacement from equilibrium. The wavelength (λ) is the shortest distance between two points in phase on a wave, e.g. crest to crest. The period (T) is the time for one complete oscillation, and frequency (f) is the number of oscillations per unit time: f = 1/T.
位移 (y) 表示质元偏离平衡位置的距离。振幅 (A) 是位移的最大值。波长 (λ) 是波上两个同相位点之间的最短距离，例如相邻波峰。周期 (T) 是完成一次全振动的时间，频率 (f) 是单位时间内的振动次数：f = 1/T。
The speed of a wave (v) is the distance travelled by the wave profile per unit time. For mechanical waves, speed depends on the medium properties (tension, density, temperature) and is not altered by changes in frequency or amplitude. Important relationships are linked through the wave equation.
波速 (v) 是波形在单位时间内传播的距离。对机械波而言，波速取决于介质性质（张力、密度、温度），不会因频率或振幅改变而改变。重要的相互关系通过波动方程联系起来。
3. Phase and Phase Difference | 相位与相位差
Phase is a measure of the fraction of a cycle that has been completed by an oscillating particle, often expressed in radians where one complete cycle equals 2π rad. Two points on a wave are in phase if their phase difference is an integer multiple of 2π, meaning they have identical displacement and velocity. They are in antiphase if the phase difference is π rad (or odd multiples of π).
相位用来衡量振动的质元完成一个周期中的比例，常以弧度表示，一个完整周期对应 2π rad。若两点的相位差为 2π 的整数倍，则两者同相，即位移与速度完全相同。若相位差为 π rad（或 π 的奇数倍），则两者反相。
Phase difference Δφ between two points separated by distance Δx is given by:
Δφ = (2π / λ) Δx
相距 Δx 的两点之间的相位差 Δφ 由以下公式给出：
Δφ = (2π/λ) Δx
This equation is fundamental for explaining interference patterns and standing waves. Exam questions frequently ask you to calculate phase difference from path difference or to identify when constructive (Δφ = 0, 2π, 4π…) or destructive (Δφ = π, 3π, 5π…) interference occurs.
该方程是解释干涉图样和驻波的基础。考题经常要求根据波程差计算相位差，或者判断何时出现相长 (Δφ = 0, 2π, 4π…) 或相消 (Δφ = π, 3π, 5π…) 干涉。
4. The Wave Equation | 波动方程
The most frequently used equation in waves is:
v = f λ
波中最常用的方程是：
v = f λ
Because T = 1/f, the speed can also be written as v = λ / T. This equation applies to all progressive waves. When a wave passes from one medium to another, its speed and wavelength change, but its frequency remains constant because the source determines frequency.
由于 T = 1/f，波速也可写作 v = λ / T。该式适用于所有行波。当波从一种介质进入另一种介质时，其速度和波长会改变，但频率保持不变，因为频率由波源决定。
In ripple-tank experiments, you may be required to measure wavelength and frequency (using a stroboscope or video analysis) to determine wave speed. Typical exam tasks include rearranging the equation and predicting the effect of changing tension on speed for a stretched string, where v = √(T/μ) and μ is the mass per unit length.
在波槽实验中，你可能需要测量波长和频率（用频闪仪或视频分析）来求波速。典型的考题包括方程变形，以及预测改变张力对弦上波速的影响，其中 v = √(T/μ)，μ 为线密度。
5. Energy and Intensity | 能量与强度
A wave carries energy. For a sinusoidal mechanical wave, the energy transmitted per unit time (power) is proportional to the square of the amplitude and the square of the frequency. Intensity I is the power per unit area incident on a surface. For a point source emitting waves uniformly in all directions, the intensity follows the inverse-square law: I ∝ 1/r², where r is the distance from the source.
波携带能量。对正弦机械波，单位时间传递的能量（功率）与振幅的平方和频率的平方成正比。强度 I 是单位面积上入射的功率。对于均匀向各方向发射波的点源，强度遵循平方反比律：I ∝ 1/r²，其中 r 为到源的距离。
For any sinusoidal wave, the intensity is directly proportional to the square of the amplitude:
I ∝ A²
对任意正弦波，强度与振幅平方成正比：
I ∝ A²
This relationship explains why doubling the amplitude quadruples the intensity, and why seismic waves become less destructive with distance. It is also used in examinations to compare amplitudes after energy loss or when waves superpose constructively.
这一关系解释了为何振幅加倍会使强度变为四倍，以及为什么地震波随距离增加破坏力减弱。考试中也会用它来比较能量损失后或波相长叠加后的振幅。
6. Electromagnetic Waves and Polarisation | 电磁波与偏振
All electromagnetic waves are transverse and travel at speed c = 3.00 × 10⁸ m s⁻¹ in a vacuum. The electromagnetic spectrum, in order of increasing frequency (decreasing wavelength), includes radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma rays. CIE expects you to recall approximate wavelength ranges and typical sources or detectors for each region.
所有电磁波均为横波，在真空中以 c = 3.00 × 10⁸ m s⁻¹ 的速度传播。电磁波谱按频率升高（波长减小）的顺序为：无线电波、微波、红外线、可见光、紫外线、X 射线和 γ 射线。CIE 要求你记住各波段的大致波长范围及典型源或探测器。
Polarisation is the process of restricting the oscillations of a transverse wave to a single plane. An unpolarised wave can be polarised by a filter (Polaroid), by reflection (Brewster’s angle), or by scattering. Longitudinal waves cannot be polarised, so polarisation provides a definitive test for transverse waves. Malus’s law states that for light passing through two polarising filters with an angle θ between their transmission axes, the transmitted intensity I = I₀ cos² θ.
偏振是指将横波的振动限制在单一平面内。非偏振波可通过偏振片（偏振滤光片）、反射（布儒斯特角）或散射实现偏振。纵波无法偏振，因此偏振成为鉴别横波的可靠方法。马吕斯定律指出，光通过透光轴夹角为 θ 的两片偏振片时，透射强度 I = I₀ cos² θ。
7. Superposition and Interference | 叠加与干涉
The principle of superposition states that when two or more waves of the same type meet at a point, the resultant displacement is the vector sum of the individual displacements. Constructive interference occurs when the waves are in phase (phase difference 0, 2π, 4π…), producing a maximum amplitude. Destructive interference happens when they are in antiphase (phase difference π, 3π…), giving a minimum amplitude, potentially zero if the amplitudes are equal.
叠加原理指出，当两个或多个同类型波在某点相遇时，合位移是各波独自位移的矢量和。若波同相（相位差 0, 2π, 4π…），发生相长干涉，振幅最大。若波反相（相位差 π, 3π…），发生相消干涉，振幅最小，若振幅相等可为零。
Path difference determines the phase relationship. If two waves from coherent sources travel distances s₁ and s₂ to a point, the path difference Δs = |s₁ − s₂|. For constructive interference: Δs = nλ (n = 0, 1, 2…); for destructive interference: Δs = (n + 1/2)λ. Coherence means the sources maintain a constant phase difference and have the same frequency.
波程差决定了相位关系。如果来自相干光源的两束波到达某点的路程分别为 s₁ 和 s₂，波程差 Δs = |s₁ − s₂|。相长干涉条件：Δs = nλ (n = 0, 1, 2…)；相消干涉条件：Δs = (n + 1/2)λ。相干性指波源保持恒定的相位差且频率相同。
8. Young’s Double-Slit Experiment | 杨氏双缝实验
Young’s double-slit experiment demonstrates the wave nature of light through interference. Monochromatic, coherent light illuminates two narrow, closely spaced slits. Each slit acts as a coherent source, producing overlapping waves that create bright and dark fringes on a distant screen. Bright fringes correspond to constructive interference (path difference = nλ), dark fringes to destructive interference (path difference = (n + 1/2)λ).
杨氏双缝实验通过干涉现象证明了光的波动性。单色相干光照射两条相距很近的窄缝，每条缝作为一个相干光源，出射波在远处屏幕上形成明暗相间的条纹。明纹对应相长干涉（波程差 = nλ），暗纹对应相消干涉（波程差 = (n + 1/2)λ）。
The fringe spacing Δy (distance between adjacent bright fringes or adjacent dark fringes) is given by:
Δy = λD / d
条纹间距 Δy（相邻明纹或暗纹距离）由以下公式给出：
Δy = λD / d
where λ is the wavelength, D is the distance from the slits to the screen, and d is the slit separation. This equation is valid when D >> d. Common examination tasks include measuring Δy from a diagram, calculating λ, and discussing how the pattern changes if slit separation or screen distance is altered. Using white light produces a central white fringe with coloured fringes on either side, because λ varies for different colours.
式中 λ 为波长，D 为双缝到屏幕的距离，d 为缝间距。该式在 D >> d 时成立。常见考题包括从图中测量 Δy、计算波长，以及讨论改变缝距或屏幕距离对条纹的影响。使用白光时，中央为白色条纹，两侧出现彩纹，因为不同颜色的波长不同。
9. Diffraction and Diffraction Gratings | 衍射与衍射光栅
Diffraction is the spreading of waves when they pass through a narrow aperture or around an obstacle. The amount of spreading is most noticeable when the size of the aperture or obstacle is comparable to the wavelength. For a single slit of width a, the central maximum spans an angle roughly given by sinθ ≈ λ / a. On a screen, minima occur at a sinθ = nλ for n = 1, 2, 3… .
衍射是波通过窄缝或绕过障碍物时发生的扩散现象。当缝隙或障碍物尺寸与波长相当时，扩散最为明显。对于宽度为 a 的单缝，中央明纹的角度范围大致满足 sinθ ≈ λ / a。在屏幕上，极小值出现在 a sinθ = nλ (n = 1, 2, 3…)。
A diffraction grating consists of many equally spaced parallel slits. It produces a series of sharp, bright maxima. The condition for principal maxima is:
d sinθ = nλ
衍射光栅由大量等间距的平行狭缝构成，产生一系列锐利的亮条纹。主极大条件为：
d sinθ = nλ
where d is the grating spacing (the distance between adjacent slits, d = 1/N with N the number of lines per metre), θ is the angle of the nth-order maximum, and n = 0, 1, 2, … (order). Grating spectrometers are used to measure wavelengths with high precision. Exam questions often require you to calculate d from lines per mm, find unknown wavelengths, and determine the maximum order observable.
式中 d 为光栅常数（相邻缝间距，d = 1/N，N 为每米刻线数），θ 为第 n 级明纹的角度，n = 0, 1, 2, … (级次)。光栅光谱仪用于高精度测量波长。考题常要求根据每毫米刻线数计算 d，求未知波长，以及确定可观察到的最高级次。
10. Standing Waves | 驻波
A standing wave forms when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. Unlike progressive waves, standing waves do not transfer energy; energy is stored in the vibrating system. They consist of nodes (points of zero displacement) and antinodes (points of maximum displacement). The distance between adjacent nodes is λ/2, and between a node and adjacent antinode is λ/4.
驻波由两列频率和振幅相同的行波沿相反方向传播并叠加而形成。与行波不同，驻波不传递能量，能量存储在振动系统中。驻波由节点（位移始终为零的点）和腹点（位移最大的点）组成。相邻节点间距离为 λ/2，节点与相邻腹点间距为 λ/4。
Standing waves are exhibited in strings fixed at both ends (e.g. guitar) and in air columns in pipes. For a string of length L fixed at both ends, the resonant frequencies are fₙ = n v / (2L) for n = 1, 2, 3…, where v is the wave speed on the string. For a pipe closed at one end and open at the other, only odd harmonics exist: fₙ = n v / (4L) for n = 1, 3, 5… . A pipe open at both ends produces all harmonics like a string.
驻波出现在两端固定的弦（如吉他）和管中空气柱中。对于两端固定、长度为 L 的弦，谐振频率为 fₙ = n v/(2L)，n = 1, 2, 3…，其中 v 为弦上的波速。对于一端封闭一端开口的管，只存在奇数倍频：fₙ = n v/(4L)，n = 1, 3, 5…。两端开口的管则像弦一样可产生所有谐频。
Practical demonstrations include Melde’s experiment, vibrating strings, and Kundt’s tube. Exam questions often ask you to sketch the shape of standing waves for given harmonics, label nodes and antinodes, and relate the mode number to wavelength and length.
演示实验包括梅尔德实验、弦振动与昆特管。考题常要求画出给定谐频的驻波形状，标注节点和腹点，并将振动模式与波长和长度联系起来。
11. 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 f’ is higher when the source and observer approach, and lower when they recede. For an observer moving towards a stationary source at speed v_o, f’ = f (v + v_o) / v, where v is the speed of sound. For a source moving towards a stationary observer at speed v_s, f’ = f v / (v − v_s).
多普勒效应是指波源和观察者相对运动时，观察到的频率发生变化的现象。对于声波，当源头与观察者相互靠近时，观察频率 f’ 增大；相互远离时减小。对于以速率 v_o 靠近静止声源的观察者，f’ = f (v + v_o)/v，其中 v 为声速。对于以速率 v_s 靠近静止观察者的声源，f’ = f v/(v − v_s)。
For electromagnetic waves (e.g. light), the Doppler shift for speeds much less than c is approximated by Δλ / λ ≈ v / c, where Δλ = λ’ − λ and v is the relative speed of recession (positive v means redshift). This effect is used in radar speed traps, medical ultrasound, and astrophysical measurements (redshift of galaxies).
对于电磁波（如光），当相对速度远小于光速 c 时，多普勒频移的近似公式为 Δλ/λ ≈ v/c，其中 Δλ = λ’ − λ，v 为相对退行速度（正 v 表示红移）。该效应在雷达测速、医用超声和天体物理（星系红移）中均有应用。
Exam questions typically involve quantitative calculations using the sound-wave equations and qualitative explanations of how pitch changes. You should be ready to identify situations where the medium is at rest and when relative motion changes the detected frequency.
考题通常涉及用声波公式进行定量计算，以及定性解释音调如何变化。你应能识别介质静止且相对运动改变接收频率的各种情景。
12. Practical Techniques and Common Pitfalls | 实验技巧与常见错误
Common experimental tasks include measuring the speed of sound using a resonance tube or an oscilloscope, determining the wavelength of light with a double slit or diffraction grating, and investigating standing waves on a string or in air columns. Always ensure measurements are taken with appropriate instruments (e.g. metre rule, vernier callipers, stroboscope) and repeat readings to reduce random errors.
常见实验任务包括使用共振管或示波器测量声速、用双缝或衍射光栅测定光的波长，以及探究弦上的驻波或空气柱驻波。务必使用合适的仪器（如米尺、游标卡尺、频闪仪）测量，并重复读数以减少随机误差。
When drawing wave diagrams, clearly indicate amplitude, wavelength, nodes, antinodes, and direction of vibrations. A common mistake is confusing the distance between a node and an antinode (λ/4) with the node-to-node distance (λ/2). Also, in interference and diffraction, remember that the equations Δy = λD/d and d sinθ = nλ rely on small-angle approximations; always check that angles are small or use precise trigonometric relationships if required.
绘制波形图时，应清晰标明振幅、波长、节点、腹点以及振动方向。一个常见错误是将节点与腹点间距 (λ/4) 与节点间距 (λ/2) 混淆。此外，在干涉和衍射中，注意方程 Δy = λD/d 和 d sinθ = nλ 基于小角度近似；一定要检验角度是否较小，否则需使用精确的三角函数关系。
Ensure you can convert units confidently: nanometres to metres (1 nm = 10⁻⁹ m), grams per metre to kg m⁻¹, etc. Finally, always relate the experimental setup back to the wave equation and the specific conditions for constructive interference or resonance, as this is how CIE markers award explanation marks.
确保能熟练转换单位：纳米转米 (1 nm = 10⁻⁹ m)，克每米转 kg m⁻¹ 等。最后，始终将实验装置与波动方程及相长干涉或共振的具体条件联系起来，这正是 CIE 阅卷时给解释分的依据。
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