📚 IB & Edexcel Science: Waves Key Points | IB与Edexcel科学：波考点精讲
Waves are a fundamental topic in both IB Physics and Edexcel Science specifications. From the properties of transverse and longitudinal waves to interference, standing waves, and the Doppler effect, a solid grasp of wave behaviour is essential for exam success. This revision guide distils the key concepts, equations, and common applications you need to master, with clear explanations in both English and Chinese.
波动是 IB 物理和 Edexcel 科学课程中的基础课题。从横波与纵波的性质到干涉、驻波及多普勒效应，牢固掌握波的行为对于考试成功至关重要。本考点精讲浓缩了核心概念、方程和常见应用，以中英双语提供清晰解释，帮助你全面复习。
1. Types of Waves | 波的类型
Waves transfer energy without transferring matter. Mechanical waves (e.g., sound, water waves) require a material medium to travel, whereas electromagnetic waves (e.g., light, radio waves) can propagate through a vacuum. All waves can be classified as either transverse or longitudinal based on the direction of particle oscillation relative to energy propagation.
波传递能量而不传递物质。机械波（如声波、水波）需要物质介质传播，而电磁波（如光、无线电波）可以在真空中传播。根据质点振动方向相对于能量传播方向，所有波都可以分为横波或纵波。
In transverse waves, the displacement of the medium is perpendicular to the direction of energy transfer. Examples include light, all electromagnetic waves, and ripples on water. Key features: crests, troughs, amplitude, and wavelength.
在横波中，介质的位移方向与能量传递方向垂直。例子包括光、所有电磁波和水波涟漪。关键特征：波峰、波谷、振幅和波长。
In longitudinal waves, the displacement of the medium is parallel to the direction of energy transfer. Sound waves in air are longitudinal, consisting of compressions (high pressure) and rarefactions (low pressure). The distance between successive compressions equals the wavelength.
在纵波中，介质的位移方向与能量传递方向平行。空气中的声波是纵波，由压缩（高压）和稀疏（低压）组成。相邻压缩区之间的距离等于波长。
2. Wave Parameters | 波参数
Displacement (y) is the distance of a point on the wave from its equilibrium position. Amplitude (A) is the maximum displacement from equilibrium; it determines the energy carried by the wave. Wavelength (λ) is the distance between two consecutive points in phase (e.g., crest to crest).
位移 (y) 是波上某点离平衡位置的距离。振幅 (A) 是离开平衡位置的最大位移；它决定了波携带的能量。波长 (λ) 是相邻两个同相点之间的距离（例如波峰到波峰）。
Period (T) is the time taken for one complete oscillation, measured in seconds. Frequency (f) is the number of oscillations per second, measured in hertz (Hz). They are inversely related: T = 1/f.
周期 (T) 是一次完整振动所需的时间，单位是秒。频率 (f) 是每秒振动的次数，单位是赫兹 (Hz)。两者互为倒数：T = 1/f。
Wave speed (v or c for light) describes how fast the wave profile moves. It depends on the medium and, for all waves, can be linked to frequency and wavelength. Phase difference between two points is often expressed in radians or degrees, where one full cycle is 2π radians.
波速 (v，光速用 c) 描述波形轮廓移动的快慢。它取决于介质，对所有波都可以与频率和波长关联。两点之间的相位差常以弧度或度表示，一个完整周期为 2π 弧度。
3. The Wave Equation | 波速公式
The fundamental wave equation: wave speed = frequency × wavelength. This applies to all types of wave.
基本波速公式：波速 = 频率 × 波长。这适用于所有类型的波。
v = f λ
In electromagnetic waves, the speed c = 3.00 × 10⁸ m s⁻¹ in a vacuum. The equation becomes c = f λ. This relationship shows that frequency and wavelength are inversely proportional for a given wave speed.
对于电磁波，真空中波速 c = 3.00 × 10⁸ m s⁻¹，公式变为 c = f λ。这一关系表明，在给定波速下，频率与波长成反比。
Worked example: A radio wave has a frequency of 100 MHz. Its wavelength is λ = c / f = 3.00×10⁸ / 1.00×10⁸ = 3.00 m. For a sound wave with λ = 0.68 m and speed 340 m s⁻¹, f = v / λ = 500 Hz.
计算示例：频率为 100 MHz 的无线电波，波长 λ = c / f = 3.00×10⁸ / 1.00×10⁸ = 3.00 m。对于波长 0.68 m、速度为 340 m s⁻¹ 的声波，f = v / λ = 500 Hz。
4. Reflection and Refraction | 反射与折射
When waves reach a boundary between two media, they can be reflected, refracted, or both. The law of reflection states: angle of incidence (i) = angle of reflection (r), with all angles measured relative to the normal.
当波到达两种介质的边界时，可能发生反射、折射或两者兼有。反射定律：入射角 (i) = 反射角 (r)，所有角度均相对于法线测量。
Refraction occurs when waves change speed as they cross a boundary at an angle, causing a change in direction. The refractive index n of a medium is n = c / v, where c is the speed of light in a vacuum and v is the speed in the medium.
当波以一定角度穿过边界并改变速度时，会发生折射，导致方向改变。介质的折射率 n 定义为 n = c / v，其中 c 是光在真空中的速度，v 是介质中的速度。
Snell’s law governs the angles: n₁ sinθ₁ = n₂ sinθ₂. If a wave enters a denser medium (higher n), it bends towards the normal. Total internal reflection occurs when light travels from a denser to a less dense medium and the angle of incidence exceeds the critical angle C, where sin C = 1 / n (provided the outside medium is air).
斯涅耳定律描述角度关系：n₁ sinθ₁ = n₂ sinθ₂。若波进入光密介质（较高 n），它会向法线偏折。全内反射发生在光从光密介质射向光疏介质且入射角大于临界角 C 时，其中 sin C = 1 / n（假设外部为空气）。
5. Diffraction | 衍射
Diffraction is the spreading of waves around obstacles or through gaps. The extent of spreading depends on the size of the gap relative to the wavelength: maximum diffraction occurs when the gap size is comparable to λ. This is why sound diffracts around doorways but light casts sharp shadows.
衍射是波绕过障碍物或穿过缝隙时发生的扩散现象。扩散程度取决于缝隙尺寸与波长的相对大小：当缝隙尺寸与波长 λ 相近时衍射最显著。这就是为何声波能绕门传播而光会投射清晰阴影的原因。
For a single slit, the central maximum is broad, and minima occur at angles θ given by a sinθ = nλ, where a is slit width and n = ±1, ±2, … (IB notation may use b). For a diffraction grating with slit spacing d, constructive interference produces bright fringes at angles satisfying d sinθ = nλ, where n is the order number (0, 1, 2…).
对于单缝衍射，中央亮纹较宽，暗纹出现在满足 a sinθ = nλ 的角度，其中 a 为缝宽，n = ±1, ±2,…。对于光栅常数为 d 的衍射光栅，相长干涉产生亮纹，满足 d sinθ = nλ，n 为级次 (0, 1, 2…)。
Smaller wavelength and smaller slit width produce narrower, more closely spaced fringes in single‑slit diffraction. In gratings, a greater number of slits per metre (smaller d) leads to larger angular separation of maxima, useful in spectroscopy.
波长越短、缝宽越小，单缝衍射的条纹越窄、间距越密。在光栅中，每毫米刻线越多（d 越小），各级亮纹的角间距越大，这在光谱学中十分有用。
6. Superposition and Interference | 叠加与干涉
The principle of superposition: when two or more waves meet, the resultant displacement is the vector sum of individual displacements. Interference can be constructive (amplitudes add) or destructive (amplitudes subtract), depending on phase difference.
叠加原理：两列或更多波相遇时，合位移是各列波位移的矢量和。干涉可以是相长的（振幅相加）或相消的（振幅相减），取决于相位差。
For two coherent sources (same frequency and constant phase relationship), a stable interference pattern is observed. Constructive interference occurs when path difference = nλ (n = 0, 1, 2…), and destructive when path difference = (n + ½)λ.
对于两个相干源（同频率且相位差恒定），可观察到稳定的干涉图样。路径差为 nλ 时相长干涉 (n = 0, 1, 2…)，路径差为 (n + ½)λ 时相消干涉。
Young’s double‑slit experiment demonstrates light interference. Fringe separation Δx on a screen at distance D from slits with separation a is given by:
杨氏双缝实验演示了光的干涉。在距离双缝 D 的屏幕上，条纹间距 Δx 与缝距 a 的关系为：
Δx = λD / a
This formula allows measurement of wavelength. The central fringe is bright (n=0), with alternating dark and bright fringes on either side. For white light, a central white fringe is flanked by spectra because different wavelengths interfere at slightly different positions.
该公式可用于测量波长。中央条纹为亮纹 (n=0)，两侧交替出现暗纹和亮纹。对于白光，中央为白色亮纹，两侧出现彩色光谱，因为不同波长的光干涉位置有轻微差异。
7. Standing Waves | 驻波
A standing (or stationary) wave is formed when two identical waves travelling in opposite directions superpose. Unlike progressive waves, standing waves do not transfer energy; they store energy in nodes (zero displacement) and antinodes (maximum displacement).
驻波由两列相同但反向传播的波叠加形成。与行波不同，驻波不传递能量；它将能量储存在节点（位移为零）和腹点（位移最大）中。
On a string fixed at both ends, standing waves occur at specific frequencies where the length L = n(λ/2), with n = 1, 2, 3… The fundamental frequency f₁ = v/(2L). Harmonics are fₙ = n f₁. In pipes open at both ends, the same harmonic series exists. For a pipe closed at one end, only odd harmonics appear: L = (2n‑1)λ/4, and f = (2n‑1)v/(4L).
在两端固定的弦上，当长度 L = n(λ/2) (n = 1,2,3…) 时产生驻波。基频 f₁ = v/(2L)，谐波 fₙ = n f₁。两端开口的管乐器遵循相同的谐波序列。一端封闭的管子只出现奇次谐波：L = (2n‑1)λ/4，且 f = (2n‑1)v/(4L)。
Nodes and antinodes are spaced λ/2 apart. Adjacent segments vibrate in antiphase. Standing wave patterns are used in musical instruments, microwave ovens, and laser cavities.
相邻节点或腹点间距为 λ/2。相邻段相位相反。驻波模式应用于乐器、微波炉和激光谐振腔。
8. Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency due to relative motion between the wave source and the observer. When source and observer move towards each other, observed frequency increases; when they move apart, frequency decreases.
多普勒效应是由于波源和观察者之间有相对运动而引起的观测频率变化。当波源与观察者相互靠近时，观测频率升高；相互远离时频率降低。
For sound waves, the observed frequency f’ is given by the formula:
对于声波，观测频率 f’ 由下列公式给出：
f’ = f × (v ± vₒ) / (v ∓ vₛ)
where f is the source frequency, v is the speed of sound, vₒ is observer velocity, and vₛ is source velocity. Sign conventions: use + in numerator when observer moves towards source; use – in denominator when source moves towards observer (consistent with relative approach increasing f’).
其中 f 是源频率，v 是声速，vₒ 是观察者速度，vₛ 是源速度。符号约定：观察者靠近源时分子用 +；源靠近观察者时分母用 –（确保相互靠近时 f’ 增大）。
For electromagnetic waves (light), the relativistic Doppler shift applies. For IB/Edexcel, the approximation for speeds much less than c is: Δλ/λ ≈ v/c, where Δλ is the change in wavelength, v is relative speed away from observer (positive for redshift). This is used in radar speed guns and astronomy (redshift of galaxies).
对于电磁波（光），需要使用相对论多普勒频移。在 IB/Edexcel 中，远低于光速时可近似为：Δλ/λ ≈ v/c，其中 Δλ 是波长变化量，v 是远离观察者的相对速度（正值为红移）。这应用于雷达测速仪和天文学（星系红移）。
9. Electromagnetic Spectrum | 电磁波谱
The electromagnetic spectrum orders all EM waves by frequency or wavelength. In order of increasing frequency: radio waves, microwaves, infrared, visible light, ultraviolet, X‑rays, gamma rays. All travel at speed c in vacuum and are transverse.
电磁波谱将所有电磁波按频率或波长排序。按频率递增顺序：无线电波、微波、红外线、可见光、紫外线、X 射线、伽马射线。所有电磁波在真空中以光速 c 传播，且都是横波。
Table of EM bands and typical wavelengths:
| EM Wave / 电磁波 | Approx. Wavelength / 近似波长 | Main Uses / 主要应用 |
|---|---|---|
| Radio | > 0.1 m | Communications, broadcasting |
| Microwave | 1 mm – 0.3 m | Cooking, radar, satellite links |
| Infrared | 700 nm – 1 mm | Thermal imaging, remote controls |
| Visible | 400 – 700 nm | Human vision, photography |
| Ultraviolet | 10 – 400 nm | Sterilisation, security markings |
| X‑rays | 0.01 – 10 nm | Medical imaging, crystallography |
| Gamma rays | < 0.01 nm | Cancer treatment, sterilisation |
Higher frequency means higher photon energy (E = h f). This explains why UV, X‑rays and gamma rays are ionising and can damage living tissue. In IB, you may be asked to link applications to wavelength/frequency.
频率越高，光子能量越大 (E = h f)。这解释了为何紫外线、X 射线和伽马射线具有电离能力并能损伤生物组织。在 IB 考试中，可能需要将应用与波长/频率联系起来。
10. Polarisation | 偏振
Polarisation is a property exclusive to transverse waves. It refers to the restriction of oscillations to a single plane. Longitudinal waves (e.g., sound) cannot be polarised, providing strong evidence that light is a transverse wave.
偏振是横波独有的性质，指将振动限制在一个平面内。纵波（如声波）无法偏振，这为光是一种横波提供了有力证据。
Unpolarised light has oscillations in all planes perpendicular to the direction of travel. A polarising filter transmits only the component of the electric field parallel to its transmission axis. If two polarisers are placed with their axes at an angle θ, Malus’s law gives the transmitted intensity:
非偏振光的振动存在于垂直于传播方向的所有平面内。偏振片只允许电场分量平行于其透射轴的部分通过。当两个偏振片的透射轴夹角为 θ 时，马吕斯定律给出透射强度：
I = I₀ cos² θ
where I₀ is the intensity after the first polariser. When θ = 0°, intensity is maximum; when θ = 90°, no light is transmitted (crossed polarisers).
其中 I₀ 是透过第一片偏振片后的光强。当 θ = 0° 时强度最大；θ = 90° 时无光透过（正交偏振）。
Polarisation has practical uses: reducing glare in sunglasses, stress analysis in photoelasticity, and LCD screens. In IB, you may be asked to describe how a microwave transmitter and receiver with a metal grid detector can demonstrate polarisation.
偏振的实际应用包括：太阳眼镜减少眩光、光弹性法进行应力分析，以及 LCD 屏幕。在 IB 中，可能需要描述如何利用微波发射器、接收器和金属栅探测器来演示偏振。
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