📚 IB WJEC Physics: Waves Key Points | IB WJEC 物理：波 考点精讲
Waves lie at the heart of many physical phenomena, from the light we see to the sound we hear. In both IB and WJEC physics specifications, wave behaviour is examined through a blend of conceptual understanding and mathematical application. This revision guide distils the essential points: wave types, descriptors, the wave equation, superposition, standing waves, interference, diffraction, the Doppler effect, and polarisation. Each section carries paired English and Chinese explanations to reinforce key ideas.
波是许多物理现象的核心，从我们所见的光到所听的声。在 IB 和 WJEC 物理大纲中，波的行为通过概念理解与数学应用的结合进行考查。本复习指南提炼出核心考点：波的类型、描述参数、波速方程、叠加原理、驻波、干涉、衍射、多普勒效应以及偏振。每个小节配有中英对照讲解，以强化核心概念。
1. Types of Waves | 波的种类
Waves are broadly classified as mechanical or electromagnetic. Mechanical waves, such as sound and water waves, require a medium to propagate; electromagnetic waves, like light and radio waves, can travel through a vacuum. The direction of particle oscillation relative to energy transfer defines whether a wave is transverse (oscillation perpendicular to propagation) or longitudinal (oscillation parallel to propagation). All electromagnetic waves are transverse, while sound in fluids is longitudinal. Seismic waves include both types: P-waves are longitudinal, S-waves are transverse.
波大致分为机械波与电磁波。机械波，例如声波和水波，需要介质才能传播；电磁波，如光波和无线电波，可以在真空中传播。介质粒子的振动方向与能量传递方向的相对关系决定了波是横波（振动垂直于传播方向）还是纵波（振动平行于传播方向）。所有电磁波都是横波，流体中的声波是纵波。地震波包含两种类型：P 波是纵波，S 波是横波。
2. Wave Parameters | 波的基本参数
Key descriptors include amplitude (A), wavelength (λ), frequency (f), period (T) and wave speed (v). The amplitude is the maximum displacement from the equilibrium position, measured in metres. The wavelength is the shortest distance between two consecutive points in phase, such as crest to crest. Frequency, in hertz (Hz), is the number of complete oscillations per second, while the period is the time for one full cycle: T = 1/f. The wave speed v is the rate at which energy is transmitted by the wave.
关键描述参数包括振幅（A）、波长（λ）、频率（f）、周期（T）和波速（v）。振幅是离开平衡位置的最大位移，以米为单位。波长是相邻两个同相点之间的最短距离，例如波峰到波峰。频率以赫兹（Hz）为单位，是每秒完整振动的次数；周期则是一个完整循环所需的时间：T = 1/f。波速 v 是波传递能量的速率。
3. The Wave Equation | 波速方程
The fundamental relationship linking wave speed, frequency and wavelength is
v = f × λ
This holds for all periodic waves. If a wave’s frequency is 50 Hz and its wavelength is 0.34 m, the speed is 17 m s⁻¹. Rearranging gives f = v/λ or λ = v/f. Note that when a wave passes from one medium to another, its speed and wavelength may change, but the frequency remains constant because it is determined by the source.
联系波速、频率与波长的基本关系式为
v = f × λ
这对所有周期波都成立。如果一个波的频率为 50 Hz，波长为 0.34 m，则波速为 17 m s⁻¹。重新整理可得 f = v/λ 或 λ = v/f。注意，当波从一种介质进入另一种介质时，其速度和波长可能改变，但频率保持不变，因为它由波源决定。
4. Phase and Phase Difference | 相位与相位差
Phase describes the position of a point within a wave cycle, measured in radians or degrees. One complete cycle corresponds to 2π rad (360°). The phase difference between two points on a wave or between two waves is the fraction of a cycle by which they are out of step. Points separated by an integer number of wavelengths are in phase (phase difference = 0, 2π, 4π …). Points separated by half a wavelength are in antiphase (phase difference = π, 3π …). Phase relationships are critical when analysing superposition and interference patterns.
相位描述波周期内某点的位置，以弧度或度为单位。一个完整循环对应 2π rad（360°）。波上两点之间或两个波之间的相位差，是指它们不同步的循环比例。相隔整数倍波长的点同相（相位差 = 0, 2π, 4π …）；相隔半波长的点反相（相位差 = π, 3π …）。在分析叠加和干涉图样时，相位关系至关重要。
5. Reflection, Refraction and Diffraction | 反射、折射与衍射
When waves encounter a boundary or obstacle, they exhibit three key behaviours. Reflection obeys the law of reflection (angle of incidence = angle of reflection), and can occur at fixed ends (phase reversal of π) or free ends (no phase reversal). Refraction is the change in direction due to a change in wave speed when crossing a boundary between media; frequency stays the same, while wavelength and speed alter. Diffraction is the spreading of a wave after passing through a gap or around an obstacle. The effect is most noticeable when the gap width is comparable to the wavelength.
波在遇到边界或障碍物时，会表现出三种主要行为。反射遵循反射定律（入射角 = 反射角），可在固定端（相位反转 π）或自由端（无相位反转）发生。折射是由于穿过介质边界时波速变化而导致的方向改变；频率保持不变，而波长和速度改变。衍射是波穿过缝隙或绕过障碍物后的扩展现象。当缝隙宽度与波长相近时，衍射效应最显著。
6. Superposition and Interference | 叠加与干涉
The principle of superposition states that when two or more waves meet, the resultant displacement at any point is the algebraic sum of the individual displacements. Constructive interference occurs when waves arrive in phase, producing a larger amplitude; destructive interference occurs when they arrive in antiphase, reducing or cancelling the amplitude. For two coherent sources (same frequency and fixed phase relationship), a stable interference pattern is formed. The condition for constructive interference in terms of path difference Δx is Δx = nλ, and for destructive interference Δx = (n + ½)λ, where n = 0, 1, 2, …
叠加原理指出，当两个或更多波相遇时，任一点的合位移等于各波单独位移的代数和。当波同相到达时发生相长干涉，产生更大的振幅；当波反相到达时发生相消干涉，振幅减小或抵消。对于两个相干波源（频率相同、相位差恒定），会形成稳定的干涉图样。用波程差 Δx 表示，相长干涉的条件为 Δx = nλ，相消干涉的条件为 Δx = (n + ½)λ，其中 n = 0, 1, 2, …
7. Standing Waves | 驻波
A standing wave is formed when two identical waves travel in opposite directions along a medium and superpose. The pattern features nodes, points of zero displacement, and antinodes, points of maximum displacement. Adjacent nodes are separated by half a wavelength (λ/2). Standing waves can be set up on strings (e.g. guitar), in air columns (e.g. organ pipes) and in microwave cavities. For a string fixed at both ends, the resonant frequencies are given by fn = n (v / 2L), where n = 1, 2, 3 … and L is the string length. For a pipe open at both ends, the same formula applies; for a pipe closed at one end, fn = n (v / 4L) with n = 1, 3, 5 …
当两个相同的波在同一介质中相向传播并叠加时，便形成驻波。驻波图样包含波节（位移为零的点）和波腹（位移最大的点）。相邻波节相距半个波长（λ/2）。驻波可以在弦（如吉他）、气柱（如管风琴）和微波腔中形成。对于两端固定的弦，共振频率为 fn = n (v / 2L)，其中 n = 1, 2, 3 …，L 为弦长。对于两端开口的管，同样适用此公式；对于一端封闭的管，fn = n (v / 4L)，n = 1, 3, 5 …
8. Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency due to relative motion between the source and observer. For sound, the observed frequency f’ is higher when the source approaches and lower when it recedes. For a stationary observer and moving source, f’ = f × v / (v ± vs), where v is the wave speed, vs is the source speed, and the minus sign is used for approach, plus for recession. For light, the Doppler shift produces a red shift (lower frequency) for receding sources and blue shift for approaching sources; this effect is pivotal in cosmology as evidence for the expanding universe.
多普勒效应是指由于波源与观察者之间的相对运动而导致观测频率改变的现象。对于声波，当波源靠近时，观测频率 f’ 升高，远离时降低。对于静止观察者和运动波源，f’ = f × v / (v ± vs)，其中 v 为波速，vs 为波源速度，波源靠近时取减号，远离时取加号。对于光波，多普勒频移使远离的光源发生红移（频率降低），靠近的光源发生蓝移；这一效应在宇宙学中是宇宙膨胀的关键证据。
9. Polarisation | 偏振
Polarisation is a property exclusive to transverse waves. An unpolarised wave vibrates in many planes perpendicular to the direction of propagation; a polarised wave oscillates in only one plane. Light can be polarised by filters (e.g. Polaroid), by reflection (Brewster’s angle), and by scattering. Malus’s law states that the intensity I of plane-polarised light after passing through an analyser is I = I0 cos² θ, where θ is the angle between the transmission axes of the polariser and analyser. Polarisation provides clear evidence for the transverse nature of electromagnetic waves, as longitudinal waves cannot be polarised.
偏振是横波独有的性质。非偏振波在垂直于传播方向的许多平面内振动；偏振波只在一个平面内振荡。光可以通过偏振片（如 Polaroid）、反射（布儒斯特角）和散射产生偏振。马吕斯定律指出，平面偏振光通过检偏器后的强度 I = I0 cos² θ，其中 θ 为起偏器与检偏器透振轴之间的夹角。偏振为电磁波的横波特性提供了明确证据，因为纵波无法被偏振。
10. Intensity and Amplitude | 强度与振幅
Wave intensity I is the power transmitted per unit area, measured in W m⁻². For a wave travelling in three dimensions, intensity is proportional to the square of the amplitude: I ∝ A². This relationship underpins many exam questions, such as how the amplitude of a water wave changes when it spreads out from a point source. Since power is spread over a larger area, spherical waves obey the inverse square law: I = P / (4πr²) and thus A ∝ 1/r. In interference patterns, the resulting intensity depends on the superposition of amplitudes taking phase into account.
波的强度 I 是单位面积传递的功率，单位为 W m⁻²。对于在三维空间中传播的波，强度与振幅的平方成正比：I ∝ A²。这一关系是许多考题的基础，例如水波从点源向外扩展时振幅如何变化。由于功率散布在更大的面积上，球面波遵循平方反比定律：I = P / (4πr²)，因此 A ∝ 1/r。在干涉图样中，合成强度取决于考虑相位后的振幅叠加。
11. Diffraction Gratings and Spectra | 衍射光栅与光谱
A diffraction grating consists of many closely spaced slits. When monochromatic light is incident, the grating produces sharp maxima at angles given by d sin θ = nλ, where d is the grating spacing, n is the order (0, 1, 2 …), and θ is the angle to the normal. The larger the number of slits, the sharper and brighter the maxima. White light is split into its component wavelengths, producing a continuous spectrum, except for the central n = 0 maximum which remains white. Diffraction gratings are used in spectrometers to analyse light from stars and identify elements through their spectral lines.
衍射光栅由许多紧密排列的狭缝构成。当单色光入射时，光栅在满足 d sin θ = nλ 的角度上产生锐利的极大，其中 d 为光栅间距，n 为级次（0，1，2 …），θ 为与法线的夹角。狭缝数量越多，极大越锐利、越明亮。白光被分解为不同波长的成分，形成连续光谱，但中央 n = 0 的极大仍为白色。衍射光栅用于光谱仪中，分析恒星光并通过光谱线识别元素。
12. Exam Tips and Common Pitfalls | 应试技巧与常见误区
Students often confuse the terms in phase and in antiphase, or misapply the λ/2 distance between adjacent nodes and antinodes (the distance between a node and the adjacent antinode is λ/4). When drawing standing waves, ensure nodes and antinodes are correctly spaced. Always convert units to SI before using the wave equation. For Doppler effect calculations, pay attention to the sign convention: observed frequency increases when the relative motion reduces the distance. In polarisation questions, remember that intensity is halved when unpolarised light passes through a single polariser. Use diagrams to support your answers, particularly for reflection, refraction and interference.
学生常混淆“同相”与“反相”的术语，或错误应用相邻波节与波腹之间的距离（波节与相邻波腹的距离为 λ/4，而非 λ/2）。绘制驻波图样时，要确保波节和波腹间距正确。在使用波速方程前，始终将单位换算为国际单位制。处理多普勒效应计算时，注意符号法则：当相对运动使距离减小时，观测频率升高。在偏振问题中，记住非偏振光通过单个偏振片后强度减半。善用示意图辅助作答，尤其是反射、折射和干涉问题。
Published by TutorHao | Physics Revision Series | aleveler.com
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