A2 Physics: Waves Key Points | A2 物理：波考点精讲

📚 A2 Physics: Waves Key Points | A2 物理：波考点精讲

This article delivers a focused revision of essential A2 wave concepts, including simple harmonic motion, resonance, progressive and standing waves, interference, diffraction, the Doppler effect and polarisation. Each section pairs a concise English explanation with its Chinese equivalent to support bilingual mastery for A-level Physics exams.

本文针对 A2 波的核心考点进行精讲，涵盖简谐运动、共振、行波与驻波、干涉、衍射、多普勒效应和偏振等内容。每小节均提供简明扼要的英文讲解与中文对照，帮助考生高效复习 A-Level 物理考试。

1. Simple Harmonic Motion (SHM) | 简谐运动

Simple harmonic motion is oscillatory motion where the acceleration a is directly proportional to the displacement x from equilibrium and always directed towards it: a = -ω²x. The displacement varies sinusoidally in time: x = A sin(ωt) or x = A cos(ωt), with ω = 2πf = 2π/T.

简谐运动是一种振荡运动，其加速度 a 与相对于平衡位置的位移 x 成正比且方向总是指向平衡位置：a = -ω²x。位移随时间呈正弦变化：x = A sin(ωt) 或 x = A cos(ωt)，其中角频率 ω = 2πf = 2π/T。

The maximum speed in SHM is vₘₐₓ = ωA, occurring at the equilibrium position. The maximum acceleration is aₘₐₓ = ω²A, achieved at the extreme displacements.

简谐运动的最大速度 vₘₐₓ = ωA，出现在平衡位置；最大加速度 aₘₐₓ = ω²A，出现在最大位移处。

2. Energy in SHM | 简谐运动中的能量

The total mechanical energy in an undamped simple harmonic oscillator is constant and is the sum of kinetic and potential energies: Eₜₒₜ = ½ m ω²A². Kinetic energy K = ½ m v² = ½ m ω²(A² − x²), and potential energy U = ½ m ω²x².

无阻尼简谐振子的总机械能保持恒定，为动能与势能之和：Eₜₒₜ = ½ m ω²A²。动能 K = ½ m v² = ½ m ω²(A² − x²)，势能 U = ½ m ω²x²。

Energy exchanges continuously between kinetic and potential forms; at the equilibrium point all energy is kinetic, while at the extremes all energy is potential.

能量在动能和势能之间连续转换；在平衡位置全部为动能，在最大位移处全部为势能。

3. Damped Oscillations and Resonance | 阻尼振动与共振

Damping removes energy from an oscillating system, reducing amplitude over time. Light damping (underdamping) produces a gradual decrease in amplitude with a slightly lowered frequency; critical damping returns the system to equilibrium in the shortest time without overshooting; heavy damping (overdamping) gives a slow return with no oscillation.

阻尼会消耗振动系统的能量，使振幅随时间减小。轻阻尼（欠阻尼）使振幅逐渐减小，频率略微降低；临界阻尼使系统在最短时间内回到平衡位置而不发生超调；重阻尼（过阻尼）则缓慢回到平衡，不产生振荡。

Resonance occurs when a periodic driving force matches the natural frequency of the system, leading to a dramatic increase in amplitude. The sharpness of the resonance peak depends on the degree of damping; less damping yields a sharper, higher peak.

当周期性驱动力频率与系统的固有频率相等时发生共振，振幅急剧增大。共振峰的尖锐程度取决于阻尼大小；阻尼越小，共振峰越尖锐、峰值越高。

4. Progressive Waves | 行波

A progressive wave transfers energy from one point to another without permanent displacement of the medium. The wave equation links speed v, frequency f and wavelength λ: v = f λ. For a transverse wave, particles vibrate perpendicularly to the direction of energy transfer; for a longitudinal wave, particles vibrate parallel to it.

行波将能量从一点传递到另一点，介质没有永久性的位移。波速 v、频率 f 和波长 λ 满足方程 v = f λ。在横波中，质点振动方向与能量传递方向垂直；在纵波中，质点振动方向与能量传递方向平行。

The phase difference Δφ between two points separated by distance Δx is Δφ = (2π/λ) Δx. In-phase points have Δφ = 2πn, anti-phase points have Δφ = (2n+1)π.

相距 Δx 的两点的相位差 Δφ = (2π/λ) Δx。同相点 Δφ = 2πn，反相点 Δφ = (2n+1)π。

5. Wave Properties: Reflection, Refraction, Diffraction | 波的特性：反射、折射、衍射

Waves undergo reflection when they bounce back from a boundary, obeying the law of reflection (angle of incidence = angle of reflection). In refraction, waves change speed and direction when entering a new medium due to a change in wave speed; Snell’s law relates the angles and speeds: sin θ₁ / sin θ₂ = v₁ / v₂.

波在边界上发生反射，遵循反射定律（入射角等于反射角）。折射时，波进入新介质后由于波速变化而改变传播方向；斯涅尔定律描述了角度与波速的关系：sin θ₁ / sin θ₂ = v₁ / v₂。

Diffraction is the spreading of waves around obstacles or when passing through apertures. The effect is most pronounced when the size of the obstacle or gap is comparable to the wavelength.

衍射是波遇到障碍物或通过缝隙时发生的展宽现象。当障碍物或缝隙的尺寸与波长相近时，衍射效应最为显著。

6. Superposition and Interference | 叠加与干涉

The principle of superposition states that when two or more waves meet, the resultant displacement is the vector sum of the individual displacements. Interference can be constructive (resultant amplitude larger) when waves are in phase, or destructive (amplitude reduced) when they are in anti-phase.

叠加原理指出，当两个或多个波相遇时，合位移等于各波位移的矢量和。当波同相时产生相长干涉（合成振幅增大），当波反相时产生相消干涉（合成振幅减小）。

For sustained interference, sources must be coherent — maintaining a constant phase difference and the same frequency. Path difference is key: constructive interference occurs when path difference = nλ, destructive when path difference = (n + ½)λ.

要形成稳定的干涉图样，波源必须相干——保持恒定的相位差和相同的频率。路程差是关键：路程差 = nλ 时相长干涉，路程差 = (n + ½)λ 时相消干涉。

7. Young’s Double-Slit Experiment | 杨氏双缝实验

Young’s double-slit experiment demonstrates the wave nature of light through interference. Coherent light passing through two narrow slits produces a pattern of bright and dark fringes on a screen. The fringe separation y is given by y = λD / d, where D is the slit‑to‑screen distance and d is the slit separation.

杨氏双缝实验通过干涉现象证明了光的波动性。相干光通过两条狭缝后在屏幕上形成明暗相间的条纹。条纹间距 y = λD / d，其中 D 为双缝到屏幕的距离，d 为双缝间距。

Bright fringes correspond to constructive interference (path difference = nλ), dark fringes to destructive interference (path difference = (n + ½)λ). Using this relationship, the wavelength of light can be measured precisely.

亮条纹对应相长干涉（路程差 = nλ），暗条纹对应相消干涉（路程差 = (n + ½)λ）。利用这一关系可以精确测量光的波长。

8. Diffraction Grating | 衍射光栅

A diffraction grating consists of many equally spaced slits. When monochromatic light is incident, sharp bright maxima are formed at angles given by the grating equation: d sin θ = nλ, where d is the slit spacing, n is the order number (0, ±1, ±2, …).

衍射光栅由许多等间距的狭缝组成。单色光入射时，在满足光栅方程 d sin θ = nλ 的角度处产生锐利的亮纹，其中 d 为缝间距，n 为级次 (0, ±1, ±2, …)。

Gratings produce much sharper and more widely spaced maxima than double slits, making them ideal for spectroscopy and precise wavelength measurement. The maximum possible order is nₘₐₓ = d/λ (since sin θ ≤ 1).

光栅产生的亮纹比双缝干涉条纹更加锐利、间距更大，因此广泛应用于光谱分析和精密波长测量。可能的最大级次 nₘₐₓ = d/λ（因为 sin θ ≤ 1）。

9. Standing (Stationary) Waves | 驻波

A standing wave is formed when two identical progressive waves travel in opposite directions and superpose. Unlike travelling waves, standing waves store energy and have fixed nodes (zero displacement) and antinodes (maximum displacement).

驻波由两列相同的行波沿相反方向传播并叠加而形成。与行波不同，驻波储存能量且具有固定的波节（位移为零）和波腹（位移最大）。

The distance between adjacent nodes or adjacent antinodes is λ/2. In a string fixed at both ends, the allowed wavelengths are λₙ = 2L/n (n = 1, 2, 3, …). The frequencies are fₙ = n v/(2L).

相邻波节或相邻波腹之间的距离为 λ/2。两端固定的弦上允许的波长为 λₙ = 2L/n (n = 1, 2, 3, …)，相应的频率为 fₙ = n v/(2L)。

10. Sound Waves and Harmonics | 声波与谐波

In pipes, standing sound waves produce resonance. For a pipe open at both ends, the harmonics follow fₙ = n v/(2L), with n = 1, 2, 3, … giving both even and odd harmonics. The displacement antinodes are at the open ends.

在管乐器中，驻波声波产生共振。两端开口的管，谐波频率为 fₙ = n v/(2L)，n = 1, 2, 3, … 包含所有奇偶次谐波，位移波腹位于开口端。

For a pipe closed at one end, the fundamental has a node at the closed end and an antinode at the open end, giving fₙ = n v/(4L) but only odd values of n (n = 1, 3, 5, …).

一端封闭的管，基频在闭端形成位移波节、开端形成位移波腹，频率为 fₙ = n v/(4L)，但仅限奇数 n 值 (n = 1, 3, 5, …)。

11. Doppler Effect | 多普勒效应

The Doppler effect is the apparent change in frequency of a wave due to relative motion between the source and the observer. When the source moves towards the observer, the observed frequency increases; when it moves away, the frequency decreases.

多普勒效应是由于波源与观察者之间的相对运动而引起的波的频率表观变化。当波源向观察者运动时，观测频率增大；当远离时，观测频率减小。

For sound waves, the observed frequency f’ is given by f’ = f (v ± vₒ)/(v ± vₛ), where v is the speed of sound, vₒ the observer’s speed and vₛ the source’s speed. Signs are chosen according to the direction of relative motion.

对于声波，观测频率 f’ 由 f’ = f (v ± vₒ)/(v ± vₛ) 给出，其中 v 为声速，vₒ 为观察者速度，vₛ 为波源速度。符号根据相对运动方向选取。

In light, the Doppler shift results in red‑shift when a source recedes and blue‑shift when it approaches, with the shift Δλ/λ ≈ v/c for speeds much less than c.

光的 Doppler 效应表现为：光源远离时产生红移，靠近时产生蓝移，当速度远小于光速 c 时，波长变化 Δλ/λ ≈ v/c。

12. Polarisation | 偏振

Polarisation is a phenomenon exclusive to transverse waves, where oscillations are confined to a single plane. Unpolarised light has vibrations in many planes; a polarising filter transmits only the component of the electric field parallel to its transmission axis.

偏振是横波特有的现象，意指振动被限制在单一平面内。非偏振光在多个平面内振动；偏振滤光片只允许与其透射轴平行的电场分量通过。

When unpolarised light passes through a polariser, its intensity is halved: I = I₀/2. If a second polariser (analyser) is placed after the first, the transmitted intensity is given by Malus’s law: I = I₀ cos²θ, where θ is the angle between their transmission axes.

非偏振光通过偏振片后强度减半：I = I₀/2。若在第一个偏振片后再放置分析器，透射光强度遵循马吕斯定律：I = I₀ cos²θ，其中 θ 为两透射轴之间的夹角。

Polarisation provides direct evidence that light is a transverse wave and has many applications, including glare‑reducing sunglasses, LCD screens and stress analysis using photoelasticity.

偏振为光是一种横波提供了直接证据，并广泛应用于防眩光太阳镜、液晶显示屏以及利用光弹性进行应力分析等领域。

Published by TutorHao | Physics Revision Series | aleveler.com

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A2 Physics: Waves Key Points | A2 物理：波 考点精讲