题型概述：波动、光学与近代物理Question types: waves, optics and modern physics
Top在我们整理的 15 套真题里，机械波 36 道、振动 26 道、光的性质 29 道、成像 20 道、相对论 14 道、原子核 14 道、光电效应 11 道，合计 150 道（约 20%）。这类题计算量小，考的是概念是否清楚：波速由介质决定、驻波的边界条件、透镜成像的几个区间、相对论因子 γ。本模块选 5 道 2024–2025 年的题。In the 15 papers of our bank: mechanical waves 36, oscillation 26, principles of optics 29, image formation 20, relativity 14, nuclear 14 and photoelectric 11, 150 in all (about 20%). These questions need little arithmetic; they test whether the concepts are clean: wave speed is set by the medium, boundary conditions fix standing waves, a lens has a few distinct regions, and relativity runs on the factor γ. This module works five questions from 2024–2025.
1行波与驻波Travelling and standing waves
v = fλ 永远成立。弦上波速 v = √(T/μ)，由张力和线密度决定，与频率无关。两端固定的弦上，驻波的"腹"（loop）数 n 满足 L = nλ/2。横波上质点只上下振动，不随波前进；它的速度方向可以用"下一刻波形往前挪一点"来判断。v = fλ always. On a string v = √(T/μ): tension and linear density set it, frequency does not. On a string fixed at both ends, the number of loops n satisfies L = nλ/2. In a transverse wave each point only moves up and down, never along with the wave; its direction follows from "shift the waveform a little forward".
两端开口的管：f = n·v/(2L)，n = 1, 2, 3, … 一端封闭的管：f = n·v/(4L)，n = 1, 3, 5, …Open at both ends: f = n·v/(2L), n = 1, 2, 3, … Closed at one end: f = n·v/(4L), n = 1, 3, 5, …2凸透镜成像的三个区间The three regions of a converging lens
| 物距Object distance | 像Image |
|---|---|
| u > 2F | 实像、倒立、缩小real, inverted, smaller |
| F < u < 2F | 实像、倒立、放大real, inverted, larger |
| u < F | 虚像、正立、放大（放大镜）virtual, upright, larger (a magnifying glass) |
3狭义相对论：γ 因子Special relativity: the factor γ
γ = 1/√(1 − v²/c²)。运动的物体沿运动方向缩短：L = L₀/γ；运动的钟变慢：Δt = γΔt₀。L₀、Δt₀ 是在物体自己的参考系里测得的"固有"量。v = 0.97c 时 γ ≈ 4.1，远不是 1/0.97。γ = 1/√(1 − v²/c²). A moving object is shorter along its motion, L = L₀/γ; a moving clock runs slow, Δt = γΔt₀. L₀ and Δt₀ are the "proper" values measured in the object's own frame. At v = 0.97c, γ ≈ 4.1, nowhere near 1/0.97.
4本模块的 5 道题The five questions in this module
| 题型Type | 核心一步Core step | 真题Paper |
|---|---|---|
| 横波上质点的速度方向Velocity of a point on a travelling wave | 把波形向前挪一点Shift the waveform forward | 2025 Q41 |
| 换弦后的驻波Standing wave on a new string | n ∝ √μ | 2025 Q34 |
| 开管变闭管Open pipe closed at one end | 只有奇次谐波Odd harmonics only | 2024 Q18 |
| 凸透镜：物在焦点内Converging lens, object inside F | 放大镜：虚、正、大Magnifier: virtual, upright, larger | 2025 Q43 |
| 长度收缩Length contraction | L = L₀√(1 − β²) | 2025 Q48 |