振动与波：从周期到能量Oscillations and waves: periods to energy
所选 2022–2024 真题中，振动与波每年均有题目，单题 3–5 分。本模块用五题练习读波形、找共振和处理能量，常见失分点是混淆角频率、把波速当成质点速率，以及质量或百分比的基准选错。In the selected 2022–2024 papers, oscillations and waves appear each year, with 3–5 marks per question. These five exercises practise wave parameters, resonance and energy; common errors involve angular frequency, wave versus particle speed, and the mass or percentage baseline.
1简谐振动：回复力决定周期SHM: restoring force sets the period
以平衡位置为原点，回复力与位移成正比、方向相反。弹簧系统的惯性来自总振动质量；小角度摆的回复力来自重力切向分量。From equilibrium, restoring force is proportional and opposite to displacement. Spring inertia comes from the total moving mass; a small-angle pendulum is restored by tangential gravity.
静态增量给出 k = Δmg/Δx；共振条件为激励频率接近固有频率。Static increments give k = Δmg/Δx; resonance requires forcing near the natural frequency.
2波形：空间与时间分开读Read space and time separately
行波表达式中的时间系数是角频率，位置系数是波数。质点只在局部振动，波形却在介质中传播。In a travelling wave the time coefficient is angular frequency and the position coefficient is wavenumber. Particles oscillate locally while the pattern travels.
拍频为两频率之差的绝对值；空气中的声速与绝对温度平方根成正比。Beat frequency is the absolute frequency difference; sound speed in air scales as the square root of absolute temperature.
3驻波与能量：不能全按波腹计算Standing waves: amplitude varies with position
两端固定的弦只能容纳整数个半波。能量与振幅平方成正比，因此整根弦的能量必须沿位置求和。A fixed-end string fits an integer number of half-wavelengths. Energy scales with amplitude squared, so contributions must be summed along the string.
这里 A 是波腹振幅；动能与弹性势能的周期平均值各为总能量的一半。Here A is the antinode amplitude; cycle-averaged kinetic and elastic potential energies each equal half the total.
| 子题型Sub-type | 识别特征Clue | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 行波与质点振动Travelling wave and particle motion | 给出 y(x,t)Given y(x,t) | 读出 ω、kRead ω and k | 2024 (h) |
| 拍频与空气温度Beats and air temperature | 拍频先减后增Beat decreases then increases | f ∝ √Tf ∝ √T | 2024 (i) |
| 三次谐波的能量Third-harmonic energy | 两端固定，第三谐波Fixed ends, third harmonic | 空间平均 sin²Spatial average of sin² | 2023 (o) |
| 路面激励与共振Road forcing and resonance | 额外下沉，周期路面Extra sag, periodic road | v/d = f₀v/d = f₀ | 2022 (q) |
| 摆钟的快慢校准Pendulum clock calibration | 每天慢十分钟Ten minutes lost per day | 长度比等于周期比平方Length ratio = period ratio squared | 2022 (r) |