📚 Oscillations and Waves | 振动与波
Oscillations and waves are fundamental concepts in IB Physics, underlying many natural phenomena and technological applications. An oscillation is a repeated back-and-forth motion about an equilibrium point, while a wave is a disturbance that transfers energy without transferring matter. Understanding these topics requires exploring simple harmonic motion, damping, resonance, wave properties, superposition, standing waves, diffraction, and the Doppler effect. This article provides a comprehensive bilingual revision guide that follows the IB syllabus sequence and highlights the definitions, equations and graph shapes that examiners reward.
振动与波是IB物理中的基础概念，是许多自然现象和技术应用的根基。振动是围绕平衡点的重复往复运动，而波则是一种传递能量而不传递物质的扰动。要理解这些主题，需要探究简谐运动、阻尼、共振、波的性质、叠加、驻波、衍射和多普勒效应。本文提供一份全面的双语复习指南，依照IB大纲顺序展开，并突出考官最看重的定义、公式与图像特征。
1. Describing Oscillations and SHM | 描述振动与简谐运动
um, a vector), amplitude A (maximum displacement), time period T (time for one complete cycle) and frequency f (number of cycles per second). Frequency and period are always linked by f = 1/T, and the angular frequency is ω = 2πf = 2π/T.
振动是围绕平衡位置的重复往复运动。核心物理量包括：位移（离开平衡位置的距离，是矢量）、振幅 A（最大位移）、周期 T（完成一次完整循环所需时间）与频率 f（每秒循环次数）。频率与周期始终满足 f = 1/T，角频率则定义为 ω = 2πf = 2π/T。
ds the equilibrium point. In symbols this gives a = −ω²x, or equivalently a restoring force F = −kx where k is a positive constant. The minus sign is the whole point: it guarantees the motion is restoring rather than runaway.
简谐运动（SHM）是一种特殊情形：加速度与位移成正比，且方向始终指向平衡位置。用符号表示即 a = −ω²x，或等价地写成回复力 F = −kx，其中 k 为正的常量。负号正是关键所在：它保证运动是回复性的，而非发散性的。
= −ω²x F = −kx ω = 2πf = 2π/T
The two classic examples are a mass on a spring and a simple pendulum swinging through a small angle (typically under about 10°, so that sinθ ≈ θ in radians). Their periods are given below; notice that the spring period depends on mass and stiffness, while the pendulum period depends only on length and gravitational field strength.
两个经典例子是弹簧振子与作小角度摆动的单摆（通常小于约10°，此时以弧度计 sinθ ≈ θ）。它们的周期公式如下；注意弹簧周期只取决于质量与劲度，而单摆周期只取决于摆长与重力场强度。
T = 2π√(m/k) T = 2π√(l/g)
- Displacement x: vector, measured from equilibrium, unit m / 位移 x：矢量，从平衡位置量起，单位 m
- Amplitude A: maximum value of |x| / 振幅 A：|x| 的最大值
- Period T: unit s; Frequency f: unit Hz / 周期 T：单位 s；频率 f：单位 Hz
- SHM test: is a proportional to −x? / 判据：a 是否与 −x 成正比？
2. Equations and Graphs of SHM | 简谐运动的方程与图像
ga;t); if it starts at equilibrium moving in the positive direction, x = A sin(ωt). The velocity and acceleration follow by differentiation, which gives the two most useful relations in this topic.
任何简谐运动都可写成正弦或余弦函数，形式取决于初始条件。若物体从最大位移处开始，x = A cos(ωt)；若从平衡位置向正方向运动，则 x = A sin(ωt)。对时间求导可得速度与加速度，从而得到本节最常用的两个关系式。
v = ±ω√(A²; − x²) v₈₃₃ = ωA a₈₃₃ = ω²A
Graphically, the displacement-time graph is a sine curve, the velocity-time graph is shifted by a quarter of a period (90°) ahead, and the acceleration-time graph is exactly out of phase with displacement by 180°. Examiners frequently ask you to identify where each quantity is zero and where it is maximum, so memorise the pattern in the table below.
在图像上，位移-时间图是正弦曲线，速度-时间图相对位移图提前四分之一周期（90°），而加速度-时间图与位移图恰好反相 180°。考官常要求你指出各物理量何时为零、何时最大，因此请记住下表的规律。
| Position / 位置 | x | v | a |
| Equilibrium / 平衡位置 | 0 | maximum ωA / 最大 | 0 |
| Maximum displacement / 最大位移 | ±A | 0 | maximum ω²A / 最大 |
A useful check: the period of the velocity graph and of the acceleration graph is the same as that of displacement, but their amplitude scales with ω and ω² respectively. The direction of acceleration always points back to the centre of the motion.
一个有用的检验方法：速度图与加速度图的周期与位移图相同，但振幅分别按 ω 与 ω² 放大。加速度的方向始终指向运动的中心点。
3. Energy in Simple Harmonic Motion | 简谐运动中的能量
ant. At the equilibrium position all the energy is kinetic; at maximum displacement all of it is potential. The potential energy of a spring system is ½kx², and since ω² = k/m this can also be written as ½mω²x².
在简谐运动中，能量在动能与势能之间不断转换；若没有阻尼，总机械能保持不变。在平衡位置，全部能量都是动能；在最大位移处，全部能量都是势能。弹簧系统的势能为 ½kx²，由于 ω² = k/m，也可写成 ½mω²x²。
Eₖ = ½mv² = ½mω²(A²; − x²) Eₚ = ½mω²x²; Eₜₒₓₐₗ = ½mω²A²;
Two subtle points are worth marks. First, the kinetic energy and the potential energy each complete two full cycles for every single cycle of the motion, so their graphs oscillate with period T/2. Second, the total energy is proportional to the square of the amplitude, which is why doubling the amplitude quadruples the energy stored.
有两个易考细节值得记牢。第一，动能与势能在运动的一个完整周期内各自完成两次完整循环，因此它们的图像以 T/2 为周期振荡。第二，总能量与振幅的平方成正比，所以振幅加倍时储能变为原来的四倍。
- Kinetic energy is maximum at x = 0 / 动能最大值出现在 x = 0
- Potential energy is maximum at x = ±A / 势能最大值出现在 x = ±A
- Total energy is constant without damping / 无阻尼时总能量恒定
- Energy ∝ A² / 能量与 A² 成正比
4. Damping | 阻尼
Damping is the progressive loss of energy from an oscillating system, usually because of friction, air resistance or internal dissipation, so that the amplitude decreases with time. Damping does not normally change the period very much, but it reduces the amplitude, and the reduction is exponential rather than linear if the damping is light.
阻尼是指振动系统能量逐渐损失的过程，通常由摩擦、空气阻力或内耗引起，其结果是振幅随时间减小。阻尼通常不会明显改变周期，但会减小振幅；在轻阻尼情形下，振幅的衰减是指数式的，而非线性的。
There are three cases you must be able to describe. In light damping the system oscillates many times while the amplitude decays gradually. In critical damping the system returns to equilibrium in the shortest possible time without overshooting. In heavy damping the system creeps back to equilibrium slowly and never crosses it. Critical damping is the design target for car suspension, door closers, galvanometers and many measuring instruments.
必须能描述三种情形。轻阻尼时，系统振动多次而振幅缓慢衰减。临界阻尼时，系统在最短时间内回到平衡位置且不越过平衡点。重阻尼时，系统缓慢爬回平衡位置，永不越过。临界阻尼是汽车悬挂、闭门器、电流计以及许多测量仪器的设计目标。
| Type / 类型 | Behaviour / 表现 | Example / 例子 |
| Light / 轻阻尼 | Slow exponential decay, many oscillations / 缓慢指数衰减，多次振动 | Pendulum in air / 空气中的单摆 |
| Critical / 临界阻尼 | Fastest return, no overshoot / 最快返回，无过冲 | Car shock absorber / 汽车减震器 |
| Heavy / 重阻尼 | Slow return, no oscillation / 缓慢返回，不振动 | Paddle in thick oil / 浓油中的桨 |
5. Resonance | 共振
Resonance occurs when a system is driven at a frequency equal to its natural frequency of vibration, producing a maximum amplitude of oscillation. The phenomenon requires a periodic driving force (forced oscillation) and it is the amplitude, not the frequency, that peaks. Energy transfer from the driver to the oscillator is most efficient at this matching frequency.
当驱动频率等于系统的固有振动频率时，就会出现共振，此时振动振幅达到最大。共振需要周期性驱动力（受迫振动），峰值出现在振幅上，而不是频率上。在频率匹配时，驱动力向振动系统传递能量的效率最高。
Damping strongly affects a resonance curve. Increasing the damping lowers the peak amplitude and broadens the curve, shifting the maximum slightly to lower frequency. This is why a well-designed building is damped so that it does not respond violently to earthquake or wind excitation, and why a singer can shatter a glass by matching its natural frequency.
阻尼对共振曲线影响显著。增大阻尼会降低峰值振幅、使曲线变宽，并使最大值略向低频移动。这正是建筑要做阻尼设计、以免对地震或风激励产生剧烈响应的原因，也是歌唱家能通过与玻璃固有频率匹配而震碎它的原因。
- Musical instruments: resonance of strings and air columns / 乐器：弦与气柱的共振
- Microwave oven: resonant absorption by water molecules / 微波炉：水分子共振吸收
- MRI: nuclear magnetic resonance of hydrogen nuclei / 核磁共振：氢核的磁共振
- Risk: Tacoma Narrows bridge collapse (a classic case study) / 风险：塔科马海峡大桥倒塌（经典案例）
6. Wave Properties and Classification | 波的性质与分类
A wave is a disturbance that transfers energy and information through a medium or through space without transferring matter. The particles of the medium oscillate about fixed positions rather than travelling with the wave, which is the single most important sentence in this topic.
波是一种扰动，它通过介质或空间传递能量与信息，但不传递物质。介质中的粒子只在固定位置附近振动，而不是随波一起迁移，这是本主题中最重要的一句话。
Waves are classified in two ways. By the direction of vibration relative to propagation, we have transverse waves (vibration perpendicular to propagation, for example light and waves on a string) and longitudinal waves (vibration parallel to propagation, for example sound and compression waves in a spring). By whether a medium is needed, we have mechanical waves, which require a material medium, and electromagnetic waves, which propagate through a vacuum.
波有两种分类方式。按振动方向与传播方向的关系，可分为横波（振动方向垂直于传播方向，如光和绳波）与纵波（振动方向平行于传播方向，如声波及弹簧中的压缩波）。按是否需要介质，可分为机械波（需要物质介质）与电磁波（可在真空中传播）。
- Wavelength λ: distance between two consecutive points in phase / 波长 λ：相邻同相点之间的距离
- Frequency f: determined by the source only / 频率 f：只由波源决定
- Wave speed v: determined by the medium / 波速 v：由介质决定
- Amplitude and phase: shape and position within the cycle / 振幅与相位：波形的形状与在周期中的位置
7. The Wave Equation and Phase | 波方程与相位
The wave equation relates the three central quantities. Because the wave travels one wavelength in one period, the speed is simply wavelength divided by period, which is identical to wavelength multiplied by frequency. For electromagnetic waves in a vacuum this speed is the speed of light.
波方程把三个核心物理量联系起来。由于波在一个周期内前进一个波长，波速等于波长除以周期，也等于波长乘以频率。对真空中的电磁波而言，这个速度就是光速。
v = fλ = λ/T c = 3.00 × 10⁸ m s⁻¹
h difference Δx gives a phase difference Δφ = 2πΔx/λ. Two points separated by a whole number of wavelengths are in phase; a separation of an odd number of half wavelengths means they are in antiphase.
相位描述某点在自身周期中所处的位置。2π 弧度的相位差对应一个波长的路程差，因此任意路程差 Δx 对应的相位差为 Δφ = 2πΔx/λ。相距整数个波长的两点同相；相距奇数个半波长的两点反相。
Δφ = 2πΔx/λ
A common exam trap is to confuse intensity with amplitude. Intensity is proportional to the square of the amplitude, so if amplitude doubles at the same frequency, intensity quadruples. For a point source spreading uniformly in three dimensions, intensity follows an inverse-square law with distance.
常见考试陷阱是把强度与振幅混淆。强度与振幅的平方成正比，因此在频率不变时振幅加倍意味着强度变为四倍。对在三维空间中均匀辐射的点波源，强度随距离满足平方反比定律。
8. Superposition and Interference | 叠加与干涉
The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. Constructive interference happens when the waves arrive in phase and their displacements add; destructive interference happens when they arrive in antiphase and cancel.
叠加原理指出：当两列或更多波在某点相遇时，合位移等于各波在该点位移的矢量之和。当两列波同相到达时发生相长干涉，位移相加；当它们反相到达时发生相消干涉，位移相互抵消。
difference = nλ Destructive: path difference = (n + ½)λ
In the Young double-slit experiment, light from two coherent slits overlaps and forms a pattern of bright and dark fringes on a screen. Coherence means the sources maintain a constant phase difference, which is why a single laser or a single source with a double slit is used rather than two ordinary lamps. The fringe spacing is proportional to wavelength and screen distance, and inversely proportional to slit separation.
在杨氏双缝实验中，来自两条相干狭缝的光发生叠加，在屏上形成明暗相间的条纹图样。相干意味着两个波源保持恒定相位差，这正是实验中要使用同一激光或同一光源加双缝，而不用两盏普通灯的原因。条纹间距与波长和屏距成正比，与双缝间距成反比。
s = λD/d
Interference is not limited to light: noise-cancelling headphones generate antiphase sound waves, radio antennas use arrays of sources, and soap films show coloured interference because different wavelengths are reinforced at different thicknesses.
干涉并不限于光：降噪耳机产生反相声波，射电天线使用波源阵列，肥皂膜则因不同厚度对不同波长产生不同程度的增强而呈现彩色干涉条纹。
9. Standing Waves | 驻波
A standing (stationary) wave forms when two waves of the same frequency and amplitude travel in opposite directions through the same medium, for example when a wave reflects back on itself. The pattern appears to be stationary because the nodes, where destructive interference always occurs, remain fixed in space.
当两列同频率、同振幅的波在同一介质中沿相反方向传播时，例如波被反射回自身路径时，就会形成驻波。由于相消干涉始终发生在固定位置，节（波节）在空间上不动，整个图样看起来是静止的。
Between every two consecutive nodes there is an antinode, so the distance from a node to the next antinode is one quarter of a wavelength, and the distance between adjacent nodes is half a wavelength. Boundary conditions then decide which wavelengths are allowed, and therefore which frequencies the system can support.
每两个相邻波节之间有一个波腹，因此节点到相邻波腹的距离是四分之一波长，相邻节点之间的距离是半个波长。边界条件决定哪些波长被允许存在，从而决定系统能够支持的频率。
String fixed at both ends: fₙ = nv/(2L) Pipe open at one end: fₙ = (2n − 1)v/(4L)
| System / 系统 | Fundamental / 基频 | Harmonics / 谐频 |
| String, both ends fixed / 两端固定弦 | v/(2L) | All integer multiples of f₁ / 全部整数倍 |
| Pipe, both ends open / 两端开口管 | v/(2L) | All integer multiples of f₁ / 全部整数倍 |
| Pipe, one end closed / 一端封闭管 | v/(4L) | Odd multiples only / 仅奇数倍 |
10. Diffraction | 衍射
Diffraction is the spreading of a wave when it passes through an opening or around an obstacle. It occurs for every kind of wave, but it becomes noticeable only when the size of the gap or obstacle is comparable to the wavelength of the wave. This is why you can hear sound around a corner but cannot see around it.
衍射是波通过狭缝或绕过障碍物时发生的扩散现象。任何波都会衍射，但只有当缝宽或障碍物尺寸与该波的波长可比时，衍射才明显。这就是为什么能绕过墙角听到声音，却无法绕过墙角看到物体。
The quantity that matters is the ratio λ/d. If the wavelength is much smaller than the gap, the wave travels almost in straight lines and diffraction is negligible. If the gap is about one wavelength wide, the wave spreads out in almost semicircular wavefronts. Single-slit diffraction of light produces a broad central maximum with narrower, weaker secondary maxima on each side.
起决定作用的量是比值 λ/d。若波长远小于缝宽，波几乎沿直线传播，衍射可以忽略。若缝宽约等于一个波长，波几乎以半圆形
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