📚 Oscillations and Waves: Complete A-Level Physics Revision Guide | 振动与波：A-Level 物理完整复习指南
Oscillations and waves appear across the AQA A-level Physics specification, from mass-spring systems and pendulums to sound and light. This unit links circular motion, energy conservation and wave behaviour into one coherent model. Mastering the definitions, graphs and equations in this topic is essential for both Paper 1 and Paper 2 questions, especially the written and practical skills sections.
振动与波贯穿 AQA A-level 物理考纲，从弹簧振子、单摆到声波与光波。本单元将圆周运动、能量守恒和波动行为整合为一个连贯的模型。掌握本主题中的定义、图像与方程，对 Paper 1 和 Paper 2 的题目都至关重要，尤其是文字题和实验技能部分。
This guide follows the order of the specification: it starts with the vocabulary of oscillations, builds up simple harmonic motion and its equations, then extends the same ideas to travelling waves, stationary waves, interference and diffraction. Every section ends with the exam-style points that markers actually look for, so you can use it as a checklist before the exam.
本指南按照考纲顺序展开：先建立振动的基本词汇，再构建简谐运动及其方程，随后把同样的思想扩展到行波、驻波、干涉与衍射。每一节都会指出阅卷人真正关注的高分要点，方便你在考前当作清单使用。
1. Describing Oscillations | 描述振动
An oscillation is a repeated motion about a fixed equilibrium position. The key quantities are displacement x, amplitude A, period T, frequency f, and angular frequency ω. These five quantities appear constantly in both calculation questions and definition questions, so you must be able to state them precisely and to use them without hesitation.
振动是围绕固定平衡位置的重复运动。关键物理量包括位移 x、振幅 A、周期 T、频率 f 和角频率 ω。这五个物理量在计算题和定义题中反复出现，因此必须能准确表述并熟练使用。
Displacement is the distance from equilibrium at any instant, while amplitude is the maximum displacement. Period is the time for one complete cycle, and frequency is the number of cycles per second. Note that displacement is a vector quantity measured from the equilibrium position, not from the starting point of the motion.
位移是任意时刻距平衡位置的距离，振幅是最大位移。周期是完成一次完整循环所需的时间，频率是每秒完成的循环次数。注意位移是从平衡位置量起的矢量，而不是从运动起点量起的距离。
The frequency and period are reciprocals, and angular frequency relates to ordinary frequency through 2π. Angular frequency is measured in radians per second, and it is the bridge between oscillation theory and circular motion.
频率与周期互为倒数，角频率通过 2π 与普通频率联系。角频率的单位是弧度每秒，它是振动理论与圆周运动之间的桥梁。
f = 1 / T , ω = 2πf , ω = 2π / T
The defining feature of a full cycle is not the distance travelled but the return of the system to the same displacement moving in the same direction. A common exam trap is to count a half-swing as one oscillation; a pendulum going from one extreme to the other and back counts as exactly one complete oscillation.
一个完整循环的判断标准不是经过的路程，而是系统回到相同位移并沿相同方向运动。常见陷阱是把半个摆动当成一次全振动；单摆从一侧极端位置摆到另一侧再摆回来，才算一次完整振动。
2. Simple Harmonic Motion and Its Conditions | 简谐运动及其条件
Simple harmonic motion (SHM) is defined by a very specific condition: the acceleration of the body is directly proportional to its displacement from the equilibrium position and is always directed towards the equilibrium position. This single statement is worth two marks in almost every paper, so it must be quoted in full rather than paraphrased loosely.
简谐运动（SHM）由一个非常明确的条件定义：物体加速度与它离开平衡位置的位移成正比，且方向始终指向平衡位置。这句话几乎每份卷子都值两分，必须完整表述，不能含糊改写。
a = − ω² x , where ω² = k / m
The negative sign is not decoration: it encodes the restoring nature of the force. When x is positive the acceleration is negative, and when x is negative the acceleration is positive, so the acceleration always opposes the displacement. This is why SHM is often described as motion under a linear restoring force.
负号不是装饰：它表示力的回复性质。当 x 为正时加速度为负，当 x 为负时加速度为正，加速度总与位移方向相反。这正是简谐运动常被称为"线性回复力作用下的运动"的原因。
The corresponding restoring force follows directly from Newton' ant k is the force per unit displacement, sometimes called the stiffness of the system. Its unit is N m⁻¹, and it is a fixed property of the oscillator rather than of the particular oscillation.
相应的回复力由牛顿第二定律直接得到，即 F = − k x。常数 k 是单位位移所需的力，有时称为系统的劲度系数，单位为 N m⁻¹。它是振子的固有属性，而不是某次具体振动的属性。
Many real systems only approximate SHM for small displacements. For larger amplitudes, the restoring force is no longer proportional to displacement, so the period starts to depend on the amplitude. A pendulum is the classic example: its period is almost constant up to about 10°, but drifts noticeably beyond that, which is why practical instructions always warn you to keep the angle small.
许多真实系统只在小位移下近似为简谐运动。振幅较大时，回复力不再与位移成正比，周期便开始随振幅改变。单摆是最典型的例子：在约 10° 以内周期几乎不变，超过后明显偏移，这也是实验指导总是提醒你保持摆角很小的原因。
3. Displacement, Velocity and Acceleration Graphs | 位移、速度与加速度图像
The graphs of x, v and a against time are the fastest way to identify SHM. The displacement curve is sinusoidal, and the velocity and acceleration curves are also sinusoidal but shifted in phase, so they are described as being π/2 and π out of phase with displacement respectively.
位移、速度、加速度随时间变化的图像是识别简谐运动最快的方法。位移曲线是正弦型曲线，速度与加速度曲线也是正弦型曲线，但相位发生移动，因此分别与位移相差 π/2 和 π。
At the extremes of the motion, where displacement is maximum, the velocity is momentarily zero but the acceleration is maximum. At the equilibrium position, the displacement is zero so the acceleration is zero, while the speed reaches its maximum. Recognising this pairing allows you to check any graph instantly.
在运动的极端位置，位移最大而速度瞬间为零，但加速度最大；在平衡位置，位移为零因而加速度为零，而速率达到最大。掌握这一对应关系，就能迅速检查任何图像是否正确。
- v = 0 when |x| = A, and |a| is maximum there. / 当 |x| = A 时 v = 0，此时 |a| 最大。
- a = 0 when x = 0, and |v| is maximum there. / 当 x = 0 时 a = 0，此时 |v| 最大。
- The x–t graph and the a–t graph are mirror images turned upside down. / x–t 图与 a–t 图是上下翻转的镜像关系。
A useful mathematical shortcut links speed and displacement without any reference to time, and it follows from the energy argument as well as from the equations of SHM.
有一个很有用的数学捷径可以跳过时间，直接联系速率与位移，它既可由能量推导，也可由简谐运动方程得到。
v = ± ω √A² − x² , v(max) = ωA , a(max) = ω²A
In graph questions, always state the amplitude and period first and then describe the phase relationship. Writing "the velocity graph leads the displacement graph by π/2 radians, and the acceleration graph is in antiphase with displacement" earns full marks far more reliably than saying the curves "look shifted".
在图像题中，先写出振幅和周期，再描述相位关系。写出"速度图比位移图超前 π/2 弧度，加速度图与位移图反相"比笼统地说曲线"看起来错开了"更容易拿满分。
4. Equations of Simple Harmonic Motion | 简谐运动的方程
written with a cosine or a sine, depending on the starting conditions. Choosing the right one saves time and prevents sign errors, because the initial value of the displacement is then built into the equation itself.
a = − ω² x 的通解可以写成余弦形式或正弦形式，取决于初始条件。选对形式既省时间又能避免符号错误，因为位移的初始值已经包含在方程里了。
x = A cos(ωt) if x = A at t = 0 ; x = A sin(ωt) if x = 0 at t = 0
owing because it lets you reconstruct the acceleration equation in the exam instead of memorising it separately.
对这些表达式求导，先得速度再得加速度，结果始终与 a = − ω² x 一致。这一点值得掌握，因为考试时你可以自己把它推出来，而不必单独记忆。
The angular frequency is determined by the physical properties of the oscillator, not by how far you displace it. This is the meaning of the term "natural angular frequency": it is fixed by the system itself. Two identical pendulums with different amplitudes still share the same period, provided both amplitudes remain small.
角频率由振子的物理性质决定，而不是由你把它拉开多远决定。"固有角频率"一词的含义就在于此：它由系统自身固定。两个相同但振幅不同的单摆，只要振幅都较小，周期就相同。
fore using any equation containing ω. Forgetting this factor of 2π is one of the most common sources of lost marks in this topic.
计算时一定要检查 ω 的单位。如果题目给出的是以 Hz 为单位的频率，在使用任何含 ω 的公式之前，必须用 ω = 2πf 换算。忘记这个 2π 是本主题最常见的失分原因之一。
5. Energy in Simple Harmonic Motion | 简谐运动中的能量
Energy in SHM is exchanged continuously between kinetic and potential forms, but the total remains constant if damping is negligible. The kinetic energy is maximum at the equilibrium position and zero at the extremes; the potential energy behaves in exactly the opposite way.
简谐运动中能量在动能与势能之间不断转换，若阻尼可忽略，总能量保持不变。动能和势能在平衡位置最大、在两端为零；势能的变化趋势与之完全相反。
E(k) = ½ m ω² (A² − x²) , E(p) = ½ m ω² x² , E(total) = ½ m ω² A² = ½ k A²
Plotting energy against displacement gives two parabolas that cross at x = ± A/√2, where the kinetic and potential energies are equal. This specific value appears in multiple-choice questions surprisingly often, so it is worth remembering as a landmark on the graph.
把能量对位移作图，得到两条抛物线，它们在 x = ± A/√2 处相交，此时动能与势能相等。这个特殊值在选择题中出现频率出奇地高，值得作为图像上的标志点记住。
Plotting energy against time gives smooth curves that oscillate at twice the frequency of the displacement, since both E(k) and E(p) depend on the square of a sinusoidal quantity. The total energy line is a straight horizontal line if no energy is lost.
把能量对时间作图，得到光滑曲线，其振荡频率是位移频率的两倍，因为动能与势能都依赖于正弦量的平方。若没有能量损失，总能量线是一条水平直线。
A key consequence is that total energy is proportional to the square of the amplitude, E ∝ A². So doubling the amplitude quadruples the energy stored, and this relationship is the basis of many "energy transfer" questions in Paper 1.
一个关键推论是总能量与振幅的平方成正比，即 E ∝ A²。因此振幅加倍会使储存的能量变为四倍，这一关系是 Paper 1 中许多"能量传递"题目的基础。
6. Mass-Spring Systems and Pendulums | 弹簧振子与单摆
Two oscillators dominate the specification: the mass on a spring and the simple pendulum. Both obey SHM for small displacements, but the period depends on different physical quantities in each case, and the two formulas must not be confused.
考纲中有两个核心振子：弹簧振子和单摆。两者在小位移下都满足简谐运动，但周期依赖的物理量不同，两个公式绝不能混淆。
T = 2π √m / k (mass–spring) , T = 2π √l / g (simple pendulum)
For a mass on a spring, the period depends on the mass and the spring constant but not on the amplitude or on the value of g. For a pendulum, the period depends on the length and on g but not on the mass of the bob. Questions frequently test exactly these independence statements.
对弹簧振子，周期取决于质量和劲度系数，而与振幅及 g 无关；对单摆，周期取决于摆长和 g，而与摆球质量无关。考题经常正是考查这些"无关性"的表述。
In the standard practical, you measure the time for 20 or more oscillations with a fiducial marker at the equilibrium position and divide by the number of oscillations to find T. Timing many oscillations reduces the percentage uncertainty in T, since human reaction time is then spread over a longer interval.
在标准实验中，你在平衡位置设置标记线，测量 20 次或更多次振动的总时间，再除以振动次数得到 T。测量多次振动可以减小 T 的百分比不确定度，因为这把人体的反应时间分摊到了更长的时间间隔上。
For the pendulum, plotting T² against l gives a straight line through the origin with gradient 4π² / g, so g can be found from the gradient. Plotting T² rather than T is essential because only T² versus l is linear; this is a standard requirement in practical skills papers.
对单摆，作 T² 对 l 的图像可得到过原点的直线，斜率为 4π² / g，因此可由斜率求 g。必须作 T² 而不是 T，因为只有 T² 与 l 的关系是线性的；这是实验技能卷的标准要求。
- Measure l from the pivot to the centre of mass of the bob. / 摆长 l 应从悬点到摆球质心测量。
- Use a small angular displacement, below about 10°. / 使用小于约 10° 的小角度位移。
- Repeat and average to reduce random error. / 重复测量取平均以减小随机误差。
7. Damping and Resonance | 阻尼与共振
Damping is the deliberate or incidental removal of energy from an oscillating system, usually by resistive forces such as friction or air resistance. The amplitude decreases over time and the energy is transferred to the surroundings, normally as thermal energy.
阻尼是振荡系统中能量被有意或无意耗散的过程，通常通过摩擦或空气阻力等阻力实现。振幅随时间减小，能量传递给周围环境，通常转化为热能。
| Type / 类型 | Behaviour / 表现 |
| Light damping / 轻阻尼 | Amplitude decays gradually; the period is almost unchanged. / 振幅逐渐衰减，周期几乎不变。 |
| Heavy (critical) damping / 重阻尼（临界阻尼） | Returns to equilibrium in the shortest possible time without oscillating. / 以最短时间回到平衡位置且不振动。 |
| Overdamping / 过阻尼 | Returns to equilibrium slowly without oscillating. / 缓慢回到平衡位置，不发生振动。 |
Critical damping is used in car suspension, door closers and moving-coil meters, where you want the system to settle as quickly as possible without overshooting. If you are asked to justify a design choice, name the application and state that overshoot or repeated oscillation must be avoided.
临界阻尼应用于汽车悬架、闭门器和动圈式电表中，因为这些场合希望系统尽快稳定而不出现超调。若题目要求解释设计选择，应说出具体应用，并指出必须避免超调或反复振荡。
Free oscillations occur at the natural frequency of the system. Forced oscillations occur when a periodic driving force is applied, and the system then oscillates at the driving frequency, not at its natural frequency. Resonance happens when the driving frequency equals the natural frequency, and the amplitude of vibration reaches a sharp maximum.
自由振动发生在系统的固有频率上。受迫振动发生在施加周期性驱动力时，此时系统以驱动频率而不是固有频率振动。当驱动频率等于固有频率时发生共振，振幅达到尖锐最大值。
Resonance is useful in microwave ovens and in musical instruments, but destructive in bridges and buildings, where damping is added to reduce the peak amplitude. Note that adding damping does more than reduce the peak: it also slightly lowers and broadens the resonance curve.
共振在微波炉和乐器中是有益的，但在桥梁和建筑中具有破坏性，因此需要增加阻尼以降低峰值振幅。注意增加阻尼不只是降低峰值：它还会使共振曲线略微降低并变宽。
8. Transverse and Longitudinal Waves | 横波与纵波
A wave is a means of transferring energy and information without transferring matter. The particles of the medium oscillate about fixed positions, passing energy along the wave but not travelling with it. This distinction is the single most important idea in the whole waves topic.
波是在不转移物质的前提下传递能量和信息的方式。介质中的质点围绕固定位置振动，把能量沿波传递出去，但质点本身并不随波前进。这一区别是整个波动主题中最重要的思想。
In a transverse wave, the particle displacement is perpendicular to the direction of energy transfer; polarisation is possible only for transverse waves. In a longitudinal wave, the displacement is parallel to the direction of energy transfer, producing compressions and rarefactions.
在横波中，质点位移方向与能量传播方向垂直；只有横波才能发生偏振。在纵波中，位移方向与能量传播方向平行，形成疏密相间的压缩与稀疏区。
- Transverse: light, waves on a string, S-waves. / 横波：光波、绳波、S 波。
- Longitudinal: sound in air, P-waves, ultrasound. / 纵波：空气中的声波、P 波、超声波。
- Electromagnetic waves are transverse and can travel through a vacuum. / 电磁波是横波，可以在真空中传播。
All waves obey the universal wave equation, which links speed, frequency and wavelength. Frequency is set by the source alone and does not change when the wave enters a new medium; the speed changes, so the wavelength must change to compensate.
所有波都遵循通用的波速公式，它把速度、频率和波长联系起来。频率仅由波源决定，当波进入新介质时频率不变；波速改变，因此波长必须相应改变。
v = f λ , and for light in a vacuum c = 3.00 × 10⁸ m s⁻¹
When light passes from air into glass, the frequency stays the same, the speed decreases, and therefore the wavelength decreases. Candidates who state that "the frequency decreases" lose the mark immediately, so write the reasoning in the correct order every time.
当光从空气进入玻璃时，频率不变，速度减小，因此波长减小。如果考生写"频率减小"，会立刻失分，所以务必按正确顺序写出推理。
9. Phase, Path Difference and Superposition | 相位、光程差与叠加
Phase difference describes how far one oscillation is ahead of another, measured in radians or degrees. Path difference is the corresponding difference in the distance travelled from two sources to the same point.
相位差描述一个振动比另一个超前多少，用弧度或度表示。光程差则是两列波从各自波源到达同一点所经过的距离之差。
phase difference = (2π / λ) × path difference
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 occurs when the waves arrive in phase and reinforce each other; destructive interference occurs when they arrive in antiphase and cancel.
叠加原理指出，当两列或更多波在同一点相遇时，合位移等于各分位移的矢量和。当波同相到达并相互加强时发生相长干涉；当波反相到达并相互抵消时发生相消干涉。
- Constructive: path difference = n λ, where n = 0, 1, 2, … / 相长：光程差 = n λ，n = 0, 1, 2, …
- Destructive: path difference = (n + ½) λ / 相消：光程差 = (n + ½) λ
- Coherence requires a constant phase difference, which in practice means a single source or two sources from the same laser. / 相干要求相位差恒定，实际中意味着使用单一波源或来自同一激光的两个波源。
Two waves of the same frequency can only produce a stable interference pattern if they are coherent. Two separate lamps never produce visible fringes because their phase relationship changes randomly many millions of times per second.
只有相干的两列同频波才能产生稳定的干涉图样。两盏独立的灯永远无法产生可见条纹，因为它们的相位关系每秒会随机变化数百万次。
Polarisation is the restriction of a transverse wave's oscillations to a single plane. Light can be polarised by reflection, by transmission through a polarising filter, or by scattering. Unpolarised light passing through one filter becomes plane polarised, and a second filter at 90° to the first transmits no light.
偏振是把横波的振动限制在单一平面内。光可以通过反射、通过偏振片透射或散射而偏振。非偏振光通过第一片偏振片后成为平面偏振光，与第一片成 90° 的第二片偏振片则不透过任何光。
10. Stationary Waves | 驻波
A stationary (standing) wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions through the same medium. Energy is not transferred along a stationary wave; instead it is stored and exchanged within the regions between the nodes.
驻波由两列频率和振幅相同、沿相反方向在同一介质中传播的行波叠加而成。驻波不沿波传播能量，而是把能量储存在波节之间的区域内并不断转换。
A node is a point of permanently zero displacement, while an antinode is a point of maximum displacement. The distance between adjacent nodes, or between adjacent antinodes, is always half a wavelength, and the distance from a node to the nearest antinode is a quarter of a wavelength.
波节是位移始终为零的点，波腹是位移最大的点。相邻波节之间或相邻波腹之间的距离总是半个波长，而波节到最近波腹的距离是四分之一波长。
node spacing = λ / 2 , antinode spacing = λ / 2 , node–antinode spacing = λ / 4
All points between two nodes oscillate in phase with one another, and points in adjacent loops oscillate in antiphase. This is why a stationary wave has a fixed envelope rather than the moving pattern of a progressive wave.
两个波节之间的所有点同相振动，而相邻波腹区间的点反相振动。这就是驻波具有固定包络、而不像行波那样整体移动的原因。
ngth. Higher harmonics correspond to an increasing number of loops, giving length = n λ / 2.
在两端固定的弦上，基频模式的波节位于两端、中间有一个波腹，因此弦长等于半个波长。更高次谐波对应更多个波腹，于是弦长 = n λ / 2。
dentify the number of loops, state that this equals the number of half wavelengths, and calculate the wave speed from v = f λ.
经典实验使用振动发生器和细绳，通过调节张力或长度直到出现清晰的波节图样。你应识别波腹的个数，说明它等于半波长的个数，再由 v = f λ 计算波速。
11. Interference and the Double Slit | 干涉与双缝实验
In Young's double slit experiment, monochromatic light passes through two narrow slits separated by a small distance, and an interference pattern of bright and dark fringes is observed on a screen. The bright fringes are formed where the path difference between the two slits is a whole number of wavelengths.
在杨氏双缝实验中，单色光通过两条间距很小的狭缝，在屏幕上形成明暗相间的干涉条纹。当两缝到某点的光程差为波长的整数倍时，形成明纹。
= ( a x ) / D , where a = slit separation, x = fringe spacing, D = slit-to-screen distance
The fringe spacing x is measured from the centre of one bright fringe to the centre of the next, and it is best obtained by measuring across many fringes and dividing. This reduces the percentage uncertainty significantly, exactly as with the pendulum timing.
条纹间距 x 从一条明纹中心量到相邻明纹中心，最好测量多条条纹的总宽度再除以条纹数。这与单摆计时同理，可以显著降低百分比不确定度。
To increase the fringe spacing, you can increase the wavelength, increase the distance from slits to screen, or decrease the slit separation. Do not confuse slit separation a with slit width: the slit width controls diffraction, while the slit separation controls the fringe spacing.
要增大条纹间距，可以增大波长、增大缝到屏的距离，或减小缝间距。切勿把缝间距 a 与缝宽混为一谈：缝宽决定衍射的展宽程度，而缝间距决定条纹间距。
Using white light instead of monochromatic light produces a central white fringe with coloured fringes on either side, because different wavelengths have different fringe spacings. Using a laser makes the pattern much brighter and clearly visible, and its coherence makes the fringes stable.
若用白光代替单色光，中央是白色明纹，两侧是彩色条纹，因为不同波长的条纹间距不同。使用激光可使图样亮得多、清晰可见，且激光的相干性使条纹稳定。
12. Diffraction, Diffraction Gratings and Refraction | 衍射、衍射光栅与折射
Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. The amount of spreading increases as the gap width becomes comparable to the wavelength, and becomes most pronounced when the gap width is approximately equal to the wavelength.
衍射是波通过狭缝或绕过障碍物时发生扩散的现象。当缝宽与波长可比时，扩散程度增大；当缝宽约等于波长时，衍射最为显著。
has a wavelength of order 5 × 10⁻⁷ m. A doorway is therefore comparable in size to a sound wavelength but enormous compared with a light wavelength.
这就是声音能明显绕过门框而光不能的原因：可听声波的波长约为米的数量级，而可见光波长约为 5 × 10⁻⁷ m。门框与声波波长相仿，但相对于光波长则极为巨大。
A diffraction grating has many parallel slits with a very small separation, producing much sharper and brighter maxima than a double slit. The condition for constructive interference for a grating is given by the grating equation.
衍射光栅有许多间距极小的平行狭缝，产生的极大值比双缝更锐利、更明亮。光栅的相长干涉条件由光栅方程给出。
n λ = d sin θ , where d = 1 / N and N is the number of lines per metre
In a practical measurement, you measure the angle between the zero-order maximum and the first-order maximum
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