📚 Describing Oscillations | 描述振荡
Oscillatory motion appears everywhere in physics, from a mass bouncing on a spring to the pendulum of a clock and the alternating current in a circuit. In CIE A-Level Physics, describing oscillations accurately means using precise quantities such as displacement, amplitude, period, frequency, angular frequency and phase. This article explains these core ideas and shows how they lead to the special case of simple harmonic motion (SHM).
振荡运动在物理学中无处不在，从弹簧上弹跳的质量块、时钟的摆锤到电路中的交流电。在 CIE A-Level 物理中，准确描述振荡需要使用精确的物理量，例如位移、振幅、周期、频率、角频率和相位。本文将解释这些核心概念，并说明它们如何引出简谐运动这一特殊情形。
1. What is an oscillation? | 什么是振荡
An oscillation is a repeated back-and-forth motion about a fixed equilibrium position. The object moves to one side, reverses, passes through equilibrium and moves to the other side, then repeats the same path. Common examples include a mass on a spring, a simple pendulum swinging through small angles, a vibrating tuning fork and the piston in a car engine.
振荡是指物体围绕一个固定平衡位置所做的往复运动。物体先向一侧运动，然后反向运动，穿过平衡位置后到达另一侧，再重复相同的路径。常见的例子包括弹簧上的质量块、小角度摆动的单摆、振动的音叉以及汽车发动机中的活塞。
If no energy is transferred away and no external driving force is applied, the system performs free oscillations at its natural frequency. In real systems, resistive forces usually remove energy, so the amplitude gradually decreases unless an external force maintains the motion.
如果没有能量散失，也没有外部驱动力作用，系统就会以其固有频率做自由振荡。在真实系统中，阻力通常会带走能量，因此如果没有外力维持运动，振幅就会逐渐减小。
2. Key quantities for describing oscillations | 描述振荡的关键物理量
To describe oscillations clearly, physicists use a standard set of quantities. The table below summarises the most important terms you must use correctly in CIE A-Level Physics.
为了清楚地描述振荡，物理学家使用一组标准物理量。下表总结了你在 CIE A-Level 物理中必须正确使用的最重要术语。
| Quantity | Symbol | Meaning | 中文含义 |
|---|---|---|---|
| Displacement | x | Distance from equilibrium in a stated direction | 相对于平衡位置的矢量距离 |
| Amplitude | x₀ | Maximum displacement from equilibrium | 离开平衡位置的最大位移 |
| Period | T | Time for one complete oscillation | 完成一次完整振荡所需的时间 |
| Frequency | f | Number of complete oscillations per second | 每秒完成完整振荡的次数 |
| Angular frequency | ω | Rate of change of phase, measured in rad s⁻¹ | 相位变化率，单位为 rad s⁻¹ |
| Phase difference | Δφ | Fraction of a cycle by which one oscillation leads or lags another | 两个振荡之间超前或滞后的周期分数 |
Using these symbols consistently will help you set up calculations and explain graphs without ambiguity.
一致地使用这些符号可以帮助你清晰地建立计算式并解释图像，避免歧义。
3. Displacement, amplitude and equilibrium position | 位移、振幅与平衡位置
The equilibrium position is the point where the net force on the oscillating object is zero. For a mass on a horizontal spring, this is the position where the spring is at its natural length. For a pendulum, it is the lowest point of the swing.
平衡位置是振荡物体所受合外力为零的位置。对于水平弹簧上的质量块，这是弹簧处于自然长度时的位置；对于单摆，这是摆动的最低点。
Displacement x is the distance from equilibrium in a specified direction. It can be positive or negative depending on which side of equilibrium the object is on. Amplitude x₀ is the maximum value of displacement, so it is always a positive quantity. Do not describe amplitude as ‘the distance between the two extreme positions’; that distance is actually twice the amplitude.
位移 x 是物体沿指定方向离开平衡位置的距离。根据物体位于平衡位置的哪一侧，位移可以是正值或负值。振幅 x₀ 是位移的最大值，因此它始终为正。不要把振幅描述为“两个极端位置之间的距离”；那个距离实际上是振幅的两倍。
4. Period, frequency and angular frequency | 周期、频率与角频率
The period T is the time taken for one complete oscillation. Frequency f is the number of complete oscillations per second, so the two quantities are reciprocals of each other.
周期 T 是完成一次完整振荡所需的时间。频率 f 是每秒完成完整振荡的次数，因此这两个量互为倒数。
T = 1 / f and f = 1 / T
The SI unit of period is the second (s), and the SI unit of frequency is the hertz (Hz), where 1 Hz = 1 s⁻¹.
周期的国际单位是秒（s），频率的国际单位是赫兹（Hz），其中 1 Hz = 1 s⁻¹。
Angular frequency ω connects frequency to circular motion. Because one complete cycle corresponds to an angle of 2π radians, ω is given by:
角频率 ω 将频率与圆周运动联系起来。由于一个完整周期对应 2π 弧度的角度，因此 ω 由下式给出：
ω = 2πf = 2π / T
The unit of ω is rad s⁻¹. Even though the motion may be linear, ω is very useful because it simplifies the equations of simple harmonic motion.
ω 的单位是 rad s⁻¹。尽管运动可能是直线运动，但 ω 非常有用，因为它可以简化简谐运动的方程。
5. Phase and phase difference | 相位与相位差
Phase describes the stage of an oscillation within its cycle. If the displacement is given by x = x₀ sin(ωt), then the phase at time t is the angle ωt, measured in radians. Phase tells you how far through the cycle the oscillator has progressed.
相位描述振荡在周期中所处的阶段。如果位移由 x = x₀ sin(ωt) 表示，那么 t 时刻的相位就是角度 ωt，以弧度为单位。相位告诉你在周期中振荡器已经进行了多少。
Phase difference Δφ compares two oscillations at the same frequency. If one oscillation reaches a given stage before the other, it leads; if it reaches that stage later, it lags. The phase difference is related to the time difference Δt by:
相位差 Δφ 用于比较两个同频率的振荡。如果一个振荡比另一个更早到达某一阶段，则称其超前；如果更晚到达，则称其滞后。相位差与时间差 Δt 的关系为：
Δφ = 2π Δt / T
Two oscillations are in phase when Δφ = 0, and they are in anti-phase when Δφ = π rad, which is 180°. A phase difference of π/2 rad corresponds to a quarter of a cycle.
当 Δφ = 0 时，两个振荡同相；当 Δφ = π 弧度（即 180°）时，两个振荡反相。π/2 弧度的相位差对应四分之一周期。
6. Simple harmonic motion: the defining features | 简谐运动：定义特征
Simple harmonic motion is a special and very important type of oscillation. In SHM, the resultant force on the object is always directed towards the equilibrium position, and its magnitude is directly proportional to the displacement from equilibrium. This can be written as F = -kx, where k is a constant.
简谐运动是一种特殊且非常重要的振荡类型。在简谐运动中，物体所受的合力始终指向平衡位置，其大小与离开平衡位置的位移成正比。这可以写成 F = -kx，其中 k 为常量。
Since acceleration is proportional to net force, an equivalent defining condition for SHM is that acceleration is proportional to displacement and opposite in direction:
由于加速度与合外力成正比，简谐运动的一个等价定义条件是：加速度与位移成正比且方向相反：
a ∝ -x
This condition is written more precisely as a = -ω²x, where ω is the angular frequency of the motion. Any system that satisfies this equation throughout its motion is undergoing SHM.
这一条件可以更精确地写为 a = -ω²x，其中 ω 是运动的角频率。任何在整个运动过程中满足该方程的系统都在做简谐运动。
7. The acceleration equation a = -ω²x | 加速度方程 a = -ω²x
The equation a = -ω²x is the central equation of simple harmonic motion. The negative sign shows that acceleration always points towards equilibrium: when x is positive, acceleration is negative, and when x is negative, acceleration is positive. The size of the acceleration increases linearly with displacement, so it is zero at equilibrium and maximum at the extreme positions.
方程 a = -ω²x 是简谐运动的核心方程。负号表明加速度始终指向平衡位置：x 为正时加速度为负，x 为负时加速度为正。加速度的大小随位移线性增大，因此在平衡位置为零，在极端位置最大。
Solutions to this equation have the form:
该方程的解具有以下形式：
x = x₀ sin(ωt) or x = x₀ cos(ωt)
Use the sine form when timing starts at the equilibrium position with x = 0 at t = 0. Use the cosine form when timing starts at maximum positive displacement. The choice depends on the initial conditions described in the question.
如果从平衡位置开始计时，即 t = 0 时 x = 0，则使用正弦形式。如果从最大正位移处开始计时，则使用余弦形式。选择哪一种取决于题目给出的初始条件。
8. Displacement, velocity and acceleration graphs | 位移、速度与加速度图像
For x = x₀ sin(ωt), the velocity is the gradient of the displacement-time graph, and the acceleration is the gradient of the velocity-time graph. This gives:
对于 x = x₀ sin(ωt)，速度是位移-时间图像的斜率，加速度是速度-时间图像的斜率。由此可得：
x = x₀ sin(ωt)
v = ωx₀ cos(ωt)
a = -ω²x₀ sin(ωt)
The velocity graph leads the displacement graph by π/2 rad. The acceleration graph is in anti-phase with displacement, meaning it leads displacement by π rad. In exam answers, state these phase relationships clearly rather than only saying the graphs are ‘shifted’.
速度图像比位移图像超前 π/2 弧度。加速度图像与位移图像反相，也就是说它比位移图像超前 π 弧度。在考试答案中，要清楚地说明这些相位关系，而不要只说图像“发生了平移”。
The maximum speed occurs at the equilibrium position and is given by v_max = ωx₀. The maximum acceleration occurs at the extreme positions and is given by a_max = ω²x₀.
最大速度出现在平衡位置，大小为 v_max = ωx₀。最大加速度出现在极端位置，大小为 a_max = ω²x₀。
9. Energy changes in oscillations | 振荡中的能量变化
In an ideal free oscillation with no damping, the total mechanical energy remains constant. Energy continuously changes between kinetic energy and potential energy. At the equilibrium position, displacement is zero, so all the energy is kinetic and the speed is maximum. At the extreme positions, displacement equals the amplitude, so all the energy is potential and the speed is zero.
在无阻尼的理想自由振荡中，总机械能保持不变。能量不断地在动能和势能之间转化。在平衡位置，位移为零，因此所有能量都是动能，速度最大。在极端位置，位移等于振幅，因此所有能量都是势能，速度为零。
For a mass-spring system, the energy expressions are:
对于弹簧-质量系统，能量表达式为：
K = ½mω²(x₀² – x²)
U = ½mω²x²
E_total = ½mω²x₀²
These expressions show that kinetic energy and potential energy each vary sinusoidally with time, but their sum is constant. In a real system, damping causes this total energy to decrease gradually unless an external source supplies energy.
这些表达式表明，动能和势能都随时间按正弦规律变化，但它们的总和保持不变。在真实系统中，阻尼会使总能量逐渐减少，除非有外部能量源提供能量。
10. Phase difference calculations and exam skills | 相位差计算与考试技巧
To calculate phase difference from two oscillations, find the time interval Δt between corresponding points, such as two successive peaks. Then use:
要计算两个振荡之间的相位差，先找出对应点之间的时间间隔 Δt，例如两个相邻峰值之间的时间。然后使用：
Δφ = 2π Δt / T
For example, if one pendulum reaches its maximum displacement 0.25 T later than another identical pendulum, the phase difference is Δφ = 2π × 0.25 = π/2 rad. The second pendulum lags the first by 90°.
例如，如果一个单摆到达最大位移的时间比另一个相同单摆晚 0.25 T，那么相位差为 Δφ = 2π × 0.25 = π/2 弧度。第二个单摆比第一个滞后 90°。
Common exam mistakes include confusing frequency f with angular frequency ω, forgetting to use radians when calculating ωt, treating amplitude as negative, and saying that acceleration is zero at maximum displacement. Always check whether a graph shows x, v or a against t before identifying phase relationships.
常见的考试错误包括：混淆频率 f 与角频率 ω；计算 ωt 时忘记使用弧度；把振幅当作负值；认为加速度在最大位移处为零。在判断相位关系之前，一定要先确认图像表示的是 x、v 还是 a 随时间变化。
11. Summary and key equations | 总结与关键公式
When describing oscillations, start with the equilibrium position and state whether the motion is a free oscillation. Define displacement, amplitude, period, frequency and angular frequency accurately, and use phase difference to compare two oscillators. Simple harmonic motion is defined by a = -ω²x, and its solutions are sinusoidal.
描述振荡时，首先要说明平衡位置，并说明这是否为自由振荡。准确描述位移、振幅、周期、频率和角频率，并使用相位差来比较两个振荡器。简谐运动由 a = -ω²x 定义，其解为正弦或余弦函数。
The essential equations are:
关键公式如下：
T = 1 / f
ω = 2πf = 2π / T
a = -ω²x
x = x₀ sin(ωt) or x = x₀ cos(ωt)
v = ±ω√(x₀² – x²)
Master these relationships and you will be able to describe oscillations both qualitatively and quantitatively, which is exactly what CIE exam questions expect.
掌握这些关系式后，你就能从定性和定量两个方面描述振荡，这正是 CIE 考试题目所要求的。
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