📚 Harmonic Oscillations | 简谐振动
In IB Mathematics, harmonic oscillations provide an elegant framework for modelling periodic phenomena such as tides, sound waves, and mechanical vibrations. Building directly on trigonometric functions and calculus, this topic introduces amplitude, angular frequency, phase shift, and the second-order differential equation that governs simple harmonic motion. Mastering harmonic oscillations is essential for success in both Analysis & Approaches and Applications & Interpretation, particularly when tackling real-world modelling and calculus-based problems.
在 IB 数学中，简谐振动为潮汐、声波和机械振动等周期性现象提供了一个优雅的建模框架。本专题直接建立在三角函数和微积分的基础上，引入了振幅、角频率、相移以及支配简谐运动的二阶微分方程。掌握简谐振动对分别在分析与方法和应用与解释两门课程中取得成功至关重要，尤其是在处理现实世界建模和基于微积分的问题时。
1. What is Simple Harmonic Motion? | 什么是简谐运动？
Simple harmonic motion (SHM) is a special type of periodic oscillation in which the restoring 'force' is directly proportional to the displacement from a fixed equilibrium position and acts in the opposite direction. In mathematical modelling, we work purely with the displacement function x(t) and its derivatives, without referring to physical forces. The motion is characterised by symmetry about the centre, with acceleration always pointing towards equilibrium.
简谐运动 (SHM) 是一种特殊的周期性振荡，其恢复“力”与偏离固定平衡位置的位移成正比，且方向相反。在数学建模中，我们仅处理位移函数 x(t) 及其导数，而不涉及物理力。该运动的特点是关于中心对称，加速度始终指向平衡位置。
The defining feature in calculus terms is that the second derivative of the displacement is proportional to the negative of the displacement itself: x''(t) ∝ −x(t). This simple relationship generates sine and cosine solutions.
用微积分术语来说，其定义特征是位移的二阶导数与位移本身的负值成正比：x''(t) ∝ −x(t)。这种简单的关系生成了正弦和余弦解。
2. The General Displacement Equation | 一般位移方程
The displacement x of a particle undergoing SHM can be written in either of two equivalent forms:
进行简谐运动质点的位移 x 可以用以下两种等价形式之一表示：
x(t) = A sin(ωt + φ)
x(t) = A sin(ωt + φ)
x(t) = A cos(ωt + φ)
x(t) = A cos(ωt + φ)
al; if it starts at maximum displacement, the cosine form with φ=0 is more convenient.
两种表达式可互换使用。正弦和余弦形式仅相差 π/2 的相移，因为 cos(θ) = sin(θ + π/2)。选择取决于初始条件：如果在 t=0 时质点从平衡位置 (x=0) 开始运动，那么 φ=0 的正弦形式是自然的；如果从最大位移处开始，则 φ=0 的余弦形式更方便。
3. Key Parameters: Amplitude, Angular Frequency, Phase | 关键参数：振幅、角频率、相位
Amplitude A is the maximum absolute displacement from the equilibrium. It is a positive constant representing the size of the oscillation. In a pure SHM model, A remains unchanged over time.
振幅 A 是相对于平衡位置的最大绝对位移。它是一个代表振动大小的正常数。在纯简谐运动模型中，A 不随时间变化。
Angular frequency ω (omega) is measured in radians per unit time (rad s−¹ if time is in seconds). It determines how rapidly the oscillation cycles occur. The higher the ω, the more cycles are completed in a given interval.
角频率 ω（欧米伽）以每单位时间的弧度来度量（若时间为秒，则为 rad s−¹）。它决定了振动循环发生的快慢。ω 越高，在给定时间间隔内完成的循环次数就越多。
Phase shift φ at t=0. For example, if φ = π/3, the sine curve is shifted left by π/(3ω).
相移 φ（phi），也称为初相，会将整个图形水平移动。它规定了系统在 t=0 时的状态。例如，若 φ = π/3，正弦曲线向左移动 π/(3ω)。
A quick reference table connects these parameters to the physical meaning:
一个快速参考表格将这些参数与物理含义联系起来：
| Symbol | Name | Role in SHM |
|---|---|---|
| A | Amplitude | Maximum displacement |
| ω | Angular frequency | Rate of oscillation |
| φ | Phase shift | Starting position on wave |
| 符号 | 名称 | 在简谐运动中的作用 |
|---|---|---|
| A | 振幅 | 最大位移 |
| ω | 角频率 | 振动速率 |
| φ | 相移 | 波上的起始位置 |
4. Period and Frequency | 周期与频率
The period T is the time taken to complete one full oscillation, after which the motion repeats. It is related to the angular frequency by the formula T = 2π/ω. The frequency f, measured in hertz (Hz) or cycles per unit time, is the reciprocal of the period: f = 1/T = ω/(2π).
周期 T 是完成一次完整振动所需的时间，之后运动重复。它与角频率的关系由公式 T = 2π/ω 给出。频率 f 以赫兹 (Hz) 或每单位时间循环数来度量，是周期的倒数：f = 1/T = ω/(2π)。
g in the ocean with a period of 4 seconds yields ω = 2π/4 = π/2 rad s−¹.
在许多 IB 问题中，你会被给出周期或频率，必须在建立模型之前确定 ω。例如，一个在海中上下浮动的浮标周期为 4 秒，可得 ω = 2π/4 = π/2 rad s−¹。
5. Velocity and Acceleration | 速度与加速度
By differentiating the displacement function, we obtain expressions for velocity and acceleration. Starting from x(t) = A sin(ωt + φ), we have:
通过对位移函数求导，我们可以得到速度和加速度的表达式。从 x(t) = A sin(ωt + φ) 出发，有：
v(t) = x'(t) = Aω cos(ωt + φ)
v(t) = x'(t) = Aω cos(ωt + φ)
a(t) = x''(t) = −Aω² sin(ωt + φ) = −ω² x(t)
a(t) = x''(t) = −Aω² sin(ωt + φ) = −ω² x(t)
The maximum speed is vmax = Aω, and it occurs when the particle passes through equilibrium. The maximum magnitude of acceleration is amax = Aω², occurring at the extreme points of motion. Notice that acceleration is always opposite in sign to displacement, confirming that it is directed towards equilibrium.
最大速率为 vmax = Aω，当质点通过平衡位置时达到。加速度的最大大小为 amax = Aω²，出现在运动的端点处。注意加速度的符号总是与位移相反，这证实了它指向平衡位置。
6. Differential Equation of SHM | 简谐运动的微分方程
From the acceleration expression we obtain the fundamental second-order differential equation that defines SHM:
由加速度表达式，我们得到定义简谐运动的基本二阶微分方程：
x''(t) = −ω² x(t) or equivalently x'' + ω² x = 0
x''(t) = −ω² x(t) 或等价地 x'' + ω² x = 0
must be a linear combination of sin(ωt) and cos(ωt). Students often encounter this form when verifying that a given function models SHM: simply differentiate twice and check that x'' = −ω²x.
这个方程表明 x 的二阶导数等于 x 本身乘以一个负常数。任何满足该微分方程的函数都必定是 sin(ωt) 和 cos(ωt) 的线性组合。学生在验证某个给定函数是否构成简谐运动模型时经常会遇到这种形式：只需对函数求两次导，并检查是否满足 x'' = −ω²x 即可。
For example, show that x = 5 cos(3t + 1) represents SHM: x' = −15 sin(3t + 1), x'' s(3t + 1) = −9 × 5 cos(3t + 1) = −9x. Here ω² = 9, so ω = 3.
例如，证明 x = 5 cos(3t + 1) 代表简谐运动：x' = −15 sin(3t + 1)，x'' = −45 cos(3t + 1) = −9 × 5 cos(3t + 1) = −9x。此处 ω² = 9，因此 ω = 3。
7. Graphical Analysis | 图形分析
n a(t) = −Aω² sin(ωt) is exactly out of phase with displacement by π.
简谐运动的位移–时间、速度–时间和加速度–时间图形均为正弦曲线。对于 x(t) = A sin(ωt)，速度 v(t) = Aω cos(ωt) 超前位移 π/2 弧度（四分之一周期），而加速度 a(t) = −Aω² sin(ωt) 与位移恰好反相 π。
Below is a summary of the key relationships among the three graphs:
以下是三个图形之间关键关系的总结：
| Graph | Amplitude | Phase relative to displacement |
|---|---|---|
| Displacement x | A | Reference |
| Velocity v | Aω | Leads by π/2 |
| Acceleration a | Aω² | Out of phase by π (opposite) |
| 图形 | 振幅 | 相对于位移的相位 |
|---|---|---|
| 位移 x | A | 参考 |
| 速度 v | Aω | 超前 π/2 |
| 加速度 a | Aω² | 反相 π（相反） |
When sketching, ensure the velocity curve crosses the time axis when x is at a maximum or minimum; the acceleration curve reaches its peaks when x is at its extremes. These graphical features are often assessed in Paper 1 and Paper 2.
在画草图时，请确保当 x 处于最大值或最小值时，速度曲线穿越时间轴；当 x 处于端点时，加速度曲线达到
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