📚 Representing s.h.m. graphically | 简谐运动的图形表示
Graphs provide a clear and immediate way to understand how displacement, velocity, acceleration and energy change during simple harmonic motion. In CIE A-Level Physics, you must be able to sketch, interpret and compare these graphs, and use them to deduce key quantities such as amplitude, period, angular frequency and phase differences. This article works through each important graph, explains their shapes in terms of the underlying equations, and shows how they link together to describe the complete behaviour of an s.h.m. system.
图形提供了清晰而直观的方式，帮助我们理解简谐运动中位移、速度、加速度和能量如何变化。在 CIE A-Level 物理中，你需要能够绘制、解读和比较这些图形，并利用它们推导出振幅、周期、角频率和相位差等关键量。本文将逐一介绍各种重要图形，结合基础方程解释它们的形状，并展示它们如何联系在一起，完整描述简谐运动系统的行为。
1. Introduction to Graphical Representation | 图形表示简介
Simple harmonic motion is defined by the condition that the restoring acceleration is directly proportional to the displacement from equilibrium and always directed towards that equilibrium. Mathematically, a ∝ −x. When this condition is satisfied, the displacement, velocity and acceleration all vary sinusoidally with time. Plotting these quantities against time or against displacement reveals fundamental relationships that are easy to interpret, such as the phase lead of velocity over displacement, the antiphase relationship between acceleration and displacement, and the constant total energy of an undamped system.
简谐运动的定义是恢复加速度与离开平衡位置的位移成正比，并且始终指向平衡位置，即 a ∝ −x。当满足这一条件时，位移、速度和加速度都随时间作正弦变化。将这些物理量对时间或位移作图，可以直观地揭示它们之间的基本关系，例如速度超前位移的相位、加速度与位移的反相关系，以及无阻尼系统恒定的总能量。
2. Displacement–Time Graph | 位移–时间图
The displacement–time (x– brium position (x = 0) and moves in the positive direction at t = 0, the displacement can be written as x = A sin(ωt), where A is the amplitude and ω is the angular frequency. The graph crosses the origin and the slope at this point gives the maximum speed vmax.
理想简谐运动的位移–时间图是一条纯正弦或余弦曲线。具体是哪种波形取决于初始条件。如果振子在 t = 0 时从平衡位置 (x = 0) 开始沿正方向运动，位移可写为 x = A sin(ωt)，其中 A 是振幅，ω 是角频率。该图经过原点，此处的斜率等于最大速率 vmax。
raph then begins at a crest. In either case, the distance from the time axis to a peak (or trough) is the amplitude A. The time between two successive corresponding points – such as crest to crest – oscillations per second, related by f = 1/T, and the angular frequency is ω = 2πf = 2π/T.
如果运动从粒子位于正最大位移处 (x = +A) 并由静止释放开始，则位移遵循 x = A cos(ωt)，图形从一个波峰开始。无论哪种情况，时间轴到波峰（或波谷）的距离均为振幅 A。两个相邻的对应点（如波峰到波峰）之间的时间即为周期 T。频率 f 是每秒完成的完整振动次数，满足 f = 1/T，而角频率 ω = 2πf = 2π/T。
From an x– hifts left for positive φ and right for negative φ. In examinations, you may be asked to sketch or identify x–t graphs with different phase offsets.
从 x–t 图中还可以确定相位常数 φ（如果时间零点的选取不同）。一般表达式为 x = A sin(ωt + φ)。φ 为正时，图形向左平移；φ 为负时，图形向右平移。考试中可能会要求你画出或识别具有不同相位偏移的位移–时间图。
3. Velocity–Time Graph | 速度–时间图
Velocity in s.h.m. is the gradient of the displacement– t). This is also a sinusoidal function, but it is shifted in phase relative to the displacement. The maximum speed is vmax through equilibrium. At the extreme positions (x = ±A), the velocity is momentarily zero.
简谐运动中的速度是位移–时间图的斜率。如果 x = A sin(ωt)，求导后得到 v = dx/dt = ωA cos(ωt)。这同样是一个正弦函数，但相对于位移有一个相位移动。最大速率为 vmax = ωA，发生在振子通过平衡位置时。在两端极限位置 (x = ±A)，速度瞬时为零。
sin(ωt + ½π), showing that the velocity leads the displacement by ½π radians (90°). The v–t graph is therefore a cosine wave when the x– e wave. Its amplitude is ωA, and like the displacement graph its period is T. The velocity– changes: positive v means motion in the positive direction, negative v means the opposite.
将 v 写成 ωA cos(ωt) 也可改写为 v = ωA sin(ωt + ½π)，表明速度超前位移 ½π 弧度 (90°)。因此，当 x–t 图是正弦波时，v–t 图是余弦波。它的幅度为 ωA，且与位移图的周期相同，均为 T。从速度–时间图中可以读出任意时刻的速率并看到方向变化：正 v 表示朝正方向运动，负 v 则相反。
It is also useful to know the speed–displacement relation v = ± ω√A² − x² argest at x = 0 and zero at the turning points. When sketching graphs, remember that the v–t curve has its maximum and minimum exactly when the x– through zero, and it crosses zero exactly at the times when x is at a maximum or minimum.
还有一个有用的关系式是 v = ± ω√A² − x²，表示速率与位移的关系。虽然这不是 v – t 图，但它解释了为什么速率在 x = 0 处最大，在端点处为零。在画图时要记住：当 x–t 曲线过零点时，v–t 曲线恰好达到最大或最小值；当 x 处于最大或最小值时，v–t 曲线正好穿过零点。
4. Acceleration–Time Graph | 加速度–时间图
n as a = −ω²x. If x = A sin(ωt), then a = −ω²A sin(ωt) = ω²A sin(ωt + π). The acceleration–time graph is therefore a sine wave that is inverted relative to the x– raph, or equivalently, it is ½π ahead of the velocity graph. The maximum acceleration is amax the extreme displacements. At the equilibrium position (x = 0), the acceleration is zero.
简谐运动的定义式给出加速度 a = −ω²x。如果 x = A sin(ωt)，则 a = −ω²A sin(ωt) = ω²A sin(ωt + π)。因此，加速度–时间图相对于 x–t 图是倒置的正弦波，或者说它超前速度图 ½π。最大加速度为 amax = ω²A，出现在两个端点。在平衡位置 (x = 0) 处，加速度为零。
ally important: it tells us that acceleration is always directed towards the equilibrium position. On an a–t graph, this is seen as a mirror image of the x– placement is positive and decreasing, acceleration is negative; when displacement is negative, acceleration is positive. The period and frequency of the a–t graph are the same as those of the x–t and v–t graphs.
a = −ω²x 中的负号具有重要的物理意义：它表明加速度始终指向平衡位置。在 a–t 图上，这表现为 x–t 图关于时间轴的镜像：当位移为正且减小时，加速度为负；当位移为负时，加速度为正。a–t 图的周期和频率与 x–t 和 v–t 图相同。
5. Phase Relationships in s.h.m. | 简谐运动的相位关系
The three time-domain graphs (x–t, v–t, a– ωt + π). This means that the velocity phasor is 90° ahead of displacement, and acceleration is 180° ahead of displacement, which is equivalent to being in antiphase.
三个时域图 (x–t, v–t, a–t) 通过相位紧密联系在一起。以 x = A sin(ωt) 为基准，速度 v = ωA sin(ωt + π/2)，加速度 a = ω²A sin(ωt + π)。这意味着速度相量超前位移 90°，加速度超前位移 180°，等同于反相（即相位差为 π 弧度）。
The table below summarises the phase differences:
下表归纳了这些相位差：
| Quantity | Phase relative to x = A sin(ωt) | Phase relative to v |
|---|---|---|
| Displacement x | 0 (reference) | Lags v by π/2 |
| Velocity v | Leads x by π/2 (90°) | 0 (
Published by TutorHao | A-Level Physics Revision Series | aleveler.com 广告 / Sponsored Cambridge International AS and A Level Physics Coursebook … 物理 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥584.0更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |