📚 Edexcel A-Level Physics Topic 5.5: Stationary Waves | Edexcel A-Level 物理 5.5：驻波
Stationary waves are an essential part of the Edexcel A-Level Physics specification, appearing in both the AS and A2 units. They are also one of the most predictable topics for practical-style questions, especially those involving strings, air columns and resonance. This guide explains the key ideas, experiments, equations and exam traps you need to master for Topic 5.5, and it links each idea back to the way Edexcel actually asks the questions.
驻波是 Edexcel A-Level 物理大纲中的重要内容，在 AS 和 A2 单元中都会出现。它也是实验类题目中最常考的主题之一，尤其是涉及弦、空气柱和共振的问题。本指南将带你梳理关键概念、实验、公式和考试中需要避开的常见陷阱，并把每个知识点对应到 Edexcel 实际的出题方式上。
1. What Are Stationary Waves? | 什么是驻波？
A stationary wave, also called a standing wave, is a wave pattern that is fixed in space rather than travelling through a medium. It is produced when two progressive waves of the same frequency, wavelength and amplitude travel in opposite directions and interfere. The resulting pattern appears to stand still, with clear positions of zero displacement and positions of maximum displacement. Unlike a progressive wave, a stationary wave does not transfer energy from one place to another.
驻波，也称为定波，是一种固定在空间中而不沿介质传播的波形。它由两列频率、波长和振幅相同但传播方向相反的行波叠加干涉而产生。形成的波形看起来静止不动，具有明确的位移为零的位置和位移最大的位置。与行波不同，驻波不把能量从一个地方传递到另一个地方。
The two travelling waves that form a stationary wave are usually the incident wave and its reflection. That is why stationary waves are so closely linked to boundaries: a fixed end, a free end, a wall, or the open end of a tube. If there is no reflection, there is no stationary wave.
形成驻波的两列行波通常是入射波及其反射波。这就是驻波与边界联系如此紧密的原因：固定端、自由端、墙壁或管子的开口端都可以产生反射。没有反射，就没有驻波。
2. Nodes and Antinodes | 波节与波腹
A node is a point in a stationary wave where the displacement is always zero. An antinode is a point where the displacement oscillates with maximum amplitude. At a node the two interfering waves always cancel because they arrive exactly out of phase, while at an antinode they always reinforce because they arrive in phase.
波节是驻波中位移始终为零的点。波腹是位移以最大振幅振荡的点。在波节处，两列干涉波总是相互抵消，因为它们到达时恰好反相；而在波腹处，它们总是相互加强，因为它们到达时同相。
Two neighbouring nodes are separated by half a wavelength, and the distance from a node to the nearest antinode is one quarter of a wavelength. These two facts are the fastest route to almost every calculation question in this topic.
相邻两个波节之间的距离是半个波长，而从波节到最近波腹的距离是四分之一个波长。把握这两点，是解决本主题几乎所有计算题的最快方法。
Node-to-node spacing = λ ÷ 2 | Node-to-antinode spacing = λ ÷ 4
- Nodes: zero amplitude, maximum phase difference between the two waves. 波节：振幅为零，两波相位差最大。
- Antinodes: maximum amplitude, waves always in phase. 波腹：振幅最大，两波始终同相。
- Amplitude varies along the wave; wavelength and frequency are the same everywhere. 振幅沿波形变化；波长和频率处处相同。
3. Stationary Waves vs Progressive Waves | 驻波与行波的区别
Edexcel frequently asks you to compare the two types of wave directly, so learn the comparison as a table rather than as separate facts. The most heavily marked differences are energy transfer, amplitude behaviour and the phase relationship between neighbouring points.
Edexcel 经常要求你直接比较这两类波，所以最好把对比记成表格，而不是分散的知识点。得分最高的区别点是能量传递、振幅行为和相邻点之间的相位关系。
| Feature 特征 | Stationary wave 驻波 | Progressive wave 行波 |
| Energy 能量 | No net transfer 无净传递 | Transfers energy 传递能量 |
| Amplitude 振幅 | Varies from 0 to maximum 从 0 到最大变化 | Same for all points 所有点相同 |
| Phase 相位 | All points between two nodes in phase 两波节之间同相 | Points have a continuous phase difference 相位差连续变化 |
| Wavelength 波长 | Twice the node separation 为波节间距的两倍 | Distance between two successive crests 相邻两波峰间距 |
A common exam statement to remember: in a stationary wave, points on opposite sides of a node are in antiphase, which means they are 180° or π radians out of phase.
一个常考的结论要记住：在驻波中，位于波节两侧的点处于反相状态，即相位差为 180° 或 π 弧度。
4. Formation by Superposition | 叠加形成机制
A stationary wave forms by the principle of superposition: when two waves meet, the resultant displacement at any instant is the vector sum of the individual displacements. Where the waves are in phase the displacements add, and where they are in antiphase they cancel.
驻波通过叠加原理形成：当两列波相遇时，任意时刻的合位移是各波位移的矢量和。两波同相处位移相加，两波反相处位移相互抵消。
For a stable stationary wave you need the following conditions to be satisfied:
要形成稳定的驻波，需要满足以下条件：
- The two waves must have the same frequency. 两列波必须频率相同。
- They must have the same wavelength, which follows from having the same speed in the same medium. 它们必须波长相同，这由同一介质中波速相同决定。
- They must have approximately the same amplitude. 它们必须具有大致相同的振幅。
- They must travel in opposite directions. 它们必须沿相反方向传播。
In a real experiment the amplitudes are not perfectly equal because some energy is lost at the boundary, so the nodes show a small residual vibration rather than a perfect zero. Examiners accept this as long as you state that the amplitudes are approximately equal.
在实际实验中，振幅不会完全相等，因为部分能量在边界处损失，所以波节处会有轻微的残余振动，而不是完美的零。只要说明振幅近似相等，考官就认可这一点。
5. Stationary Waves on a Stretched String | 弦上的驻波
A string fixed at both ends reflects waves at each end, producing a stationary wave with a node at each fixed end. The simplest pattern, called the first harmonic or fundamental mode, has a single antinode at the centre, so the length of the string equals half a wavelength.
两端固定的弦会在两端反射波，产生在两端各有一个波节的驻波。最简单的波形称为第一谐波或基频模式，中心只有一个波腹，因此弦长等于半个波长。
L = n λ ÷ 2 → λ = 2L ÷ n → f = n v ÷ 2L
higher harmonic is an integer multiple of the fundamental, which is why a plucked string produces a musical note with a definite pitch plus a set of overtones.
这里的 n = 1, 2, 3, ... 是谐波数。基频对应 n = 1，等于 v ÷ 2L。每一个更高的谐波都是基频的整数倍，这就是拨动琴弦会产生具有确定音高的音以及一组泛音的原因。
The speed of a transverse wave on a string depends on the tension T and the mass per unit length μ:
弦上横波的速度取决于张力 T 和单位长度质量 μ：
v = √T ÷ μ
Combining the two equations shows that increasing tension raises the frequency and increasing mass per unit length lowers it. This is exactly why a guitar is tuned by tightening the strings and why thick bass strings sound lower than thin ones.
把两个公式结合可以看出：增大张力会提高频率，增大单位长度质量会降低频率。这正是吉他通过拧紧琴弦来调音、以及粗的低音弦听起来比细弦低沉的原因。
6. Stationary Waves in Air Columns | 空气柱中的驻波
In a tube, the reflection happens at the ends. At a closed end there must be a displacement node, because the air cannot move against a solid barrier. At an open end there is approximately a displacement antinode, because the air is free to vibrate.
在管中，反射发生在端部。在封闭端必然存在位移波节，因为空气无法贴着固体障碍物运动。在开口端则近似存在位移波腹，因为空气可以自由振动。
This gives two standard cases that you must be able to write down instantly:
由此得到两种必须能立即写出的标准情形：
| Tube 管 | Ends 端 | Length condition 长度条件 | Fundamental 基频 |
| Open pipe 开管 | Antinode at both ends 两端为波腹 | L = n λ ÷ 2 | f₁ = v ÷ 2L |
| Closed pipe 闭管 | Node at closed end, antinode at open end 闭端波节、开口端波腹 | L = (2n − 1) λ ÷ 4 | f₁ = v ÷ 4L |
A closed pipe therefore has a fundamental frequency one octave lower than an open pipe of the same length, and it only supports odd harmonics: 1st, 3rd, 5th and so on. Ask yourself at every step: how many nodes and antinodes does this mode have?
因此，同样长度的闭管基频比开管低一个八度，而且闭管只支持奇次谐波：第 1、第 3、第 5 等。每做一步都问自己：这个模式有几个波节和几个波腹？
7. Resonance, Fundamental Frequency and Harmonics | 共振、基频与谐波
Resonance occurs when a system is driven at one of its natural frequencies, so that energy is transferred efficiently and the amplitude of vibration reaches a maximum. A stationary wave pattern only exists at these natural frequencies, which are set by the length, tension or geometry of the system.
共振发生在系统被以其固有频率之一驱动时，此时能量高效传递，振动幅度达到最大。驻波图样只在这些固有频率下存在，而这些频率由系统的长度、张力或几何形状决定。
- Fundamental mode (1st harmonic): the lowest frequency at which a stationary wave forms. 基频模式（第 1 谐波）：形成驻波的最低频率。
- Harmonics: frequencies that are integer multiples of the fundamental in strings and open pipes. 谐波：在弦和开管中为基频整数倍的频率。
- Overtones: any harmonic above the fundamental. 泛音：基频以上的任何谐波。
In a resonance tube experiment, the air column length is adjusted until a loud sound is heard. At that length the driving frequency matches a natural frequency of the column, so a stationary wave with a large amplitude builds up.
在共振管实验中，调节空气柱长度直到听到响亮的声音。在该长度下，驱动频率与空气柱的固有频率匹配，因此形成大幅度驻波。
8. Using the Wave Equation | 波方程的使用
All the calculations in this topic combine the wave equation with a length condition. The wave equation links speed, frequency and wavelength, and it applies to both travelling waves and stationary waves because the wavelength and frequency are the same everywhere in the pattern.
本主题的所有计算都把波方程与长度条件结合起来。波方程联系波速、频率和波长，它对行波和驻波都适用，因为在整个图样中波长和频率处处相同。
v = f λ
A reliable working method for any numerical question is:
任何数值题都可以用以下可靠步骤解决：
- Step 1: sketch the mode and mark the nodes and antinodes. 第一步：画出该模式，标出波节和波腹。
- Step 2: write the length condition, for example L = 3λ ÷ 2 or L = 3λ ÷ 4. 第二步：写出长度条件，例如 L = 3λ ÷ 2 或 L = 3λ ÷ 4。
- Step 3: rearrange to find λ, then use v = f λ to find f, v or λ. 第三步：变形求出 λ，再用 v = f λ 求 f、v 或 λ。
- Step 4: check the number of nodes and antinodes against the question. 第四步：用题目条件核对波节和波腹的数目。
= λ, so λ = 0.60 m. If the wave speed is 240 m s⁻¹, then f = v ÷ λ = 240 ÷ 0.60 = 400 Hz.
例如，一根长 0.60 m 的弦以第二谐波振动时 L = λ，所以 λ = 0.60 m。若波速为 240 m s⁻¹，则 f = v ÷ λ = 240 ÷ 0.60 = 400 Hz。
9. Core Practical: Investigating Standing Waves | 核心实验：探究驻波
The standard practical uses a vibration generator attached to a stretched string passing over a pulley with hanging masses. The generator is driven by a signal generator, and the frequency is adjusted until a clear stationary wave pattern with visible nodes appears.
标准实验使用固定在张紧弦上的振动发生器，弦跨过滑轮并悬挂砝码。发生器由信号发生器驱动，调节频率直到出现清晰的、能看到波节的驻波图样。
- Measure the distance between several nodes, then divide to find the mean node spacing. 测量多个波节之间的距离，再相除得到平均波节间距。
- Double the node spacing to obtain the wavelength. 把波节间距乘以 2 得到波长。
- Record the driving frequency f and the hanging mass, then calculate tension T = mg. 记录驱动频率 f 和悬挂质量，然后计算张力 T = mg。
- Repeat for several different tensions and plot a suitable graph. 对几个不同张力重复，绘制合适的图像。
tension, then plot v² against T. The gradient of this graph equals 1 ÷ μ, so μ is the reciprocal of the gradient.
要测出单位长度质量 μ，可用 v = f λ 求出每个张力下的波速，然后以 v² 对 T 作图。该图像的斜率等于 1 ÷ μ，所以 μ 是斜率的倒数。
The main sources of uncertainty are the measurement of node positions, the assumption that nodes are exactly at the ends, and the fact that the string is not perfectly uniform. Repeating with more nodes reduces the percentage uncertainty in the wavelength.
主要误差来源包括波节位置的测量、波节恰好在两端的假设，以及弦并非完全均匀。增加波节数量并重复测量可以减小波长的百分比不确定度。
10. Graphs and Energy in Stationary Waves | 图像与驻波中的能量
If you plot displacement against position at one instant, a stationary wave looks like a sine curve. If you plot displacement against time for a single point, it oscillates with simple harmonic motion at the driving frequency, with amplitude depending on how close the point is to an antinode.
如果在某一时刻绘制位移随位置变化的图像，驻波看起来像一条正弦曲线。如果绘制某一点的位移随时间的变化，它以驱动频率做简谐振动，振幅取决于该点离波腹有多近。
Energy in a stationary wave is not transferred along the wave but is stored and exchanged. At the instant when every point is at maximum displacement, all the energy is potential energy stored in the stretched medium. A quarter of a cycle later every point passes through its equilibrium position, and the energy is entirely kinetic. The total energy remains constant if damping is ignored.
驻波中的能量并不沿波传递，而是被储存和交换。当所有点都处于最大位移的瞬间，能量全部是以被拉伸介质储存的势能形式存在。四分之一周期后，所有点都通过平衡位置，能量完全变成动能。若忽略阻尼，总能量保持不变。
This contrasts with a progressive wave, in which energy is continuously carried away from the source at the wave speed.
这与行波形成对比：行波中的能量以波速不断从波源被带走。
11. Exam Technique and Common Traps | 考试技巧与常见陷阱
Most lost marks in this topic come from a small number of repeated mistakes rather than from a lack of understanding. Learn the traps and check for them before you finish each answer.
本主题丢分大多来自少数几个反复出现的错误，而不是理解不到位。记住这些陷阱，并在完成每道答案前逐一检查。
- Confusing displacement nodes with pressure nodes in air columns. 混淆空气柱中的位移波节与压强波节。
- Forgetting that a closed pipe only produces odd harmonics. 忘记闭管只产生奇次谐波。
- Using L = λ for the fundamental instead of L = λ ÷ 2 on a string. 在弦上把基频写成 L = λ，而不是 L = λ ÷ 2。
- Stating that a stationary wave transfers energy. 错误地声称驻波传递能量。
- Forgetting units, especially m s⁻¹ and Hz, or leaving answers to too many significant figures. 漏写单位（尤其是 m s⁻¹ 和 Hz），或者有效数字过多。
- Drawing a stationary wave without nodes at the fixed boundaries. 画驻波时没有在固定边界处画出波节。
Definitions are worth easy marks, so quote them precisely: a node is a point of zero displacement, an antinode is a point of maximum displacement, and a stationary wave is formed by two waves of the same frequency travelling in opposite directions.
定义题是容易拿分的，所以要准确引用：波节是位移为零的点，波腹是位移最大的点，驻波由两列频率相同、传播方向相反的波叠加形成。
12. Quick Revision Checklist | 快速复习清单
Use this checklist in the final week of revision. If you can answer every line from memory and back it up with one equation or example, you are ready for any Topic 5.5 question Edexcel can set.
在复习的最后一周使用这份清单。如果你能凭记忆回答每一条，并举出一个公式或例子支撑，那么你已经准备好应对 Edexcel 关于 5.5 的任何题目。
- Define stationary wave, node and antinode. 定义驻波、波节和波腹。
- Explain how a stationary wave forms by superposition of incident and reflected waves. 解释驻波如何由入射波与反射波叠加形成。
- Compare stationary and progressive waves in terms of energy, amplitude and phase. 从能量、振幅、相位三方面比较驻波与行波。
- State the length conditions for strings, open pipes and closed pipes. 说出弦、开管和闭管的长度条件。
- Derive and use v = f λ, L = n λ ÷ 2 and L = (2n − 1) λ ÷ 4. 推导并使用 v = f λ、L = n λ ÷ 2 和 L = (2n − 1) λ ÷ 4。
- Use v = √T ÷ μ for a stretched string and describe how to find μ experimentally. 使用 v = √T ÷ μ，并描述如何通过实验求 μ。
- Explain resonance and why amplitude is maximum at a natural frequency. 解释共振，以及为什么在固有频率时振幅最大。
- Sketch harmonics for strings and pipes and count nodes and antinodes correctly. 画出弦和管的谐波，并正确数出波节和波腹。
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图