📚 Nodes and Antinodes | 节点与腹点
In CIE A-Level Physics, stationary waves are central to understanding musical instruments, microwaves and even electron behaviour. The key features of any stationary wave pattern are nodes, where displacement is always zero, and antinodes, where displacement varies with maximum amplitude. This article explains their formation, spacing and phase relationships, and links them to strings and pipes.
在 CIE A-Level 物理中，驻波是理解乐器、微波甚至电子行为的重要基础。任何驻波图样的关键特征都是节点和腹点：节点处位移始终为零，腹点处位移以最大振幅变化。本文将解释它们的形成、间距和相位关系，并联系弦和管中的驻波。
1. Standing Waves: Superposition in Confined Systems | 驻波：受限系统中的叠加
A stationary wave is formed when two progressive waves of the same frequency, wavelength and amplitude travel in opposite directions and interfere. In a confined medium, such as a stretched string or an air column, the reflected wave overlaps the incident wave. Under the right conditions, the resultant wave pattern appears to stay in place rather than travel.
驻波由两个频率、波长和振幅相同但传播方向相反的进行波相互叠加形成。在拉伸的弦或空气柱等受限介质中，反射波与入射波重叠。在合适的条件下，合成波的图样似乎停留在原处，而不是向前传播。
Unlike a progressive wave, a stationary wave does not transfer energy along the medium. Energy remains localised, oscillating between kinetic and potential forms at different positions. The positions of zero disturbance are the nodes, and the positions of maximum disturbance are the antinodes.
与进行波不同，驻波不会沿介质传递能量。能量被局限在本地，在不同位置以动能和势能的形式相互转换。零扰动的位置就是节点，最大扰动的位置就是腹点。
2. Defining a Node | 节点的定义
A node is a point in a stationary wave where the amplitude of oscillation is always zero. At a node, the two superposing progressive waves always meet in antiphase, so their displacements cancel at every instant.
节点是驻波中振荡幅度始终为零的点。在节点处，两个叠加的进行波总是反相相遇，因此它们在任意时刻的位移都相互抵消。
On a string, nodes are the points that remain stationary; in a sound wave in a pipe, they are planes where the pressure variation is maximum but the particle displacement is minimum. For a displacement graph, nodes appear as fixed zero crossings.
在弦上，节点是保持不动的点；在管中的声波里，节点是压力变化最大但质点位移最小的平面。在位移图像中，节点表现为固定的零交点。
3. Defining an Antinode | 腹点的定义
An antinode is a point in a stationary wave where the amplitude of oscillation is a maximum. At an antinode, the two superposing progressive waves arrive in phase, so their displacements add constructively at every instant.
腹点是驻波中振荡幅度最大的点。在腹点处，两个叠加的进行波同相到达，因此它们的位移在任意时刻都发生相长叠加。
The maximum displacement at an antinode is twice the amplitude of each individual progressive wave. For example, if each travelling wave has amplitude A, the antinode oscillates between +2A and −2A.
腹点处的最大位移是每个单独进行波振幅的两倍。例如，如果每个行波的振幅为 A，则腹点在 +2A 和 −2A 之间振荡。
A_max = 2A
4. Node and Antinode Spacing | 节点与腹点的间距
Adjacent nodes are separated by half a wavelength. Since an antinode lies midway between two adjacent nodes, the distance from a node to the next antinode is one quarter of a wavelength.
相邻节点之间的距离为半个波长。由于腹点位于两个相邻节点的正中间，因此节点到下一个腹点的距离为四分之一波长。
node-to-node distance = λ/2
node-to-antinode distance = λ/4
These fixed spacings provide a direct method to measure wavelength: if the positions of consecutive nodes are found, the distance between them is λ/2, so λ = 2 × separation.
这些固定间距提供了一种直接测量波长的方法：如果确定相邻节点的位置，它们之间的距离为 λ/2，因此 λ = 2 × 间距。
5. Phase Relationship | 相位关系
All points between two adjacent nodes oscillate in phase with each other, although they may have different amplitudes. Points on opposite sides of a node oscillate in antiphase.
两个相邻节点之间的所有点彼此同相振荡，尽管它们的振幅可能不同。节点两侧的点则以相反相位振荡。
This phase property is useful when comparing particle motion on a stationary wave: at a given moment, particles in one loop move in the same direction, while particles in adjacent loops move in opposite directions.
这一相位特性在比较驻波上的质点运动时很有用：在某一时刻，同一波腹段中的质点沿相同方向运动，而相邻波腹段中的质点则沿相反方向运动。
6. Energy Distribution | 能量分布
In a stationary wave, energy is not transmitted along the medium. At antinodes, kinetic energy varies between maximum and zero during each cycle, while at nodes, particles never move, so kinetic energy is always zero there.
在驻波中，能量不会沿介质传播。在腹点处，动能每个周期内在最大值和零之间变化；而在节点处，质点从不运动，因此动能始终为零。
Potential energy has a different distribution depending on the medium. For a stretched string, potential energy is stored by tension and displacement, so it is largest at antinodes when displacement is greatest. In a sound wave in air, pressure variation is greatest at displacement nodes, so potential energy is concentrated near nodes.
势能的分布因介质而异。对于拉伸的弦，势能由张力和位移储存，因此当位移最大时，势能在腹点处最大。在空气中的声波里，压力变化在位移节点处最大，因此势能集中在节点附近。
7. Standing Waves on Stretched Strings | 拉伸弦上的驻波
For a string fixed at both ends, nodes must form at the two fixed ends. The simplest standing wave therefore has one antinode in the middle and is called the first harmonic or fundamental mode.
对于两端固定的弦，两个固定端处必须形成节点。因此，最简单的驻波在中间有一个腹点，称为第一谐波或基频模式。
L = λ/2
f₁ = v/(2L) = (1/2L)√(T/μ)
Higher harmonics appear when the string length equals a whole number of half-wavelengths. The nth harmonic has n antinodes and n + 1 nodes, including both fixed ends.
当弦长等于半波长的整数倍时，会出现更高的谐波。第 n 次谐波有 n 个腹点和 n + 1 个节点，包括两个固定端。
L = nλ/2
fₙ = n f₁
8. Standing Waves in Pipes | 管中的驻波
In a pipe open at both ends, both ends are displacement antinodes, because air particles are free to move at the open ends. The first harmonic has a node at the centre, and the length equals half a wavelength.
在两端开口的管中，两个开口端都是位移腹点，因为空气质点在开口端可以自由运动。第一谐波在管中央有一个节点，管长等于半个波长。
L = λ/2
fₙ = n v/(2L)
In a pipe closed at one end, the closed end must be a displacement node and the open end an antinode. The first harmonic therefore has a quarter wavelength in the pipe length, so only odd harmonics are present.
在一端封闭的管中，封闭端必须是位移节点，开口端是腹点。因此第一谐波的管长对应四分之一波长，所以只有奇数谐波存在。
L = (2n−1)λ/4
fₙ = (2n−1)v/(4L)
9. Harmonics and Overtones | 谐波与泛音
The term harmonic refers to the integer multiples of the fundamental frequency. The first overtone is the second harmonic for strings and open pipes, but for a closed pipe, the first overtone is the third harmonic because even harmonics are missing.
谐波一词指基频的整数倍。对于弦和开口管，第一泛音是第二谐波；但对于闭口管，第一泛音是第三谐波，因为偶数谐波缺失。
Exam questions often ask you to compare the number of nodes and antinodes for a given harmonic, or to identify which harmonic is shown in a diagram. Always count from the fixed boundaries or open ends.
考试题常要求比较给定谐波的节点和腹点数量，或根据图像判断显示的是哪一次谐波。始终从固定边界或开口端开始计数。
| System | Harmonic present | Frequency relation |
|---|---|---|
| String fixed at both ends | n = 1, 2, 3, … | fₙ = n f₁ |
| Pipe open at both ends | n = 1, 2, 3, … | Published by TutorHao | A-Level Physics Revision Series | aleveler.com
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