Stationary Waves | 驻波

📚 Stationary Waves | 驻波

Stationary waves, also called standing waves, occur when two progressive waves of the same frequency and amplitude travel in opposite directions and interfere. They are central to understanding musical instruments, microwave experiments and many CIE A-level physics practical questions.

驻波，又称 standing waves，是两列频率和振幅相同、传播方向相反的波叠加干涉时形成的。驻波是理解乐器、微波实验以及许多 CIE A-level 物理实验题的核心。

1. What Are Stationary Waves? | 什么是驻波？

A stationary wave is a wave pattern produced by the superposition of two waves of the same type, same frequency and same amplitude travelling in opposite directions. Unlike a progressive wave, the wave profile does not appear to move along the medium; instead, fixed points of zero displacement and points of maximum oscillation are produced.

驻波是由两列同类型、同频率、同振幅但传播方向相反的波叠加产生的波形。与行波不同，驻波的波形似乎不沿介质传播，而是形成位移始终为零的固定点和振动最大的点。

The name stationary does not mean the particles are at rest; the particles oscillate, but the pattern of nodes and antinodes stays fixed in space.

驻波的 stationary 并不表示质点静止；质点仍在振动，只是波节和波腹的图案在空间中保持固定。

2. Formation: Superposition and Reflection | 形成：叠加与反射

A stationary wave is usually formed when a progressive wave is reflected at a boundary and the reflected wave overlaps with the incident wave. If the incident and reflected waves have the same frequency and similar amplitude, their superposition can produce a stable stationary pattern.

驻波通常由一列行波在边界处反射，反射波与入射波重叠而形成。如果入射波和反射波频率相同、振幅接近，叠加后就能产生稳定的驻波图案。

Common boundaries include the fixed end of a stretched string, the closed end of a pipe, or an impedance mismatch in a microwave guide.

常见的边界包括张紧弦的固定端、管子的封闭端或微波波导中的阻抗突变处。

3. Nodes and Antinodes | 波节与波腹

Nodes are positions where the displacement is always zero because the two superposing waves cancel exactly at those points. Antinodes are positions where the displacement oscillates between maximum positive and maximum negative values.

波节是位移始终为零的位置，因为两列叠加波在这些点恰好完全相消。波腹是位移在正最大值和负最大值之间振动的位置。

Adjacent nodes are separated by half a wavelength, λ/2, and a node is separated from the adjacent antinode by λ/4.

相邻波节之间的距离为半个波长 λ/2，波节与相邻波腹之间的距离为 λ/4。

4. Standing Waves on a Stretched String | 张紧弦上的驻波

For a string of length L fixed at both ends, stationary waves can only exist when the string length contains an integer number of half-wavelengths: L = nλ/2, where n = 1, 2, 3, …

对于两端固定、长度为 L 的弦，只有当弦长包含整数个半波长时才能形成驻波：L = nλ/2，其中 n = 1, 2, 3, …

The wave speed on a string is given by v = √(T/μ), where T is the tension and μ is the mass per unit length. Therefore the allowed frequencies are fₙ = n v / (2L) = n/(2L) √(T/μ).

弦上的波速为 v = √(T/μ)，其中 T 是张力，μ 是单位长度质量。因此允许的频率为 fₙ = n v / (2L) = n/(2L) √(T/μ)。

v = √(T/μ)

fₙ = n v / (2L) = n/(2L) √(T/μ)

5. Harmonics and Overtones on Strings | 弦上的谐波与泛音

The lowest frequency n = 1 is the fundamental frequency or first harmonic. The frequency n = 2 is the second harmonic and is also called the first overtone, because it is the first frequency above the fundamental.

最低频率 n = 1 称为基频或第一谐波。n = 2 的频率是第二谐波，也称为第一泛音，因为它是基频之上的第一个频率。

For n = 1 the string has one antinode at the centre and nodes at both ends. For n = 2 there are three nodes and two antinodes, and so on.

当 n = 1 时，弦中央有一个波腹，两端各有一个波节。当 n = 2 时，有三个波节和两个波腹，以此类推。

It is important to remember that the nth harmonic has n antinodes and n+1 nodes for a string fixed at both ends.

需要记住：两端固定的弦，第 n 次谐波有 n 个波腹和 n+1 个波节。

6. Stationary Waves in Air Columns | 空气柱中的驻波

Sound waves can also form stationary waves in air columns inside pipes. At a closed end, the air particles cannot move freely, so a displacement node is formed. At an open end, the air is free to move, so a displacement antinode is formed, slightly beyond the physical end due to the end correction.

声波也可以在管内的空气柱中形成驻波。在封闭端，空气质点不能自由运动，因此形成位移波节。在开口端，空气可以自由振动，因此形成位移波腹，且由于端部修正，波腹略超出管口。

For a pipe closed at one end and open at the other, the length L of the air column at resonance must contain an odd number of quarter-wavelengths: L = nλ/4 with n = 1, 3, 5, …

对于一端封闭、一端开口的管，共振时空气柱长度必须包含奇数个四分之一波长：L = nλ/4，其中 n = 1, 3, 5, …

7. Harmonics in Pipes: Closed and Open | 管中的谐波：闭管与开管

A closed pipe has only odd harmonics. Its fundamental frequency is f₁ = v / (4L), and the next possible frequencies are f₃ = 3v/(4L), f₅ = 5v/(4L), etc. There is no second harmonic.

闭管只有奇次谐波。其基频为 f₁ = v / (4L)，接下来的可能频率为 f₃ = 3v/(4L)、f₅ = 5v/(4L) 等。不存在第二谐波。

An open pipe has all harmonics. Its fundamental frequency is f₁ = v / (2L), and the frequencies are fₙ = n v / (2L), with n = 1, 2, 3, …

开管具有所有谐波。其基频为 f₁ = v / (2L)，频率为 fₙ = n v / (2L)，其中 n = 1, 2, 3, …

Closed pipe: fₙ = n v / (4L), n = 1, 3, 5, …

Open pipe: fₙ = n v / (2L), n = 1, 2, 3, …

8. Comparing Stationary and Progressive Waves | 驻波与行波的比较

The main differences between stationary and progressive waves are summarised below.

驻波和行波的主要区别总结如下。

Feature	Stationary wave	Progressive wave
Energy transfer	No net energy transfer along the medium	Energy is transferred in the direction of wave travel
Amplitude	Varies from zero at nodes to maximum at antinodes	All particles have the same amplitude in an ideal lossless wave
Phase	Particles between adjacent nodes oscillate in phase; adjacent segments are in antiphase	Phase varies continuously along the wave
Wave profile	Does not move along the medium	Moves with the wave speed

在驻波中没有沿介质方向的净能量传递，能量集中在波腹附近；而行波则将能量从波源向远处传递。驻波的振幅从波节到波腹变化，而在理想行波中各质点的振幅相同。相位方面，驻波中相邻波节之间的质点是同相的，而行波中相位沿传播方向连续变化。

9. Phase Relationships in a Stationary Wave | 驻波中的相位关系

All particles between two adjacent nodes oscillate in phase with each other, meaning they reach their maximum displacement at the same time. Particles in adjacent segments, separated by a node, oscillate in antiphase.

两个相邻波节之间的所有质点同相振动，它们同时到达最大位移。被一个波节隔开的相邻区段中的质点则反相振动。

At a node the phase difference is not normally defined in the same way because the amplitude is zero. This phase behaviour is different from a progressive wave, where the phase of oscillation changes continuously with position.

在波节处，由于振幅为零，相位差通常不以相同方式定义。这种相位行为与行波不同，行波的振动相位随位置连续变化。

10. Energy in Stationary Waves | 驻波中的能量

In a stationary wave, energy is not transferred along the medium from one end to the other. Instead, energy is localised: it is concentrated near antinodes and is zero at nodes.

在驻波中，能量不会沿介质从一端传递到另一端。相反，能量被局限在局部：能量集中在波腹附近，在波节处为零。

This is why a standing wave on a string does not deliver net power to a support, even though individual particles are moving.

这就是为什么弦上的驻波不会向支撑端传递净功率，尽管各个质点仍在运动。

11. Experiments: Measuring Wave Speed Using Stationary Waves | 实验：利用驻波测量波速

For a stretched string, a vibration generator can drive the string at a known frequency. By adjusting the length or tension until a clear standing wave with several antinodes is observed, the wavelength can be measured from the distance between nodes. The wave speed is then calculated using v = fλ.

对于张紧弦，可以用振动器以已知频率驱动弦。调整长度或张力，直到观察到具有几个波腹的清晰驻波。通过测量

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Stationary Waves | 驻波