📚 Formation of Stationary Waves | 驻波的形成
In A-Level physics, a stationary wave (also called a standing wave) is produced when two progressive waves of the same type, frequency, speed and amplitude travel in opposite directions and superpose. This topic is central to understanding musical instruments, resonance tubes and many wave experiments.
在 A-Level 物理中，驻波（也称定波）由两列类型相同、频率相同、波速相同、振幅相同但传播方向相反的波叠加而成。这一主题对理解乐器、共振管以及许多波动实验至关重要。
1. What is a Stationary Wave? | 什么是驻波？
A stationary wave is a wave pattern that remains confined to a region. Unlike a progressive wave, the wave profile does not move forward; instead, some points remain permanently at rest while others oscillate with maximum amplitude. Energy is stored rather than transmitted along the medium.
驻波是一种被限制在一定区域内的波形。与行波不同，驻波的波形不向前移动；相反，一些点始终保持静止，而另一些点以最大振幅振动。能量被储存起来，而不是沿介质传播。
2. How Stationary Waves Form: Superposition of Two Travelling Waves | 驻波如何形成：两列行波的叠加
When two identical progressive waves travel through the same medium in opposite directions, their displacements add by the principle of superposition. At some positions the waves always arrive in phase, producing constructive interference; at other positions they always arrive in antiphase, producing destructive interference.
当两列相同的行波在同一介质中沿相反方向传播时，它们的位移依据叠加原理相加。在某些位置，两列波总是同相到达，产生相长干涉；在另一些位置，它们总是反相到达，产生相消干涉。
3. Conditions for Formation | 形成条件
For a stationary wave to form, the two waves must have the same frequency, wavelength, speed and amplitude, and must travel in opposite directions. In practice one wave is often a reflection of the other at a boundary, such as the fixed end of a string or the closed end of a pipe.
要形成驻波，两列波必须具有相同的频率、波长、波速和振幅，且传播方向相反。在实际中，一列波往往是另一列波在边界处的反射，例如弦的固定端或管子的闭口端。
- Conditions: same frequency, same wavelength, same speed, same amplitude, opposite directions.
- 条件：相同的频率、相同的波长、相同的波速、相同的振幅、相反的方向。
4. Nodes and Antinodes | 波节与波腹
Nodes are points where the two waves always cancel, so the displacement is permanently zero. Antinodes are points where the two waves always reinforce, so the particle oscillates with maximum amplitude. Adjacent nodes are separated by half a wavelength, as are adjacent antinodes.
波节是两列波总是相互抵消的点，因此位移始终为零。波腹是两列波总是相互加强的点，因此质点以最大振幅振动。相邻波节之间的距离为半个波长，相邻波腹之间的距离也为半个波长。
Node spacing = λ/2, Antinode spacing = λ/2
The distance from a node to the adjacent antinode is one quarter of a wavelength: λ/4.
从波节到相邻波腹的距离为四分之一波长：λ/4。
5. Phase Relationship in Stationary Waves | 驻波中的相位关系
All particles between two adjacent nodes vibrate in phase with one another. Particles in adjacent loops vibrate in antiphase, meaning their phase difference is π radians or 180°. The amplitude of vibration varies from zero at a node to a maximum at an antinode.
相邻两个波节之间的所有质点彼此同相振动。相邻波腹段中的质点反相振动，即相位差为 π 弧度或 180°。振动幅度从波节处的零变化到波腹处的最大值。
6. Stationary Waves on a Stretched String | 张紧弦上的驻波
the length L must equal a whole number of half-wavelengths: L = nλ/2, where n = 1, 2, 3...
两端固定的弦在被拨动或驱动时可以形成驻波。固定端必须是波节。因此弦长 L 必须等于半波长的整数倍：L = nλ/2，其中 n = 1, 2, 3…
L = nλ/2, fₙ = nv/(2L)
frequency f₁ = v/(2L). Higher n values give the second harmonic, third harmonic and so on.
当 n = 1 时，最低频率为基频 f₁ = v/(2L)。更高的 n 值给出第二谐波、第三谐波等。
| n | Harmonic | Frequency |
|---|---|---|
| 1 | First harmonic / fundamental | f₁ = v/(2L) |
| 2 | Second harmonic | f₂ = 2v/(2L) = v/L |
| 3 | Third harmonic | f₃ = 3v/(2L) |
7. Stationary Waves in Air Columns | 空气柱中的驻波
antinode. The length is an odd number of quarter-wavelengths: L = nλ/4 for n = 1, 3, 5...
声波可以在管中形成驻波。在一端封闭的管中，封闭端是位移波节，开口端是位移波腹。管长等于四分之一波长的奇数倍：L = nλ/4，其中 n = 1, 3, 5…
L = nλ/4, fₙ = nv/(4L), n = 1, 3, 5...
pe open at both ends, both ends are displacement antinodes, so L = nλ/2 for n = 1, 2, 3...
对于两端开口的管，两端都是位移波腹，因此 L = nλ/2，其中 n = 1, 2, 3…
L = nλ/2, fₙ = nv/(2L), n = 1, 2, 3...
In a closed pipe, only odd harmonics are present; in an open pipe, all harmonics are present.
在闭管中只存在奇数谐波；在开管中所有谐波都存在。
8. Harmonics and Overtones | 谐波与泛音
The lowest frequency at which a stationary wave forms is the fundamental frequency, also called the first harmonic. Higher frequencies are called overtones. For a fixed string or an open pipe, all harmonics are present; for a pipe closed at one end, only odd harmonics are present.
形成驻波的最低频率称为基频，也称为第一谐波。更高的频率称为泛音。对于两端固定的弦或两端开口的管，所有谐波都存在；对于一端封闭的管，只存在奇数谐波。
| System | Length condition | Harmonics present |
|---|---|---|
| String fixed at both ends | L = nλ/2, n = 1, 2, 3... | All harmonics |
| Pipe open at both ends | L = nλ/2, n = 1, 2, 3... | All harmonics |
| Pipe closed at one end | L = nλ/4, n = 1, 3, 5... | Odd harmonics only |
9. Resonance and Energy in Stationary Waves | 驻波中的共振与能量
Stationary waves are set up most easily when the driving frequency matches a natural frequency of the system; this is resonance. In a stationary wave, energy is stored in the vibrating loops and is not transferred from one end to the other as in a progressive wave.
当驱动频率与系统的固有频率相匹配时，最容易形成驻波，这就是共振。在驻波中，能量储存在振动的波腹段中，并不像行波那样从一端传递到另一端。
10. Measuring the Speed of Sound Using Stationary Waves | 利用驻波测量声速
λ/4 and the next at L₂ = 3λ/4. Therefore λ = 2(L₂ - L₁), and the speed of sound is v = fλ.
共振管可以与已知频率为 f 的音叉配合使用。当调节空气柱长度时，第一次听到响亮声音的位置在 L₁ = λ/4，下一次在 L₂ = 3λ/4。因此 λ = 2(L₂ - L₁)，声速为 v = fλ。
λ = 2(L₂ - L₁), v = fλ
11. Comparing Stationary and Progressive Waves | 驻波与行波的比较
In a progressive wave, all particles have the same amplitude and energy is transferred. In a stationary wave, amplitude varies from zero to a maximum, and there is no net energy transfer. Adjacent points in a progressive wave have different phases, while all points in one loop of a stationary wave are in phase.
在行波中，所有质点具有相同的振幅，能量被传递。在驻波中，振幅从零到最大值变化，没有净能量传递。行波中相邻点的相位不同，而驻波中同一波腹段内所有点同相。
| Property | Progressive wave | Stationary wave |
|---|---|---|
| Amplitude | Same for all particles | Varies from zero at nodes to maximum at antinodes |
| Energy transfer | Energy is transferred along the wave | No net energy transfer |
