📚 Stationary Waves | 驻波
A stationary wave, also known as a standing wave, is a wave pattern that remains confined to a specific region without any net transfer of energy along the medium. It arises from the superposition of two progressive waves of the same frequency and amplitude travelling in opposite directions. Understanding stationary waves is essential for explaining musical instruments, microwave resonant cavities, and many phenomena in A-Level Physics.
驻波（也称定波）是一种被限制在特定区域内的波形，没有能量沿介质的净传输。它是由两列频率相同、振幅相等但传播方向相反的行波叠加而成的。理解驻波对于解释乐器、微波谐振腔以及A-Level物理中的许多现象至关重要。
1. What Are Stationary Waves? | 什么是驻波？
A stationary wave is produced when two identical progressive waves travel in opposite directions and interfere. The resulting pattern appears to stand still, with points of zero displacement (nodes) and points of maximum displacement (antinodes) alternating in space. Unlike a progressive wave, the profile of a stationary wave does not move forward; instead, the amplitude of oscillation varies with position.
当两列相同的行波沿相反方向传播并发生干涉时，便会产生驻波。所形成的图样看似静止不动，空间中出现位移为零的点（波节）和位移最大的点（波腹），二者交替排列。与行波不同，驻波的波形不向前移动；其振幅随位置变化。
2. Conditions for Formation | 驻波的形成条件
For a stationary wave to be established, two coherent waves of the same type, identical frequency and equal amplitude must travel in opposite directions through the same medium. In practice, this is achieved by reflecting a progressive wave from a rigid or free boundary, causing the incident and reflected waves to superpose. The medium must have fixed boundaries or ends to support standing wave patterns; otherwise, the waves dissipate.
要形成驻波，必须有两列相干波，同类型、同频率且振幅相等，在同一介质中沿相反方向传播。实际中，通过让行波从固定或自由边界反射，使入射波与反射波叠加来实现。介质必须有固定的边界或端点以维持驻波图样；否则，波会耗散掉。
3. Nodes and Antinodes | 波节与波腹
Nodes are points along the medium where the particles remain permanently at rest because destructive interference is complete at all times. Antinodes are points where the amplitude of oscillation is a maximum, owing to constructive interference. The distance between two successive nodes is half a wavelength (λ/2), as is the distance between two successive antinodes. A node and an adjacent antinode are separated by λ/4.
波节是介质中质点始终保持静止的点，因为该处始终发生完全的相消干涉。波腹是振幅最大的点，源于相长干涉。相邻两个波节之间的距离为半个波长（λ/2），相邻波腹之间的距离也是λ/2。相邻的一对波节与波腹相距λ/4。
4. Mathematical Description | 驻波的数学描述
Consider two waves travelling in opposite directions: y₁ = A sin(kx – scribes the spatial amplitude envelope, and cos(ωt) describes the time evolution. Nodes occur where sin(kx) = 0, giving x = nλ/2 with n = 0, 1, 2,... Antinodes occur where |sin(kx)| = 1, giving x = (2n+1)λ/4.
考虑两列反向传播的波：y₁ = A sin(kx – ωt) 和 y₂ = A sin(kx + ωt)。它们的叠加得出驻波方程 y = 2A sin(kx) cos(ωt)，其中 k = 2π/λ，ω = 2πf。sin(kx) 项描述空间的振幅包络，cos(ωt) 描述时间演化。波节出现在 sin(kx)=0 处，即 x = nλ/2 (n = 0, 1, 2, …)。波腹出现在 |sin(kx)|=1 处，即 x = (2n+1)λ/4。
5. Stretched Strings: Both Ends Fixed | 两端固定的弦
ength L of the string must be an integer multiple of half-wavelengths: L = nλ/2, where n = 1, 2, 3,... The frequency of the nth harmonic is fₙ = n(v/(2L)), where v = √T/μ tension and μ is the mass per unit length. The fundamental frequency (n=1) is f₁ = v/(2L).
当弦的两端均固定时，两端必须为位移波节。最简单的驻波图样在中间有一个波腹。弦的长度 L 必须是半波长的整数倍：L = nλ/2，其中 n = 1, 2, 3,…。第 n 次谐波的频率为 fₙ = n(v/(2L))，其中波速 v = √T/μ，T 为张力，μ 为线密度。基频（n=1）为 f₁ = v/(2L)。
6. Stretched Strings: One End Fixed, One End Free | 一端固定一端自由的弦
becomes an antinode, while the fixed end is a node. The length L must then satisfy L = (2n – ... The fundamental frequency is f₁ = v/(4L), and only odd harmonics are present: fₙ = (2n–1)f₁.
如果弦的一端自由（例如套在光滑杆上的轻环），则自由端成为波腹，固定端为波节。此时长度 L 必须满足 L = (2n – 1)λ/4，n = 1, 2, 3,…。基频为 f₁ = v/(4L)，且只有奇次谐波出现：fₙ = (2n–1)f₁。
7. Air Columns: Closed Pipes | 空气柱：闭管
antinode at the open end. The length L of the air column relates to wavelength by L = (2n– requency is f₁ = v/(4L), where v is the speed of sound. Only odd harmonics exist: fₙ = (2n– ure variation is a maximum at the closed end (pressure antinode) and zero at the open end.
一端封闭的管能形成纵波驻波，闭端为位移波节（空气无法运动），开口端为位移波腹。空气柱的长度 L 满足 L = (2n–1)λ/4。基频为 f₁ = v/(4L)，v 为声速。只存在奇次谐波：fₙ = (2n–1)f₁。在闭端，压强变化最大（压强波腹），开口端为零。
8. Air Columns: Open Pipes | 空气柱：开管
damental frequency is f₁ = v/(2L), and all harmonics are present: fₙ = nf₁. The pressure variation has nodes at the open ends and an antinode at the mid-point for the fundamental. This explains why flutes and organ pipes produce a full harmonic series.
在两端开口的管中，两端都形成位移波腹。长度 L 为半波长的整数倍：L = nλ/2。基频为 f₁ = v/(2L)，所有谐波均存在：fₙ = nf₁。压强变化在开口端为波节，在基频模式的管道中间为压强波腹。这解释了长笛和管风琴为何能发出完整泛音列。
9. Harmonics and Overtones | 谐波与泛音
The term 'harmonic' refers to a component frequency that is an integer multiple of the fundamental. The first harmonic is the fundamental itself. 'Overtones' are harmonics above the fundamental; the first overtone is the second harmonic. In fixed-fixed strings and open pipes, all harmonics are present. In fixed-free strings and closed pipes, only odd harmonics appear. The quality (timbre) of a musical note depends on the mixture of harmonics.
“谐波”是指频率为基频整数倍的成分，第一谐波即基频。“泛音”是指基频以上的谐波；第一泛音是第二谐波。在两端固定的弦和开口管中，所有谐波都存在。在一端固定一端自由的弦和闭管中，只出现奇次谐波。乐音的音色（音质）取决于谐波的组合。
10. Phase Relationships in Stationary Waves | 驻波中的相位关系
In a stationary wave, all particles between two successive nodes oscillate in phase with each other, but in anti-phase with particles in the adjacent segment (separated by a node). The phase difference between particles in neighbouring loops is π radians (180°). Unlike progressive waves, there is no progressive phase shift along the medium; the wave does not advance.
在驻波中，两个相邻波节之间的所有质点同相振动，但与相邻节段（被一个波节隔开）的质点反相。相邻波腹之间的相位差为 π 弧度（180°）。与行波不同，沿介质没有持续的相位移动；波形并不推进。
11. Energy in Stationary Waves | 驻波中的能量
Standing waves do not transfer energy from one place to another; instead, energy is stored alternately as kinetic and potential energy within each half-wavelength segment. At the instant when particles pass through equilibrium, energy is entirely kinetic. At maximum displacement, energy is entirely potential (elastic). The net energy flux across any node is zero, so the wave is 'trapped' between boundaries.
驻波不会将能量从一个地方传输到另一个地方；相反，能量在每个半波长节段内以动能和势能的形式交替储存。质点经过平衡位置的瞬间，能量全为动能；在最大位移时，能量全为势能（弹性能）。通过任何一个波节的净能流为零，因此波被“束缚”在边界之间。
12. Comparison with Progressive Waves | 与行波的比较
| Feature | Stationary Wave | Progressive Wave |
|---|---|---|
| Energy transfer | No net energy transfer; energy is stored in segments | Energy is transported in the direction of wave travel |
| Phase | Particles between nodes are in phase | Phase varies progressively along the wave |
| Amplitude | Varies from zero at nodes to maximum at antinodes | All points have the same amplitude (if undamped) |
| Waveform | Does not advance; the profile oscillates in place | The waveform travels through the medium |
The table summarises the fundamental distinctions. These differences are frequently examined in CIE A-Level Physics, particularly the energy and phase contrasts.
该表格总结了基本区别。这些差异是CIE A-Level物理中常考的内容，尤其是能量和相位方面的对比。
13. Experimental Determination: Melde’s Experiment | 实验测定：梅尔代实验
Melde' passes over a pulley and carries a mass to provide tension. By adjusting the tension or the length, stable standing wave patterns with clear nodes and antinodes are produced. Measuring the distance between nodes yields λ/2, and the frequency of the oscillator is known, allowing the wave speed v = fλ to be determined. The relationship v² ∝ e verified by altering the tension and observing the required length for a given harmonic.
梅尔代实验演示了弦上的驻波。弦的一端由振动音叉或振荡器驱动，另一端绕过滑轮并悬挂砝码以提供张力。通过调节张力或长度，可以产生清晰的波节和波腹的稳定驻波图样。测量相邻波节间的距离可得λ/2，已知振荡器频率，便可求出波速 v = fλ。改变张力并观察给定谐波所需的弦长，可以验证 v² ∝ T 的关系。
14. Applications of Stationary Waves | 驻波的应用
Stationary waves are fundamental to musical instruments: string instruments (violin, guitar) use fixed-fixed strings; woodwind instruments exploit air column resonance. Microwave ovens create a standing electromagnetic wave pattern to heat food, with nodes and antinodes causing uneven heating — hence the turntable. In atomic physics, electron standing waves in orbitals are described by stationary wave functions. Engineering uses the concept to avoid resonance in bridges and buildings.
驻波是乐器的基础：弦乐器（小提琴、吉他）利用两端固定的弦；木管乐器利用空气柱共振。微波炉产生驻波电磁场加热食物，波节与波腹造成加热不均——因此需要转盘。在原子物理中，轨道中的电子驻波由定态波函数描述。工程上利用驻波概念避免桥梁和建筑发生共振。
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