📚 Formation of Stationary Waves | 驻波的形成
Stationary waves, also known as standing waves, are a fundamental phenomenon in wave physics where the wave pattern does not appear to travel. They are formed by the superposition of two progressive waves of the same frequency and amplitude travelling in opposite directions. Understanding their formation is crucial for explaining resonance in musical instruments, microwave cavities, and many other systems. This article explores the principles, conditions, and characteristics of stationary waves as required by CIE A-Level Physics.
驻波（又称定波）是波动物理学中一种基本现象，波型看起来并不传播。它由两列频率相同、振幅相等、传播方向相反的波叠加形成。理解驻波的形成对于解释乐器、微波谐振腔及许多其他系统中的共振至关重要。本文将根据 CIE A-Level 物理大纲探讨驻波的原理、条件和特性。
1. What is a Stationary Wave? | 什么是驻波？
A stationary wave is a wave pattern that results from the interference of two identical progressive waves moving in opposite directions. In a stationary wave, points of maximum displacement (antinodes) and points of zero displacement (nodes) alternate in fixed positions. Unlike progressive waves, energy is not transferred along the medium; it is stored in the oscillating segments between nodes.
驻波是由两列完全相同的行波沿相反方向传播发生干涉而形成的波型。在驻波中，最大位移点（波腹）和零位移点（波节）在固定位置上交替出现。与行波不同，能量并不沿介质传递，而是储存在波节之间的振荡段内。
The entire pattern appears to stand still because the waveform does not advance; only the amplitude at any point changes with time. The positions of nodes and antinodes remain constant, making the wave easily observable in stretched strings or air columns.
整个图景看上去是静止的，因为波形不向前移动；只有各点的振幅随时间变化。波节和波腹的位置保持不变，这使得驻波在张紧的弦或空气柱中很容易被观察到。
2. Progressive Waves vs Stationary Waves | 行波与驻波的对比
Progressive waves transfer energy from one point to another; all points on the wave vibrate with the same amplitude (if undamped) and the wave profile moves forward. The phase varies continuously along the direction of propagation.
行波将能量从一点传递到另一点；波上所有质点振动振幅相同（无阻尼情况下），波形向前移动。相位沿传播方向连续变化。
In a stationary wave, there is no net transfer of energy. The amplitude varies from zero at nodes to a maximum at antinodes. Points between two adjacent nodes vibrate in phase, while points on opposite sides of a node are in antiphase (180° out of phase).
在驻波中，没有能量的净传递。振幅从波节处的零变化到波腹处的最大值。相邻两波节之间的所有质点同相振动，而波节两侧的质点反相振动（相位差 180°）。
The key differences can be summarized: progressive waves have a moving wave profile, constant amplitude, and energy transport; stationary waves have fixed nodes and antinodes, variable amplitude, and no net energy flow.
关键区别可概括为：行波波形移动，振幅恒定，传递能量；驻波波节和波腹位置固定，振幅变化，无净能量流动。
3. Principle of Superposition and Formation | 叠加原理与形成
nd amplitude A travel in opposite directions, they combine. Mathematically, the resultant displacement y at position x and time t is the sum: y = y1 + y2. If the waves are y1 = A sin(ωt - kx) and y2 = A sin(ωt + kx), superposition gives y = 2A sin(kx) cos(ωt). This is the equation of a stationary wave.
驻波直接源于叠加原理。当两列波长 λ、频率 f 和振幅 A 均相同的正弦波沿相反方向传播时，它们会叠加。数学上，位置 x 和时间 t 处的合位移 y 为两项之和：y = y1 + y2。若两列波分别为 y1 = A sin(ωt - kx) 和 y2 = A sin(ωt + kx)，叠加后得到 y = 2A sin(kx) cos(ωt)。这就是驻波方程。
in(kx) = ±1, the amplitude is maximum – these are antinodes. The time factor cos(ωt) shows that all points execute simple harmonic motion, varying in phase or antiphase depending on the sign of sin(kx).
项 2A sin(kx) 表示与位置有关的振幅。当 sin(kx) = 0 时，位移始终为零——这就是波节。当 sin(kx) = ±1 时，振幅最大——这就是波腹。时间因子 cos(ωt) 表明所有质点都作简谐振动，根据 sin(kx) 的正负号同相或反相变化。
4. Reflection and Boundary Conditions | 反射与边界条件
A stationary wave is typically formed when a progressive wave is reflected at a boundary and interferes with the incident wave. The nature of the reflection depends on the boundary type: a fixed end reflects the wave with a phase change of π radians (180°), while a free end reflects without a phase change. For example, a string fixed at both ends will have a phase reversal at each reflection.
通常，当行波在边界上反射并与入射波干涉时，便形成驻波。反射的性质取决于边界类型：固定端反射会产生 π 弧度（180°）的相位突变，而自由端反射则无相位突变。例如，两端固定的弦每次反射都会发生相位反转。
These phase conditions together with the path difference determine the locations of nodes and antinodes. At a fixed end, destructive interference creates a node; at a free end (or an open end of an air column), constructive interference typically results in an antinode.
这些相位条件与程差共同决定了波节和波腹的位置。在固定端，相消干涉形成波节；在自由端（或空气柱的开口端），相长干涉通常形成波腹。
In air columns, it is essential to distinguish between displacement nodes and pressure nodes. A closed end is a displacement node but a pressure antinode; an open end is a displacement antinode but a pressure node. Most CIE problems refer to displacement.
在空气柱中，区分位移波节和压力波节至关重要。封闭端是位移波节但为压力波腹；开口端是位移波腹但为压力波节。CIE 考题大多以位移为准。
5. Nodes and Antinodes – Spacing and Properties | 波节与波腹——间距与特性
Nodes are points of permanently zero displacement. Adjacent nodes are separated by half a wavelength (λ/2). Antinodes are points of maximum oscillation, also spaced λ/2 apart. The distance between a node and the next antinode is λ/4.
波节是位移始终为零的点。相邻波节的间距为半个波长（λ/2）。波腹是振动幅度最大的点，间距也为 λ/2。波节与下一个波腹之间的距离为 λ/4。
All points between two adjacent nodes move in phase, but the amplitude gradually increases from node to antinode. Points on either side of a node move in antiphase (180° out of phase). Thus, the wave is ‘standing’ because the energy oscillates locally rather than moving forward.
相邻两波节间的所有质点同相运动，但振幅从波节到波腹逐渐增大。波节两侧的质点反相（相位差 180°）运动。因此，波是“驻定的”，能量仅在本地振荡而不向前流动。
6. Stationary Waves on a Stretched String | 张紧弦上的驻波
Consider a string fixed at both ends. When the string is plucked or driven continuously, waves travel to the ends and reflect. The two fixed ends must be nodes. For a standing wave to be sustained, the length L of the string must accommodate an integer number of half-wavelengths:
L = n λ/2, n = 1, 2, 3, ...
考虑一根两端固定的弦。当弦被拨动或持续驱动时，
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