📚 Standing Waves | 驻波
A standing wave is a unique wave pattern that arises when two identical travelling waves move in opposite directions and interfere. Unlike a travelling wave, a standing wave does not transfer energy from one place to another but instead stores energy in a stationary pattern of nodes and antinodes. This phenomenon is fundamental in musical instruments, communication technology, and many areas of physics.
驻波是一种独特的波形，由两列完全相同但沿相反方向传播的行波发生干涉而形成。与行波不同，驻波不会将能量从一处传递到另一处，而是将能量储存在由波节和波腹构成的静止图样中。这一现象在乐器、通信技术以及物理学的众多领域中都至关重要。
1. What are Standing Waves? | 什么是驻波？
A standing wave, also called a stationary wave, is the result of the superposition of two progressive waves with the same frequency and amplitude travelling in opposite directions. The superposition creates points of no displacement (nodes) and points of maximum displacement (antinodes). The wave appears to stand still, hence the name.
驻波，又称定态波，是两列频率和振幅相同但传播方向相反的行波叠加的结果。这种叠加产生了位移为零的点（波节）和位移最大的点（波腹）。波形看起来静止不动，因而得名。
2. Nodes and Antinodes | 波节与波腹
Nodes are positions along the medium where the displacement is always zero. They are separated by half a wavelength (λ/2). Antinodes are positions where the displacement oscillates between maximum positive and negative values, occurring midway between nodes. The distance between adjacent nodes or adjacent antinodes is λ/2, while the distance between a node and the nearest antinode is λ/4.
波节是介质中位移始终为零的位置，它们之间的间隔为半个波长（λ/2）。波腹是位移在正负最大值之间振荡的位置，位于两个波节的正中间。相邻波节或相邻波腹之间的距离为 λ/2，而波节到最近波腹的距离为 λ/4。
3. Formation and Superposition | 形成与叠加
Standing waves form when a wave is reflected back upon itself. The incident and reflected waves have the same speed, frequency, and amplitude, and their superposition produces a resultant wave that varies in amplitude along the medium but does not appear to propagate. This requires that the reflected wave is in phase with the incident wave at certain boundaries, leading to constructive interference at antinodes and destructive interference at nodes.
驻波的形成源于波被反射回自身。入射波与反射波具有相同的速度、频率和振幅，它们的叠加产生了一个沿介质振幅变化但看似不传播的合成波。这就要求在特定边界处反射波与入射波同相，从而在波腹处产生相长干涉，在波节处产生相消干涉。
4. Standing Waves on a String | 弦上的驻波
When a string is fixed at both ends, standing waves can be produced by plucking or driving it at a resonant frequency. The boundary conditions require nodes at the fixed ends, so the string's length L must equal an integer number of half-wavelengths. The possible wavelengths are given by:
当弦的两端固定时，通过弹拨或以共振频率驱动弦，可以产生驻波。边界条件要求固定端必须是波节，因此弦长 L 必须等于半波长的整数倍。可能的波长为：
L = n λ/2, n = 1, 2, 3, ...
e lowest frequency (n=1) is the fundamental frequency, and higher n values give overtones.
每种驻波模式的频率为 f = v/λ = n v/(2L)，其中 v 是弦上的波速。最低的频率（n=1）是基频，更高的 n 值给出泛音。
5. Harmonics: The Fundamental and Overtones | 谐波：基频与泛音
In a standing wave system, the fundamental frequency is the lowest possible frequency of vibration. The higher frequencies that are integer multiples of the fundamental are called harmonics. The first harmonic is the fundamental itself, the second harmonic (first overtone) has twice the fundamental frequency, and so on. On a string fixed at both ends, all harmonics are present in integer ratios.
在驻波系统中，基频是振动可能的最低频率。基频的整数倍高频称为谐波。第一谐波就是基频，第二谐波（第一泛音）的频率是基频的两倍，以此类推。在两端固定的弦上，所有整数比的谐波均会出现。
6. Standing Waves in Pipes: Open and Closed | 管中的驻波：开口与闭口
to a string. For a pipe closed at one end, there must be a node at the closed end and an antinode at the open end. This gives:
驻波也可以在管内的空气柱中形成。对于两端开口的管，边界条件要求在开口端为波腹。管长 L 满足 L = n λ/2，与弦类似。对于一端封闭的管，封闭端必须是波节，开口端为波腹，因此得到：
L = (2n - 1) λ/4, n = 1, 2, 3, ...
Thus, a closed pipe supports only odd harmonics (1st, 3rd, 5th, ...), while an open pipe supports all harmonics. The fundamental frequency for a closed pipe is half that of an open pipe of the same length, assuming the same wave speed.
因此，一端封闭的管只支持奇次谐波（第1、3、5……），而两端开口的管支持所有谐波。假设波速相同，相同长度的闭管基频是开管基频的一半。
7. Resonance and Standing Wave Ratio | 共振与驻波比
Resonance occurs when a driving frequency matches a natural frequency of the system, causing the standing wave amplitude to become very large. In many contexts, the Standing Wave Ratio (SWR) is used to describe how much of the wave is reflected. A perfect standing wave has an infinite SWR (no net power transfer), while a pure travelling wave has an SWR of 1. In instruments, resonance selects specific frequencies, producing musical tones.
当驱动频率与系统的固有频率匹配时发生共振，使驻波振幅变得非常大。在许多情况下，驻波比（SWR）用于描述波被反射的程度。完美的驻波具有无穷大的驻波比（没有净功率传输），而纯行波的驻波比为1。在乐器中，共振选择特定频率，产生乐音。
8. Phase Relationship and Energy | 相位关系与能量
In a standing wave, all points between two consecutive nodes vibrate in phase with each other, but they are in antiphase (180° out of phase) with points between the next pair of nodes. Energy is not propagated along the medium; instead, it is stored alternately as kinetic and potential energy. At antinodes, kinetic energy is maximum, while at nodes, potential energy from tension or pressure is maximum, depending on the medium.
在驻波中，两个相邻波节之间的所有点同相振动，但与下一对节点之间的点反相（相位差180°）。能量不会沿介质传播，而是交替以动能和势能的形式储存。在波腹处动能最大，而在波节处，根据介质不同，张力势能或压力势能最大。
9. Mathematical Description | 数学描述
ibes the time oscillation. Nodes occur where sin(kx) = 0, i.e., x = m λ/2, with m integer.
驻波的位移可以表示为两列波的和：y = y₀ sin(kx - ωt) + y₀ sin(kx + ωt) = 2y₀ sin(kx) cos(ωt)。其中 k = 2π/λ 是波数，ω = 2πf 是角频率。sin(kx) 项给出空间振幅包络，cos(ωt) 描述时间振荡。波节出现在 sin(kx) = 0 处，即 x = m λ/2，m 为整数。
10. Experimental Determination | 实验测定
be determined from v = f λ. Similar experiments with resonance tubes allow the speed of sound to be measured.
在实验室中，可以通过将振动发生器连接到张紧的弦上来演示弦上的驻波。调整频率，直到出现稳定的波节和波腹图案。通过测量波节间的距离并结合频率，可以利用 v = f λ 求出波速。类似的共振管实验可以测量声速。
11. Applications of Standing Waves | 驻波的应用
Standing waves are central to the operation of musical instruments such as guitars, violins, flutes, and organs. They are also used in microwave cavities, lasers (optical standing waves), and in the design of antennas. In quantum mechanics, electron standing waves describe atomic orbitals, and in telecommunications, standing wave ratio is a critical parameter for impedance matching.
驻波对于吉他、小提琴、长笛和风琴等乐器的运作至关重要。它们还被用于微波腔、激光器（光学驻波）以及天线设计中。在量子力学中，电子驻波描述了原子轨道；而在电信领域，驻波比是阻抗匹配的关键参数。
12. Comparison with Travelling Waves | 与行波的比较
d. Travelling waves can be described by y = A sin(kx ± ωt), whereas standing waves involve a product of spatial and temporal functions.
与驻波不同，行波通过介质传递能量。在行波中，如果不考虑阻尼，每个质点以相同的振幅振动，但相位沿传播方向变化。而在驻波中，振幅随位置变化，能量是局域的。行波可由 y = A sin(kx ± ωt) 描述，而驻波则是空间函数与时间函数的乘积。
Published by TutorHao | Physics Revision Series | aleveler.com
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