📚 Standing Waves: Conditions and Mechanism Explained | 驻波的形成条件与机制详解
A standing wave, also called a stationary wave, is a wave pattern that appears to remain fixed in space. Unlike a travelling wave that transfers energy from one place to another, a standing wave stores energy in an oscillating pattern, with some points permanently at rest and other points vibrating with maximum amplitude.
驻波（又称定波）是一种在空间中看似固定不动的波动图样。与沿传播方向传递能量的行波不同，驻波将能量储存在振荡图样中，某些点始终保持静止，另一些点则以最大振幅振动。
1. What Is a Standing Wave? | 什么是驻波？
In a travelling wave, every particle of the medium has the same amplitude of vibration, and the phase of the oscillation changes continuously from one point to the next. In a standing wave, however, the amplitude of vibration varies with position: it is zero at nodes and maximum at antinodes.
在行波中，介质中每个质点的振动振幅相同，且振动的相位会随位置连续变化。而在驻波中，振动振幅随位置变化：波节处振幅为零，波腹处振幅最大。
The key feature of a standing wave is that the nodes and antinodes do not move along the medium. The individual particles still oscillate, but the overall wave pattern appears stationary.
驻波的关键特征是波节和波腹不沿介质移动。单个质点仍然在振动，但整个波动图样看起来是静止的。
2. The Principle of Superposition | 叠加原理
The formation of a standing wave is a direct consequence of the principle of superposition. When two or more waves meet at a point in a medium, the resultant displacement at that point is the vector sum of the displacements of the individual waves.
驻波的形成是叠加原理的直接结果。当两列或更多列波在介质中某点相遇时，该点的合位移等于各列波单独存在时位移的矢量和。
If two waves arrive in phase at a point, they reinforce each other and produce a larger amplitude. If they arrive in antiphase, they cancel each other and produce a smaller amplitude, possibly zero.
如果两列波在某点同相到达，它们相互加强，产生更大的振幅；如果反相到达，它们相互抵消，产生较小的振幅，甚至可能为零。
For a standing wave, we need two waves of equal amplitude and frequency travelling in opposite directions along the same line. Their superposition creates regions of constructive and destructive interference that are fixed in space.
要形成驻波，需要两列振幅相等、频率相同、沿同一直线反向传播的波。它们的叠加会在空间中形成固定不动的相长干涉区和相消干涉区。
3. Formation Mechanism: Two Travelling Waves | 形成机制：两列行波的叠加
Consider a stretched string fixed at one end. A continuous wave is sent along the string towards the fixed end. When it reaches the fixed end, the wave is reflected and travels back along the string in the opposite direction. The incident wave and the reflected wave now travel through the same medium in opposite directions.
设想一根一端固定的张紧弦线。一列连续波沿弦线传向固定端，到达固定端后被反射，并沿相反方向传回弦线。这时入射波和反射波在同一介质中沿相反方向传播。
Because both waves have the same speed, frequency and amplitude, they satisfy the conditions for sustained interference. At certain positions, the two waves always meet in phase, creating antinodes. At other positions, they always meet in antiphase, creating nodes.
由于两列波的波速、频率和振幅相同，它们满足持续干涉的条件。在某些位置，两列波始终同相相遇，形成波腹；在另一些位置，它们始终反相相遇，形成波节。
This interference pattern is stable because the phase difference between the two waves at any fixed point does not change with time. Therefore the nodes and antinodes remain at fixed positions.
这种干涉图样是稳定的，因为两列波在任意固定点的相位差不随时间变化，因此波节和波腹保持在固定位置。
4. Mathematical Description | 数学描述
Let the incident wave travelling to the right be represented by y₁ = A sin(kx − ωt) and the reflected wave travelling to the left be represented by y₂ = A sin(kx + ωt). Here A is the amplitude, k = 2π/λ is the angular wave number, and ω = 2πf is the angular frequency.
设向右传播的入射波为 y₁ = A sin(kx − ωt)，向左传播的反射波为 y₂ = A sin(kx + ωt)。其中 A 为振幅，k = 2π/λ 为角波数，ω = 2πf 为角频率。
Using the trigonometric identity for the sum of two sine functions, the resultant displacement is:
利用三角恒等式对两个正弦函数求和，合位移为：
y = y₁ + y₂ = A sin(kx − ωt) + A sin(kx + ωt) = 2A sin(kx) cos(ωt)
In this expression, the term 2A sin(kx) describes the amplitude of vibration at position x, while cos(ωt) describes the oscillation of every particle with the same angular frequency ω.
在这个表达式中，2A sin(kx) 表示位置 x 处的振动振幅，而 cos(ωt) 表示所有质点以相同角频率 ω 振动。
Notice that the resulting function is not of the form f(x − vt) or f(x + vt), so it does not travel. It is the product of a spatial factor and a time factor.
注意，合成函数不是 f(x − vt) 或 f(x + vt) 的形式，因此它不传播。它是空间因子与时间因子的乘积。
5. Nodes and Antinodes | 波节与波腹
Nodes occur where the amplitude factor is zero:
波节出现在振幅因子为零的位置：
sin(kx) = 0
kx = nπ, so x = nλ/2, where n = 0, 1, 2, …
Thus nodes are separated by half a wavelength, λ/2.
因此相邻波节之间的距离为半个波长 λ/2。
Antinodes occur where the amplitude factor has its maximum magnitude:
波腹出现在振幅因子达到最大绝对值的位置：
|sin(kx)| = 1
kx = (n + ½)π, so x = (2n + 1)λ/4, where n = 0, 1, 2, …
The distance between a node and the next antinode is therefore λ/4.
因此相邻波节与波腹之间的距离为 λ/4。
All particles between two consecutive nodes vibrate in phase, reaching their maximum and minimum displacements at the same time. Particles on opposite sides of a node vibrate in antiphase.
相邻两个波节之间的所有质点相位相同，同时达到最大位移和最小位移；波节两侧的质点相位相反。
6. Conditions for Standing Waves | 形成驻波的条件
For a stable standing wave to form, the following conditions must be satisfied:
要形成稳定的驻波，必须满足以下条件：
- The two waves must have the same frequency and wavelength; 两列波的频率和波长必须相同。
- The two waves must have the same amplitude if the nodes are to be perfect zeros; 如果希望波节处振幅严格为零，两列波的振幅必须相同。
- The two waves must travel in opposite directions along the same line; 两列波必须沿同一直线以相反方向传播。
- The waves must be continuous or sustained long enough to establish a steady pattern; 波必须是持续的，或有足够长时间来形成稳定图样。
- Boundary conditions must allow reflection, such as a fixed end or a free end; 边界条件必须允许反射，例如固定端或自由端。
If the amplitudes are not exactly equal, the destructive interference is incomplete. The nodes will not be true points of zero displacement, but rather points of minimum amplitude.
如果振幅不完全相等，相消干涉就不彻底。此时波节不会是完全静止的点，而是振幅最小的点。
7. Harmonics and Normal Modes | 谐波与简正模式
When a standing wave is formed on a string fixed at both ends, only certain wavelengths can fit exactly between the two fixed ends. This is because each fixed end must be a node.
当两端固定的弦线上形成驻波时，只有某些特定波长的波才能正好容纳在两固定端之间，因为每个固定端必须是波节。
For a string of length L, the allowed wavelengths satisfy:
对于长度为 L 的弦线，允许的波长满足：
L = nλ/2, so λₙ = 2L/n, where n = 1, 2, 3, …
The corresponding natural frequencies are:
对应的固有频率为：
fₙ = nv / (2L), where n = 1, 2, 3, …
Here v is the speed of the wave along the string. The n = 1 mode is called the fundamental or first harmonic; n = 2 is the second harmonic, and so on.
其中 v 是波沿弦线传播的速度。n = 1 的模式称为基频或一次谐波，n = 2 称为二次谐波，依此类推。
| Mode n | 模式 | Wavelength | 波长 | Frequency | 频率 |
| 1 | 2L | v / (2L) |
| 2 | L | v / L |
| 3 | 2L / 3 | 3v / (2L) |
8. Reflection and Boundary Conditions | 反射与边界条件
A standing wave is usually produced by the superposition of an incident wave and a wave reflected from a boundary. The type of boundary determines whether a node or an antinode forms there.
驻波通常由入射波与边界反射波的叠加产生。边界的类型决定了该处形成波节还是波腹。
At a fixed end, the reflected wave undergoes a phase change of π radians, which is equivalent to a path difference of λ/2. The incident and reflected displacements cancel at the boundary, producing a node.
在固定端，反射波会发生 π 弧度的相位突变，等价于 λ/2 的程差。入射波和反射波在边界处位移相消，形成波节。
At a free end, there is no phase change on reflection, and the reflected wave reinforces the incident wave at the boundary, producing an antinode.
在自由端，反射不发生相位突变，反射波与入射波在边界处相互加强，形成波腹。
For air columns, a closed end corresponds to a displacement node, while an open end corresponds to a displacement antinode. For a pipe closed at one end, the allowed wavelengths are odd multiples of the fundamental wavelength:
对于空气柱，封闭端对应位移波节，开口端对应位移波腹。对于一端封闭的管，允许的波长为基波波长的奇数倍：
L = (2m − 1)λ/4, so fₘ = (2m − 1)v / (4L), where m = 1, 2, 3, …
For a pipe open at both ends, the harmonic series is the same as for a string fixed at both ends: L = nλ/2.
对于两端开口的管，谐波系列与两端固定的弦相同：L = nλ/2。
9. Energy in a Standing Wave | 驻波中的能量
In an ideal standing wave, there is no net transfer of energy along the medium. The two travelling waves that form the standing wave carry equal energy in opposite directions, so their energy fluxes cancel.
在理想驻波中，能量不会沿介质发生净传递。形成驻波的两列行波携带等量能量但方向相反，因此它们的能流相互抵消。
Energy is, however, stored in the vibrating medium. It alternates between kinetic energy, when particles are moving through their equilibrium positions, and potential energy, when particles are momentarily at rest at their maximum displacements.
然而，能量储存在振动的介质中。它会在动能和势能之间交替转换：当质点经过平衡位置时动能最大，当质点到达最大位移瞬间静止时势能最大。
Because particles at the nodes do not move, energy does not flow across a node. This is why the standing wave appears to vibrate in “loops” between fixed nodes.
由于波节处的质点不运动，能量不会穿过波节。这就是驻波看起来像是在固定波节之间的“波腹段”内振动的原因。
10. Applications and Exam Tips | 应用与考试提示
Standing waves are essential to the operation of stringed instruments such as guitars and violins, and wind instruments such as organ pipes and flutes. The pitch of the note produced is determined by the fundamental frequency, which depends on the length, tension and mass per unit length of the vibrating medium.
驻波对于吉他、小提琴等弦乐器以及风琴管、长笛等管乐器至关重要。发出的音调由基频决定，而基频取决于振动介质的长度、张力和单位长度质量。
In the laboratory, stationary waves in a resonance tube are used to measure the speed of sound in air. By adjusting the length of the air column until a loud resonance is heard, the wavelength and hence the speed can be determined.
在实验室中，常利用共鸣管中的驻波来测量空气中的声速。通过调节空气柱长度，直到听到明显共鸣，即可确定波长，进而求出声速。
When solving A-Level problems, remember these key points:
在解答 A-Level 题目时，请记住以下要点：
- Draw a sketch and label nodes and antinodes clearly; 先画示意图，并清楚标出波节和波腹。
- Use the correct boundary condition: fixed end means a node, free end means an antinode; 使用正确的边界条件：固定端为波节，自由端为波腹。
- Remember that adjacent nodes are separated by λ/2, and a node and an antinode are separated by λ/4; 牢记相邻波节相距 λ/2，波节与相邻波腹相距 λ/4。
- For strings and open pipes use L = nλ/2; for pipes closed at one end use L = (2m − 1)λ/4; 对弦线和两端开口管使用 L = nλ/2；对一端封闭的管使用 L = (2m − 1)λ/4。
- Check whether the wave speed is given by v = fλ before substituting into frequency equations; 代入频率公式前，先确认是否可用 v = fλ。
By mastering the conditions and mechanism of standing waves, you can confidently solve problems involving interference, harmonics and resonance.
掌握了驻波的形成条件和机制，你就能自信地解决涉及干涉、谐波和共振的问题。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导