📚 Travelling Waves | 行波
A travelling wave is a disturbance that propagates through a medium or space, carrying energy from one location to another without the permanent displacement of matter. Understanding travelling waves is fundamental in IB Physics, as they underpin topics ranging from sound and light to water waves and seismic activity. This article explores the key characteristics, mathematical descriptions, and behaviours of travelling waves, providing a thorough revision resource for students.
行波是一种通过介质或空间传播的扰动，它将能量从一个位置传递到另一个位置，而物质本身不发生永久性位移。理解行波在IB物理中是基础，因为它们涵盖了从声音和光到水波和地震活动等主题。本文探讨行波的关键特性、数学描述和行为，为学生提供全面的复习资源。
1. What is a Travelling Wave? | 什么是行波？
A travelling wave is a moving pattern of oscillation that transfers energy through a medium or vacuum. Crucially, the particles of the medium oscillate about fixed equilibrium positions; they do not travel along with the wave. For example, when a stone is dropped into a pond, the water ripples move outward, but the water molecules themselves mainly move up and down locally.
行波是一种移动的振动模式，通过介质或真空传递能量。关键的是，介质的粒子围绕固定的平衡位置振动，它们并不随波一起移动。例如，当石头落入池塘，水波向外移动，但水分子本身只是在原地上下运动。
Waves can be mechanical, requiring a material medium (e.g., sound waves, water waves, waves on a string), or electromagnetic, which can travel through a vacuum (e.g., light, radio waves). In both cases, the wave carries energy and momentum without transporting matter over long distances.
波可以是机械波，需要物质介质（如声波、水波、绳波），也可以是电磁波，可以在真空中传播（如光、无线电波）。在这两种情况下，波携带能量和动量而不远距离输送物质。
A travelling wave is often represented by a sinusoidal profile moving with a constant speed. The direction of energy transfer is always perpendicular to the wavefronts, where wavefronts are lines connecting points of equal phase.
行波通常用一个以恒定速度移动的正弦形状来表示。能量传递方向始终与波前垂直，而波前是连接同相位点的线。
2. Transverse and Longitudinal Waves | 横波与纵波
Travelling waves are classified into two main types based on the direction of particle oscillation relative to the direction of energy transfer. In a transverse wave, the particles oscillate perpendicular to the direction of energy propagation. Common examples include waves on a string, water surface waves, and all electromagnetic waves. The up-and-down motion of a rope shaken at one end illustrates a transverse travelling wave.
行波根据粒子振动方向相对于能量传递方向分为两种主要类型。在横波中，粒子振动方向垂直于能量传播方向。常见例子包括绳上的波、水面波以及所有电磁波。在绳子一端抖动产生的上下运动就展示了横波的行波。
In a longitudinal wave, the particles oscillate parallel to the direction of energy transfer. This creates compressions (regions of high pressure/density) and rarefactions (regions of low pressure/density). Sound waves in air are the most typical example of longitudinal travelling waves. When a loudspeaker cone moves forward, it compresses the air in front, sending a compression travelling outward.
在纵波中，粒子振动方向与能量传递方向平行。这产生压缩区（高压/高密度区域）和稀疏区（低压/低密度区域）。空气中的声波是纵波最典型的例子。当扬声器纸盆向前移动时，它压缩前方的空气，将压缩区向外传播。
Some waves, such as water waves, have both transverse and longitudinal components, resulting in circular particle motion. Seismic S-waves are transverse, while P-waves are longitudinal and travel through the Earth's interior.
有些波，如水波，同时具有横向和纵向分量，导致粒子做圆周运动。地震S波是横波，而P波是纵波，它们穿过地球内部传播。
| Property | Transverse Wave | Longitudinal Wave |
|---|---|---|
| Particle oscillation direction | Perpendicular to energy transfer | Parallel to energy transfer |
| Examples | Light, string waves, water ripples | Sound, primary seismic waves |
| Can be polarised? | Yes | No |
| Visual representation | Sine curve with crests and troughs | Series of compressions and rarefactions |
（表格：横波与纵波的特性对比）
3. Wave Parameters: Amplitude, Wavelength, Frequency, Period | 波的参数：振幅、波长、频率、周期
Every travelling wave is described by several key quantities. The amplitude (A) is the maximum displacement of a particle from its equilibrium position. It is measured in metres (m) and indicates the amount of energy the wave carries – the larger the amplitude, the more energetic the wave.
每种行波都可以用几个关键量来描述。振幅（A）是粒子偏离其平衡位置的最大位移，单位为米（m），表示波携带的能量多少——振幅越大，波的能量越大。
The wavelength (λ) is the distance between two consecutive points that are in phase, such as crest to crest or trough to trough. It is also measured in metres and determines the spatial period of the wave.
波长（λ）是相邻两个同相位点之间的距离，如波峰到波峰或波谷到波谷。波长也以米为单位，决定了波的空间周期。
in hertz (Hz), where 1 Hz = 1 s⁻¹. Frequency and period are inverses: f = 1/T or T = 1/f.
周期（T）是粒子完成一次完整振动所需的时间，或一个完整的波周期经过一个固定点所需的时间。它用秒（s）来衡量。频率（f）是每秒通过某一点的完整周期数，单位为赫兹（Hz），1 Hz = 1 s⁻¹。频率和周期互为倒数：f = 1/T 或 T = 1/f。
In a travelling wave, the product of frequency and wavelength gives the wave speed, a relationship explored in the next section. All these parameters are independent of the wave's speed in a given medium, except that frequency is determined by the source and remains constant when a wave passes from one medium to another, while speed and wavelength change.
在行波中，频率和波长的乘积给出波速，下一节将探讨这一关系。所有这些参数除了频率由波源决定并且在波从一种介质进入另一种介质时保持不变外，在给定介质中与波速无关，而波速和波长则会发生改变。
4. Wave Speed Equation: v = fλ | 波速公式：v = fλ
The fundamental relationship for any travelling wave connects speed (v), frequency (f), and wavelength (λ):
任何行波的基本关系式将速度（v）、频率（f）和波长（λ）联系起来：
v = f × λ
where v is measured in m s⁻¹, f in Hz, and λ in m. This equation implies that for a constant wave speed (as in a uniform medium), higher frequency means shorter wavelength, and vice versa.
其中 v 以 m s⁻¹ 为单位，f 以 Hz 为单位，λ 以 m 为单位。该公式表明，在波速恒定的情况下（如同种均匀介质），频率越高波长越短，反之亦然。
When a wave travels from one medium to another, the frequency remains unchanged because it is set by the source. However, the wave speed changes (for example, light slows down in glass), and therefore the wavelength adjusts accordingly. This is why a straw appears bent in water – the wavelength of light changes, causing refraction.
当波从一种介质进入另一种介质时，频率保持不变，因为它由波源决定。然而，波速会改变（例如，光在玻璃中变慢），因此波长会相应调整。这就是为什么吸管在水中看起来是弯的——光的波长改变，导致折射。
On a string under tension, the wave speed v is determined by v = √T/μ e wavelengths and frequencies for standing waves, but the travelling wave speed itself is fixed by the string's properties.
在张紧的绳上，波速 v 由 v = √T/μ 决定，其中 T 是张力，μ 是单位长度质量。将此与 v = fλ 结合可以计算驻波的可能波长和频率，但行波速度本身由绳的性质固定。
5. Displacement–Time and Displacement–Distance Graphs | 位移-时间图与位移-距离图
Two graphs are essential for analysing travelling waves. A displacement–distance graph shows how the displacement of all particles varies with position along the wave at a specific instant in time. It looks like a sine wave from which you can measure the wavelength λ and amplitude A directly. The shape of the graph moves along the direction of propagation as time progresses, which is the defining feature of a travelling wave.
分析行波有两个重要的图。位移-距离图显示在某一特定时刻，所有粒子的位移如何随波的位置变化。它看起来像一个正弦波，可以直接从中测量波长 λ 和振幅 A。随着时间的推移，图形沿传播方向移动，这是行波的决定性特征。
A displacement– requency f = 1/T. The particle undergoes simple harmonic motion if the wave is sinusoidal.
另一方面，位移-时间图显示单个粒子的位移如何随时间变化。从该图中可以测量振动的周期 T 和振幅 A。知道了 T，就可以计算频率 f = 1/T。如果波是正弦波，粒子将做简谐运动。
It is vital not to confuse these two representations. In a displacement–distance graph, the horizontal axis is distance x (m), and the separation between peaks is λ. In a displacement–time graph, the horizontal axis is time t (s), and the separation between peaks is T. The two graphs are complementary and together fully define the wave.
务必不要混淆这两种表示方法。在位移-距离图中，横轴是距离 x（m），峰值间隔为 λ。在位移-时间图中，横轴是时间 t（s），峰值间隔为 T。这两种图互为补充，共同完整定义波。
6. The Wave Function | 波函数
For a sinusoidal travelling wave moving in the positive x-direction, the displacement y of a particle at position x and time t can be described mathematically as:
对于沿 x 轴正方向传播的正弦行波，粒子在位置 x 和时间 t 的位移 y 可以用数学描述为：
y(x,t) = A sin(ωt − kx + φ₀)
x) ensures that a particular wave crest moves with speed v = ω/k, which reduces to v = fλ.
这里，A 是振幅，ω 是角频率（ω = 2πf = 2π/T），k 是波数（k = 2π/λ），φ₀ 是初相常数。(ωt − kx) 这一项确保特定的波峰以 v = ω/k 的速度移动，最终化为 v = fλ。
he wave function describes the instantaneous displacement of any particle at any time, encapsulating the whole wave motion.
如果波沿 x 轴负方向传播，kx 前面的符号变为正号：y = A sin(ωt + kx + φ₀)。波函数描述了任意时刻任意粒子的瞬时位移，概括了整个波动运动。
onstrates the fundamental link between the temporal and spatial aspects of the wave. The same mathematics applies to both mechanical and electromagnetic travelling waves.
这种正弦形式是一维波动方程（∂²y/∂x² = (1/v²) ∂²y/∂t²）的解，它展示了波的时间与空间方面的基本联系。同样的数学适用于机械波和电磁波。
7. Phase and Phase Difference | 相位与相位差
The argument of the sine function in the wave equation, (ωt − kx + φ₀), is called the phase of the wave, often denoted by φ. Phase determines the stage of oscillation a particle is at in its cycle – for instance, whether it is at a crest, trough, or zero crossing. Two points on a wave that are an integer multiple of 2π radians apart in phase are said to be in phase; they have identical displacements and velocities at all times.
波动方程中正弦函数的自变量 (ωt − kx + φ₀) 被称为波的相位，通常用 φ 表示。相位决定了粒子在其周期中处于哪个振动阶段——例如，是处于波峰、波谷还是过零点。波上相位相差 2π 弧度整数倍的两点被称为同相；它们在任何时刻都具有相同的位移和速度。
The phase difference Δφ between two points separated by a distance Δx along the wave is given by:
波上相距 Δx 的两点之间的相位差 Δφ 由下式给出：
Δφ = (2π/λ) × Δx
antiphase. Phase difference is crucial in understanding interference and superposition effects.
因此，一个完整波长的路程差对应于 2π rad 的相位差，而半个波长的路程差（Δx = λ/2）给出 Δφ = π rad，意味着两点反相。相位差对于理解干涉和叠加效应至关重要。
Phase can also be described in terms of radians, degrees, or fractions of a cycle. In IB Physics, radian measure is standard. The concept of phase difference allows us to predict constructive or destructive interference when two waves meet.
相位还可以用弧度、角度或周期的分数来描述。在IB物理中，弧度是标准度量。相位差的概念使我们能够预测两波相遇时产生相长干涉还是相消干涉。
8. Energy Transport and Intensity | 能量传输与强度
A travelling wave transfers energy without transferring matter. The energy of a mechanical wave is associated with the kinetic and potential energy of the oscillating particles. For a sinusoidal wave on a string, the average power transported is proportional to the square of the amplitude and the square of the angular frequency: P ∝ A²ω².
行波传输能量而不传输物质。机械波的能量与振动粒子的动能和势能有关。对于弦上的正弦波，传输的平均功率正比于振幅的平方和角频率的平方：P ∝ A²ω²。
r²). This inverse-square law applies to sound, light, and other waves that spread in three dimensions.
波的强度（I）定义为单位面积上入射的功率。对于从点源均匀扩展开的球面波前，强度随距离源的距离的平方而减小：I = P/(4πr²)。这个平方反比定律适用于在三维空间中扩散的声波、光波和其他波。
Since intensity is proportional to the square of the amplitude (I ∝ A²), the amplitude of a spherical travelling wave decreases linearly with distance from the source (A ∝ 1/r). This is why distant sounds are quieter and distant stars are fainter – the wave energy spreads over a larger area.
由于强度正比于振幅的平方（I ∝ A²），球面行波的振幅随距波源的距离线性减小（A ∝ 1/r）。这就是为什么远处的声音较小、远处的星星较暗——波的能量散布在更大的面积上。
9. Reflection of Travelling Waves | 行波的反射
When a travelling wave encounters a boundary between two media, some of its energy is reflected and some is transmitted. The behaviour at a fixed end (such as a string tied to a wall) results in a reflected wave that is inverted – a phase change of π rad (180°) occurs. This is analogous to a pulse hitting a fixed wall and returning upside down.
当行波遇到两种介质之间的边界时，部分能量被反射，部分被透射。在固定端（如系在墙上的绳），产生的反射波是倒置的——发生 π rad（180°）的相位变化。这类似于一个脉冲撞击固定墙壁并倒置返回。
At a free end (such as a string attached to a ring that can slide frictionlessly on a rod), the reflected wave is not inverted and returns with the same orientation. Here, no phase change occurs upon reflection. These reflection principles are essential in understanding standing wave formation.
在自由端（如连接在一个可以在杆上无摩擦滑动的环上的绳），反射波不翻转并以相同取向返回。这里，反射不发生相位变化。这些反射原理对于理解驻波的形成至关重要。
When a wave moves from one medium to another, the transmitted wave experiences a change in speed and wavelength, while its frequency remains constant. The proportion of energy reflected versus transmitted depends on the impedance difference of the two media – larger mismatch causes greater reflection.
当波从一种介质进入另一种介质时，透射波经历速度和波长的变化，而频率保持不变。反射和透射的能量比例取决于两种介质的阻抗差异——差异越大，反射越强。
10. Polarisation as a Property of Transverse Travelling Waves | 偏振作为横波行波的特性
Polarisation is a unique property of transverse waves that confirms their transverse nature. A wave is polarised when the oscillations of its particles are restricted to a single plane. For instance, a vertically oscillating string produces a vertically polarised transverse wave.
偏振是横波的独特性质，证实了它们的横向本质。当波的粒子振动被限制在一个单一平面内时，该波就是偏振的。例如，垂直振动的弦产生垂直偏振的横波。
Longitudinal waves cannot be polarised because particle motion is parallel to the direction of propagation – there is no transverse plane to restrict. Polarisation of light, achieved by passing it through a polarising filter, provides strong evidence that light is a transverse electromagnetic wave.
纵波不能偏振，因为粒子运动方向平行于传播方向——不存在可以限制的横向平面。光的偏振（通过偏振滤光片实现）提供了光是横电磁波的有力证据。
In IB Physics, Malus' nsity of polarised light after passing through a second polariser, known as an analyser. This law further illustrates the transverse oscillation of electric fields in electromagnetic travelling waves.
在IB物理中，马吕斯定律（I = I₀ cos²θ）描述了偏振光通过第二个偏振器（称为检偏器）后的强度。这一定律进一步说明了电磁行波中电场的横向振动。
11. Examples of Travelling Waves in Context | 实际场景中的行波例子
Sound waves in air are a common example of longitudinal travelling waves. A vibrating source, such as a tuning fork, alternately compresses and rarefies the surrounding air, creating a longitudinal wave that travels at approximately 340 m s⁻¹ in air at room temperature. The frequency determines the pitch, while the amplitude is related to the loudness.
空气中的声波是纵波行波的常见例子。振动源（如音叉）交替压缩和稀疏周围的空气，产生纵波，在室温空气中以约 340 m s⁻¹ 的速度传播。频率决定了音调的高低，而振幅与响度有关。
he oscillating electric and magnetic fields are perpendicular to each other and to the direction of travel.
电磁波，包括可见光、无线电波和X射线，是可以在真空中以 c = 3.00 × 10⁸ m s⁻¹ 传播的横波行波。它们的波长和频率跨越巨大范围，但在真空中都遵循 v = fλ。振荡的电场和磁场彼此垂直，且都垂直于传播方向。
Water waves present a combination of transverse and longitudinal motion, resulting in elliptical particle paths. They travel along the surface of water and are influenced by depth. Tsunamis are travelling water waves with extremely long wavelengths that can carry devastating energy across entire oceans.
水波呈现横波和纵波运动的结合，导致粒子做椭圆路径运动。它们沿水面传播，并受水深影响。海啸是长波长的行波水波，可以携带毁灭性的能量穿越整个海洋。
12. Summary of Key Concepts | 关键概念总结
s help visualise wave behaviour, while the wave function y(x,t) provides a complete mathematical description.
行波的特征在于通过振动传输能量，介质不发生整体移动。它们分为横波和纵波，其空间和时间特性由 v = fλ 关联。位移图有助于可视化波的行为，而波函数 y(x,t) 提供完整的数学描述。
Phase difference explains interference, energy transport is governed by amplitude and distance, and reflection depends on boundary conditions. Polarisation proves the transverse nature of certain waves. Mastery of these concepts is essential for success in the IB Physics waves topic.
相位差解释干涉，能量传输由振幅和距离决定，反射取决于边界条件。偏振证明了某些波的横向本质。掌握这些概念对于在IB物理波动部分取得成功至关重要。
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