2 Travelling Waves | 2 行进波

📚 2 Travelling Waves | 2 行进波

Waves are one of the most fundamental ways in which energy and information are transferred from one place to another. A travelling wave is a disturbance that propagates through a medium or vacuum, carrying energy without any net transport of matter. From the ripples on a pond to the light from a distant star, travelling waves shape our understanding of the physical world. In this article, we explore the essential properties, mathematical descriptions, and characteristic behaviours of travelling waves, as required by the IB Physics syllabus and Cambridge International examinations.

波是能量和信息从一处传递到另一处的最基本方式之一。行进波是一种在介质或真空中传播的扰动，它携带能量而不引起物质的净位移。从池塘的涟漪到遥远恒星的光，行进波塑造了我们对物理世界的理解。本文根据IB物理大纲及剑桥国际考试的要求，探讨行进波的基本性质、数学描述和特征行为。

1. What is a Travelling Wave? | 什么是行进波？

A travelling wave is a self-sustaining disturbance that moves through space and time, transferring energy from a source to surrounding points. Unlike a stationary pattern, the wave profile advances continuously. Crucially, the particles of the medium (if one exists) oscillate about their equilibrium positions but do not travel with the wave. For example, when a water wave moves across the sea, the water molecules move in small circles or ellipses, but they return to their original positions as the crest passes. The energy, not the material, moves forward.

行进波是一种自持的扰动，在空间和时间中移动，将能量从波源传到周围各点。与驻波不同，行进波的波形不断前进。关键在于，介质中的粒子（若介质存在）在平衡位置附近振动，但并不随波前进。例如，当水波在海面上移动时，水分子作微小的圆周或椭圆运动，当波峰通过后它们又返回原处。前进的是能量，而非物质。

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2. Transverse and Longitudinal Waves | 横波与纵波

Travelling waves can be classified according to the direction of particle oscillation relative to the direction of energy propagation. In a transverse wave, the particles vibrate perpendicular to the direction of wave travel. Light and all other electromagnetic waves are transverse, as are waves on a stretched string and seismic S-waves. In a longitudinal wave, the particles vibrate parallel to the direction of energy transfer. Sound waves in air and seismic P-waves are common examples. Some waves, such as water surface waves, combine both transverse and longitudinal components.

行进波可以根据质点振动方向与能量传播方向的相对关系来分类。在横波中，质点振动方向垂直于波的传播方向。光及其他所有电磁波都是横波，拉紧的弦上的波和地震S波也是横波。在纵波中，质点振动方向平行于能量传递方向。空气中的声波和地震P波是常见的纵波。有些波（如水表面波）兼具横波和纵波成分。

Key characteristics:

主要特点：

• Transverse waves exhibit crests and troughs; / 横波呈现波峰和波谷；
• Longitudinal waves exhibit compressions and rarefactions. / 纵波呈现疏密（压缩和稀疏）区域。

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3. Describing Waves: Key Quantities | 描述波：关键物理量

To analyse travelling waves, physicists define several fundamental quantities. The displacement (y) is the distance of a particle from its equilibrium position at a given instant. The amplitude (A) is the maximum displacement from equilibrium. The wavelength (λ) is the shortest distance between two successive points that are in phase, e.g., two adjacent crests. The period (T) is the time taken for one complete oscillation, and the frequency (f) is the number of oscillations per second, related by f = 1/T. The wave speed (v) is the rate at which the wave profile advances.

为了分析行进波，物理学家定义了几个基本物理量。位移（y）是指给定时刻质点离平衡位置的距离。振幅（A）是距离平衡位置的最大位移。波长（λ）是相继两个同相点之间的最短距离，例如两相邻波峰。周期（T）是完成一次全振动所需的时间，频率（f）是每秒振动的次数，满足 f = 1/T。波速（v）是波形前进的速率。

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4. The Wave Equation: v = f λ | 波动方程：v = f λ

One of the most important relationships for travelling waves links speed, frequency and wavelength. The wave equation is expressed as:

对于行进波，最重要的关系式之一将波速、频率和波长联系起来。波动方程表述为：

v = f λ

In this equation, v is the wave speed measured in metres per second (m s⁻¹), f is the frequency in hertz (Hz), and λ is the wavelength in metres (m). This equation holds for all types of travelling waves. For a given medium, the wave speed is often fixed by its physical properties (e.g., tension and linear density for a string, or bulk modulus and density for sound in a fluid). Therefore, if the frequency changes, the wavelength adjusts inversely to keep the product constant.

该式中，v 为波速，单位米每秒（m s⁻¹）；f 为频率，单位赫兹（Hz）；λ 为波长，单位米（m）。此方程对所有类型的行进波均成立。对于给定的介质，波速通常由其物理性质决定（例如，弦的波速取决于张力和线密度，流体中声速取决于体弹模量和密度）。因此，若频率改变，波长会成反比调整以保持乘积不变。

Quantity	Symbol	SI unit
Wave speed	v	m s⁻¹
Frequency	f	Hz
Wavelength	λ	m

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5. Mathematical Representation | 数学表示

A travelling wave can be represented mathematically by a sinusoidal function. For a wave moving in the positive x-direction, the displacement y of a particle at position x and time t is given by:

行进波可用正弦函数进行数学表示。对于沿正x方向传播的波，位置 x、时刻 t 处质点的位移 y 由下式给出：

y = A sin (k x − ω t + φ)

Here A is the amplitude, k = 2π/λ is the wave number (angular spatial frequency), ω = 2πf is the angular frequency, and φ is the initial phase constant. The argument (k x − ω t) determines the phase of the wave. If the wave moves in the negative x-direction, the sign between the terms becomes positive: y = A sin (k x + ω t + φ). This description allows us to calculate the displacement of any particle at any instant and is fundamental to understanding interference and diffraction.

式中 A 为振幅，k = 2π/λ 为波数（角空间频率），ω = 2πf 为角频率，φ 为初相常数。(k x − ω t) 这个自变量决定了波的相位。若波沿负x方向传播，则两者之间的符号为正：y = A sin (k x + ω t + φ)。这种描述使我们能计算任一质点在任一时刻的位移，也是理解干涉和衍射的基础。

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6. Wavefronts and Huygens’ Principle | 波前与惠更斯原理

A wavefront is a surface connecting all adjacent points that are in phase. For a point source emitting waves uniformly in a homogeneous medium, the wavefronts are spherical. At a large distance from the source, these spherical surfaces approximate plane wavefronts. Rays are lines drawn perpendicular to wavefronts, indicating the direction of energy propagation. Huygens’ principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets. This construction elegantly explains reflection, refraction, and diffraction without requiring the full wave equation.

波前是连接所有相邻同相点的一个面。对于在均匀介质中均匀发射的点波源，波前为球形。在远离波源处，这些球面可近似为平面波前。射线是与波前垂直的线，指示能量传播的方向。惠更斯原理指出，波前上的每一点都可以视为次级的球面子波源。新的波前就是这些子波的包络面。这一构造巧妙地解释了反射、折射和衍射现象，无需完全依赖波动方程。

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7. Intensity and Amplitude | 强度与振幅

The intensity (I) of a travelling wave is the power transmitted per unit area perpendicular to the direction of propagation. It is measured in watts per square metre (W m⁻²). For any wave, intensity is proportional to the square of the amplitude:

行进波的强度（I）定义为单位时间内通过垂直于传播方向的单位面积的能量，单位为瓦每平方米（W m⁻²）。对任何波，强度与振幅的平方成正比：

I ∝ A²

For a point source emitting spherical waves uniformly, the same energy is spread over increasingly large spherical surfaces. The surface area of a sphere is 4πr², so the intensity obeys the inverse-square law:

对于均匀发射球面波的点波源，相同的能量分散在越来越大的球面上。球面面积为 4πr²，因此强度遵循平方反比定律：

I = P / (4π r²) ∝ 1/r²

This explains why light from a distant star appears dimmer and why the sound from a speaker becomes quieter as you move away. The relationship I ∝ A² also links loudness in sound to pressure amplitude, and brightness in light to electric field amplitude.

这解释了为何遥远恒星的光显得暗淡，以及为何远离扬声器时声音变小。I ∝ A² 的关系也将声音的响度与压强振幅、光的亮度与电场振幅联系起来。

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8. Polarisation | 偏振

Polarisation is the phenomenon that distinguishes transverse waves from longitudinal waves. Only transverse waves can be polarised. A polarised wave vibrates in a single plane, whereas an unpolarised wave has vibrations in many planes perpendicular to the direction of propagation. Light from the sun or a filament bulb is unpolarised. Passing it through a polarising filter restricts the oscillations to one direction, reducing intensity. According to Malus’s law, if an analyser (a second polariser) is rotated by an angle θ relative to the first, the transmitted intensity is:

偏振是区分横波与纵波的现象。只有横波才能被偏振。偏振波在单一平面内振动，而非偏振波的振动存在于垂直于传播方向的多个平面内。来自太阳或白炽灯的光是非偏振的。使其通过偏振片可将振动限制在一个方向上，强度随之降低。根据马吕斯定律，若检偏器（第二个偏振片）相对于第一个偏振片旋转角度 θ，则透射强度为：

I = I₀ cos² θ

Polarisation by reflection, scattering, and selective absorption has many applications, including polarising sunglasses, LCD screens, and stress analysis in materials. The fact that sound (a longitudinal wave) cannot be polarised is strong evidence for its nature.

反射偏振、散射偏振和选择性吸收偏振有多种应用，包括偏振太阳镜、液晶显示屏和材料应力分析。声波（纵波）不能被偏振，这一事实有力地证明了其纵波特性。

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9. Superposition of Travelling Waves | 行进波的叠加

When two or more travelling waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This is the principle of superposition. It holds for all linear wave phenomena. If the waves are coherent (same frequency and constant phase difference), they produce a steady interference pattern. Constructive interference occurs when waves meet in phase, giving resultant amplitude A_resultant = A₁ + A₂, and thus increased intensity. Destructive interference occurs when they meet exactly out of phase, yielding minimum amplitude and reduced intensity. The superposition of travelling waves leads to standing waves under reflection conditions, which are described by a different wave pattern.

当两列或多列行进波在某点相遇时，合位移等于各列波位移的矢量和，这就是叠加原理。该原理适用于所有线性波动现象。若波是相干的（频率相同且相位差恒定），则会产生稳定的干涉图样。当波同相相遇时发生相长干涉，合振幅 A_resultant = A₁ + A₂，强度增大。当波反相相遇时发生相消干涉，得到最小振幅和减弱的强度。行进波的叠加在反射条件下可形成驻波，驻波由不同的波形模式描述。

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10. Reflection, Refraction and Diffraction | 反射、折射与衍射

When a travelling wave encounters a boundary between two media, part of the wave is reflected and part is transmitted (refracted). The law of reflection states that the angle of incidence equals the angle of reflection. For refraction, the wave changes speed and, unless incident normally, changes direction. Snell’s law relates the angles and wave speeds or refractive indices:

当行进波遇到两种介质的界面时，一部分波被反射，一部分波被透射（折射）。反射定律：入射角等于反射角。对于折射，波速改变，除非垂直入射，方向也会改变。斯涅尔定律将角度与波速或折射率联系起来：

n₁ sin θ₁ = n₂ sin θ₂

Diffraction is the spreading of a wave around obstacles or through apertures. The amount of diffraction increases when the wavelength is comparable to the gap size. Diffraction is a key signature of wave behaviour and is explained by Huygens’ principle.

衍射是波绕过障碍物或通过狭缝时展宽的现象。当波长与缝隙大小相当时，衍射最显著。衍射是波动行为的重要标志，并可由惠更斯原理加以解释。

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11. The Doppler Effect | 多普勒效应

The Doppler effect is the change in observed frequency when there is relative motion between a wave source and an observer. For sound and other mechanical waves, the observed frequency f’ is given by:

多普勒效应是指波源与观察者之间存在相对运动时，观测频率发生变化的现象。对于声音和其他机械波，观测频率 f’ 由下式给出：

f’ = f × (v ± vₒ) / (v ∓ vₛ)

Here f is the source frequency, v is the wave speed in the medium, vₒ is the observer’s speed relative to the medium, and vₛ is the source’s speed. The upper signs apply when the source and observer move towards each other; the lower signs when they move apart. For light, a relativistic treatment is required, but at speeds much lower than c, the fractional change in frequency is approximately Δf/f ≈ v_rel/c. The Doppler effect is used in radar speed guns, medical ultrasound, and the redshift of galaxies.

式中 f 是波源频率，v 是介质中的波速，vₒ 是观察者相对于介质的速率，vₛ 是波源的速率。当波源与观察者相互靠近时取上方符号，相互远离时取下方符号。对于光，需用相对论处理，但在远低于光速时，频率的相对变化近似为 Δf/f ≈ v_rel/c。多普勒效应应用于雷达测速、医用超声波和星系红移。

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12. Summary and Applications | 总结与应用

Travelling waves form the backbone of wave physics. They are described by amplitude, wavelength, frequency, and speed, linked by v = f λ. The mathematical function y = A sin (k x − ω t) captures their sinusoidal nature. Intensity is proportional to the square of the amplitude, and for spherical waves follows an inverse-square law. Polarisation proves the transverse nature of light and other electromagnetic waves. Superposition explains interference, while Huygens’ principle unifies reflection, refraction, and diffraction. The Doppler effect has profound implications in both everyday technology and astrophysics.

行进波是波动物理学的基石。它们由振幅、波长、频率和波速来描述，通过 v = f λ 相互联系。数学函数 y = A sin (k x − ω t) 描绘了其正弦本质。强度正比于振幅的平方，球面波服从平方反比定律。偏振证明了光和其他电磁波的横波属性。叠加原理解释了干涉，而惠更斯原理统一了反射、折射和衍射。多普勒效应在日常技术和天体物理学中均具有深远意义。

From seismic wave detection to fibre optic communication, the principles of travelling waves are essential for modern science and engineering. Mastery of these concepts provides a strong foundation for understanding more advanced topics such as quantum mechanics and electromagnetic theory.

从地震波探测到光纤通信，行进波的原理对现代科学与工程至关重要。掌握这些概念，为理解量子力学与电磁理论等更高级的课题奠定了坚实基础。

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