IB Physics: Propagation and Characteristics of Travelling Waves | IB物理：行波的传播与特征

📚 IB Physics: Propagation and Characteristics of Travelling Waves | IB物理：行波的传播与特征

A travelling wave, or progressive wave, is a disturbance that carries energy and momentum from one point in space to another without the bulk transfer of matter. Understanding how waves propagate and what defines their behaviour is essential for many IB Physics topics, from sound and light to seismic waves and quantum phenomena.

行波（又称前进波）是一种扰动，它把能量和动量从空间中的一个点传递到另一个点，而并不伴随物质的整体迁移。理解波如何传播以及什么决定了波的行为，对IB物理中的许多主题都至关重要，从声波、光波到地震波和量子现象。

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

A travelling wave is created when a source oscillates and the disturbance moves away from the source with a constant speed in a medium (or through a field). Each particle of the medium vibrates about its equilibrium position, but the wave itself travels onward. The key idea is that the medium does not travel with the wave; it merely oscillates locally.

行波由振源产生，扰动以恒定速度在介质（或场）中离开振源传播。介质的每个质点都在平衡位置附近振动，但波本身继续向前传播。关键概念是：介质并不随波移动，它只是在原地振动。

• The wave transfers energy and information, not matter.
• 波传递能量和信息，而不是传递物质。
• The medium’s particles oscillate about fixed equilibrium positions.
• 介质中的质点围绕固定的平衡位置振动。
• A travelling wave can be represented as a moving disturbance in space and time.
• 行波可以表示为在空间和时间中移动的扰动。

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2. Wavefronts and Rays | 波前与波线

A wavefront is an imaginary surface joining all points of a wave that are in the same phase, for example all crests at a given instant. Rays are lines drawn perpendicular to the wavefronts, indicating the direction of energy propagation. For a point source in a uniform medium, wavefronts are spheres; far from the source, they appear as parallel planes.

波前是一个假想的曲面，它连接同一时刻处于相同相位的所有点，例如所有波峰。波线是与波前垂直的线，表示能量传播的方向。对于均匀介质中的点波源，波前是球面；在远离波源处，波前近似为平行平面。

• Plane waves have parallel, equally spaced wavefronts.
• 平面波的波前是平行且等间距的。
• Rays are always perpendicular to wavefronts in isotropic media.
• 在各向同性介质中，波线始终垂直于波前。
• In IB problems, wavefronts help visualise reflection, refraction and diffraction.
• 在IB题目中，波前有助于直观地分析反射、折射和衍射。

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3. Transverse and Longitudinal Waves | 横波与纵波

In a transverse wave, the particle displacement is perpendicular to the direction of wave propagation. Examples include waves on a string, electromagnetic waves, and the secondary waves (S-waves) of earthquakes. In a longitudinal wave, the particle displacement is parallel to the direction of propagation, such as sound waves in air and primary waves (P-waves) in earthquakes.

在横波中，质点位移垂直于波的传播方向。例如弦上的波、电磁波以及地震的S波。在纵波中，质点位移平行于波的传播方向，例如空气中的声波和地震的P波。

Property	Transverse	Longitudinal
Displacement vs propagation	Perpendicular (90°)	Parallel (0° or 180°)
Common examples	String waves, light	Sound, spring pulses
Can travel in vacuum?	Yes (EM waves)	No (needs medium)

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4. Key Quantities: Wavelength, Period, Frequency, Amplitude | 关键物理量：波长、周期、频率、振幅

Wavelength λ is the distance between two consecutive points in the same phase, measured along the direction of propagation. The period T is the time for one complete oscillation of any particle in the medium. Frequency f is the number of oscillations per second, so f = 1/T. Amplitude A is the maximum displacement of a particle from equilibrium.

波长 λ 是沿传播方向两个相邻同相点之间的距离。周期 T 是介质中任一质点完成一次全振动所需的时间。频率 f 是每秒振动的次数，因此 f = 1/T。振幅 A 是质点偏离平衡位置的最大位移。

f = 1/T

• Wavelength is measured in metres (m).
• 波长的单位是米（m）。
• Frequency is measured in hertz (Hz), where 1 Hz = 1 s⁻¹.
• 频率的单位是赫兹（Hz），1 Hz = 1 s⁻¹。
• Amplitude determines the energy carried by the wave; larger amplitude means more energy.
• 振幅决定波携带的能量；振幅越大，能量越多。

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5. The Wave Equation v = fλ | 波速公式 v = fλ

The speed of a wave is the product of its frequency and wavelength. This relationship applies to all types of travelling waves, whether mechanical or electromagnetic. For a wave moving at speed v, in one period T the wave advances by one wavelength λ, so v = λ/T = fλ.

波速等于频率与波长的乘积。这个关系适用于所有类型的行波，无论是机械波还是电磁波。对于以速度 v 传播的波，在一个周期 T 内波前进一个波长 λ，因此 v = λ/T = fλ。

v = fλ

• For a given medium, wave speed is usually constant, so increasing f decreases λ.
• 对给定介质，波速通常恒定，因此频率增大时波长减小。
• When a wave changes medium, frequency stays constant but wavelength and speed change.
• 波进入不同介质时，频率保持不变，但波长和波速会改变。
• The wave equation is valid for both transverse and longitudinal waves.
• 波速公式对横波和纵波都成立。

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6. The Shape of a Wave: Displacement–Position and Displacement–Time Graphs | 波形图：位移-位置图与位移-时间图

A displacement–position graph is a snapshot of the wave at one instant; it shows how displacement varies with distance along the wave. A displacement–time graph follows one particle over time; it shows the oscillation of that single point. Both graphs are sinusoidal for a simple harmonic wave.

位移-位置图是波在某一时刻的快照，它显示位移沿波传播方向如何变化。位移-时间图跟踪某个质点随时间的变化，它显示该点的振动。对于简谐波，两个图像都是正弦曲线。

• On a displacement–position graph, the distance between adjacent crests is λ.
• 在位移-位置图上，相邻波峰之间的距离为 λ。
• On a displacement–time graph, the time between adjacent crests is T.
• 在位移-时间图上，相邻波峰之间的时间为 T。
• The gradient of a displacement–position graph is not the wave speed.
• 位移-位置图的斜率不是波速。
• The maximum value of either graph equals the amplitude A.
• 两个图像的最大值都等于振幅 A。

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7. Phase and Phase Difference | 相位与相位差

Phase describes the position of a point on the wave cycle relative to a reference point. Two points are in phase if they have the same displacement and are moving in the same direction. Phase difference is measured in radians or degrees, or as a fraction of a wavelength. A full cycle corresponds to 2π radians or 360°.

相位描述波循环中某一点相对于参考点的位置。如果两个点具有相同的位移且运动方向相同，则它们同相。相位差用弧度或度来度量，也可以用波长的分数表示。一个完整循环对应 2π 弧度或 360°。

Phase difference = (2π Δx)/λ

• Points separated by an integer number of wavelengths are in phase.
• 相距整数个波长的点同相。
• Points separated by an odd half-wavelength are in antiphase (phase difference π).
• 相距奇数个半波长的点反相（相位差为 π）。
• Phase difference is important in interference and standing waves.
• 相位差在干涉和驻波中非常重要。

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8. Energy Transfer in Travelling Waves | 行波中的能量传递

As a wave travels, it carries energy from the source to distant locations. For a mechanical wave, the energy is the sum of kinetic energy of oscillating particles and elastic potential energy stored in the medium. The intensity of a wave is proportional to the square of its amplitude: I ∝ A².

波传播时，把能量从波源传递到远处。对于机械波，能量是质点振动动能与介质弹性势能之和。波的强度与振幅的平方成正比：I ∝ A²。

I ∝ A²

• Intensity is power per unit area, measured in W m⁻².
• 强度是单位面积上的功率，单位是 W m⁻²。
• For a spherical wave from a point source, intensity decreases as 1/r² due to spreading.
• 对于点波源发出的球面波，由于扩散，强度按 1/r² 减小。
• The wave transfers energy without transferring net mass.
• 波传递能量但不传递净质量。

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9. Huygens’ Principle and Wave Propagation | 惠更斯原理与波的传播

Huygens’ principle states that every point on a wavefront acts as a secondary source of spherical wavelets. The new wavefront is the envelope of these wavelets after a small time interval. This principle explains why waves spread out and how they change direction when passing through openings or around obstacles.

惠更斯原理指出：波前上的每一点都可以看作新的子波源，发出球形子波。下一时刻的新波前就是这些子波的包络面。该原理解释了波为什么会扩展，以及通过狭缝或绕过障碍物时方向如何改变。

• Huygens’ construction applies to all types of travelling waves.
• 惠更斯作图法适用于所有类型的行波。
• It explains diffraction: wavefronts bend at edges and openings.
• 它解释了衍射：波前在边缘和狭缝处发生弯曲。
• It also explains reflection and refraction by considering the change in wave speed.
• 它还可以通过考虑波速变化来解释反射和折射。

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

When a wave meets a boundary, it can be reflected or transmitted. The law of reflection states that the angle of incidence equals the angle of reflection. Refraction occurs when a wave changes speed as it enters a different medium, causing the wavefronts to change direction. Diffraction is the spreading of waves around obstacles; the effect is most noticeable when the slit width is comparable to the wavelength.

波遇到边界时可能被反射或透射。反射定律指出入射角等于反射角。当波进入不同介质而速度改变时，会发生折射，导致波前方向改变。衍射是波绕过障碍物而扩展的现象；当狭缝宽度与波长相近时，衍射现象最明显。

• Reflection obeys θᵢ = θᵣ.
• 反射遵循 θᵢ = θᵣ。
• Refraction changes wavelength and speed, but frequency remains constant.
• 折射改变波长和波速，但频率保持不变。
• Diffraction is significant when aperture size ≈ λ.
• 当孔径大小 ≈ λ 时，衍射显著。
• These phenomena are common evidence for wave behaviour.
• 这些现象是波行为的常见证据。

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11. Doppler Effect for Travelling Waves | 行波的多普勒效应

The Doppler effect is the apparent change in frequency of a wave when the source or observer moves relative to the medium. For sound, a moving source compresses wavefronts ahead of it, increasing the observed frequency; behind the source, the wavefronts spread out, decreasing the observed frequency. The same principle applies to light and other electromagnetic waves.

多普勒效应是当波源或观察者相对介质运动时，观察到的波频率发生表观变化的现象。对于声波，运动波源前方的波前被压缩，观察到的频率升高；波源后方的波前被拉长，观察到的频率降低。同样的原理适用于光和其他电磁波。

f’ = f(v ± vₒ)/(v ∓ vₛ)

• Use the sign convention carefully: observer moving toward the source increases f’.
• 注意符号约定：观察者朝波源运动时 f’ 增大。
• Doppler effect is used in radar speed guns, medical ultrasound and astronomy.
• 多普勒效应应用于雷达测速枪、医学超声和天文学。
• For electromagnetic waves, the relativistic Doppler formula is needed for high speeds.
• 对于电磁波，高速时需要使用相对论多普勒公式。

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12. Summary | 总结

Travelling waves transfer energy without transferring matter. They are characterised by wavefronts, rays, wavelength, frequency, period, amplitude, and speed. Transverse and longitudinal waves obey the wave equation v = fλ. Phase difference describes how points on the wave relate to one another, while Huygens’ principle explains propagation, reflection, refraction and diffraction. The Doppler effect connects wave motion to relative motion between source and observer.

行波传递能量而不传递物质。它们通过波前、波线、波长、频率、周期、振幅和速度来表征。横波和纵波都遵循波速公式 v = fλ。相位差描述了波上各点之间的关系，而惠更斯原理解释了传播、反射、折射和衍射。多普勒效应将波动与波源和观察者之间的相对运动联系起来。

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