📚 C.5 Doppler Effect | IB物理C.5 多普勒效应
The Doppler effect describes the change in observed frequency of a wave when there is relative motion between the source and the observer. In IB Physics, this phenomenon appears both in sound and in electromagnetic radiation, with applications ranging from medical ultrasound to the expansion of the Universe.
多普勒效应描述了当波源与观察者之间存在相对运动时，观测到的频率发生变化的现象。在IB物理中，这一现象既出现在声波中，也出现在电磁辐射中，其应用范围从医学超声波一直延伸到宇宙的膨胀。
1. What is the Doppler Effect? | 什么是多普勒效应？
The Doppler effect is the apparent shift in frequency (and wavelength) of a wave caused by the relative motion between the source and the observer. When the source and observer move towards each other, the observed frequency is higher than the emitted frequency; when they move apart, the observed frequency is lower.
多普勒效应是由波源与观察者之间的相对运动引起的波频率（和波长）的表观移动。当波源和观察者相互靠近时，观测到的频率高于发射频率；当它们相互远离时，观测到的频率则降低。
The effect is commonly experienced with sound waves – for example, the siren of an ambulance sounds higher pitched as it approaches and lower pitched as it moves away. For light, relative motion causes a shift towards the blue or red end of the spectrum.
这一效应在声波中普遍存在——例如，救护车警报声在接近时听起来音调更高，在远离时音调更低。对于光波，相对运动会引起光谱向蓝端或红端的移动。
2. Doppler Effect for Sound: Moving Source | 声源移动时的多普勒效应
towards the observer, the wavefronts bunch up, shortening the observed wavelength. The observed frequency is then given by f' = f × v / (v − uₛ).
考虑一个静止的观察者和一个相对于介质以速度uₛ运动的声源。介质中的波速为v。当声源朝向观察者移动时，波阵面被压缩，观测到的波长变短。观测频率为 f' = f × v / (v − uₛ)。
e source moves away from the observer, the wavefronts stretch, and the frequency becomes f' / (v + uₛ). Notice that the denominator changes sign depending on the direction of motion.
如果声源远离观察者，波阵面被拉伸，频率变为 f' = f × v / (v + uₛ)。注意分母根据运动方向改变符号。
3. Doppler Effect for Sound: Moving Observer | 观察者移动时的多普勒效应
urce, they encounter more wavefronts per unit time, so the observed frequency increases: f' = f × (v + uₒ) / v.
现在考虑一个静止的波源和一个以速度uₒ运动的观察者。当观察者朝向波源移动时，单位时间内遇到的波阵面更多，因此观测频率增加：f' = f × (v + uₒ) / v。
When the observer moves away from the source, the relative speed reduces, giving f' = f × (v − uₒ) / v. Here the numerator changes.
当观察者远离波源时，相对速度减小，得到 f' = f × (v − uₒ) / v。这里分子发生变化。
4. General Equations for Sound Waves | 声波的一般多普勒公式
The general Doppler shift for sound waves, where both source and observer may move relative to the medium, combines both effects. The formula is often written as:
对于声波，当波源和观察者都可能相对于介质运动时，一般多普勒频移公式结合了这两种效应。公式通常写为：
f' = f × (v ± uₒ) / (v ∓ uₛ)
The upper signs (+ in numerator, − in denominator) are used when the source and observer move towards each other. The lower signs apply when they move apart. Here f is the emitted frequency, f' the observed frequency, v the wave speed, uₛ the speed of the source, and uₒ the speed of the observer – all relative to the medium.
上方符号（分子取+，分母取−）用于波源和观察者相互靠近的情况。下方符号用于它们相互远离的情况。式中f为发射频率，f'为观测频率，v为波速，uₛ为波源速度，uₒ为观察者速度——所有速度均相对于介质而言。
e all positive magnitudes. IB examination questions often require careful selection of the ‘±’ or ‘∓’.
解题时，首先要判断是波源还是观察者在运动，选择合适的符号惯例，并确保速度都是正的数值。IB考试题目常常要求谨慎选择‘±’或‘∓’。
| Scenario / 情景 | Towards / 靠近 | Away / 远离 |
|---|---|---|
| Moving source, stationary observer / 移动波源，静止观察者 | f' = f v/(v − uₛ) | f' = f v/(v + uₛ) |
| Moving observer, stationary source / 移动观察者，静止波源 | f' = f (v + uₒ)/v | f' = f (v − uₒ)/v |
| Both moving / 两者都运动 | f' = f (v ± uₒ)/(v ∓ uₛ) | Use upper or lower signs consistently / 一致使用上方或下方符号 |
5. Electromagnetic Waves and the Doppler Effect | 电磁波与多普勒效应
e a medium, so the Doppler effect for light differs from that for sound. The shift depends only on the relative velocity between source and observer. For speeds much less than the speed of light, the observed frequency f' is approximated by f' the relative velocity of recession (negative if approaching) and c is the speed of light.
电磁波不需要介质，因此光的多普勒效应与声波不同。频移仅取决于波源与观察者之间的相对速度。当速度远小于光速时，观测频率f'近似为 f' ≈ f (1 − v/c)，其中v为退行相对速度（若为接近则取负值），c为光速。
In terms of wavelength, the shift Δλ = λ' − λ is often expressed by the redshift parameter z = Δλ/λ ≈ v/c for v ≪ ive z indicates a redshift (source moving away), while a negative z indicates a blueshift.
在波长方面，移动量 Δλ = λ' − λ 通常用红移参数z表示： z = Δλ/λ ≈ v/c （v ≪ c）。正z值表示红移（波源远离），负z值表示蓝移。
6. Redshift and Blueshift in Astronomy | 天文学中的红移与蓝移
When a star or galaxy moves away from Earth, its spectral lines are shifted towards longer (redder) wavelengths – a redshift. If the object moves towards Earth, the shift is towards shorter (bluer) wavelengths – a blueshift. By measuring z, astronomers can calculate the line‑of‑sight velocity of a celestial object.
当恒星或星系远离地球运动时，其谱线向更长（更红）的波长移动——即红移。如果天体朝向地球运动，则向更短（更蓝）的波长移动——即蓝移。通过测量z，天文学家可以计算出天体的视向速度。
relativistic. This simple linear relationship is a cornerstone of observational cosmology.
对于已知静止波长λ₀，利用观测波长λ可求得z = (λ − λ₀)/λ₀，然后在速度非相对论性的情况下，退行速度 v ≈ z c。这一简单的线性关系是观测宇宙学的基石。
7. Hubble's Law and Cosmic Expansion | 哈勃定律与宇宙膨胀
e Hubble constant. This provided the first direct evidence that the Universe is expanding.
埃德温·哈勃发现，星系的红移随其与我们的距离而增大。哈勃定律指出，星系的退行速度v与其距离d成正比：v = H₀ d，其中H₀为哈勃常数。这为宇宙正在膨胀提供了首个直接证据。
In an expanding Universe, the cosmological redshift is not due to motion through space in the usual Doppler sense, but rather to the stretching of space itself. However, for relatively nearby galaxies, the simple Doppler interpretation works well.
在膨胀的宇宙中，宇宙学红移并非源于通常多普勒意义上的空间运动，而是源自空间本身的拉伸。不过，对于相对邻近的星系，简单的多普勒解释仍然适用。
8. Relativistic Doppler Effect for Light | 光的相对论多普勒效应
When relative speeds approach the speed of light, the relativistic Doppler formula must be used. For a source and observer receding from each other along the line of sight, the observed frequency is:
当相对速度接近光速时，必须使用相对论多普勒公式。对于沿视线相互远离的波源和观察者，观测频率为：
f' = f √((c − v) / (c + v))
ecession. If the source and observer approach each other, the signs are reversed, giving f' = f √((c + v) / (c − v)), which results in a blueshift.
式中v是相对退行速度。如果波源和观察者相互靠近，符号要互换，即 f' = f √((c + v) / (c − v))，这会导致蓝移。
It is often convenient to define β = v/c, so these expressions become f' = f √((1 − β)/(1 + β)) for recession and f' , a purely relativistic consequence when the motion is perpendicular to the line of sight.
通常定义 β = v/c，于是退行情形的公式变为 f' = f √((1 − β)/(1 + β))，靠近情形变为 f' = f √((1 + β)/(1 − β))。相对论多普勒效应还预言了横向多普勒频移，这是运动方向与视线垂直时纯粹的相对论效应。
9. Applications: Radar Speed Traps | 应用：雷达测速
of the vehicle is determined from the beat frequency Δf, usually via v = (c × Δf) / (2 f).
警用雷达测速仪发射微波脉冲，并测量从行驶车辆反射回来的波的频率变化。由于车辆先充当移动观察者、再充当移动波源，等效的多普勒频移被加倍。车辆速度v根据拍频Δf确定，通常通过 v = (c × Δf) / (2 f) 计算。
This principle is also used in sports radar guns, traffic monitoring, and even satellite tracking. The relativistic correction is negligible at ordinary vehicle speeds but becomes important for aircraft and spacecraft.
这一原理也用于体育雷达测速枪、交通监测甚至卫星跟踪。在普通的车辆速度下，相对论修正可以忽略不计，但对于飞机和航天器则变得重要。
10. Medical Ultrasound Doppler | 医学超声多普勒
Doppler ultrasound is used to measure blood flow velocity. High‑frequency sound waves (typically 2–15 MHz) are transmitted into the body and reflected by moving red blood cells. The frequency shift Δf depends on the blood velocity v, the emitted frequency f, the speed of sound in tissue c, and the angle θ between the ultrasound beam and the blood flow.
多普勒超声被用来测量血流速度。高频声波（通常为2–15 MHz）被射入体内，并由流动的红细胞反射。频率变化Δf取决于血流速度v、发射频率f、组织中的声速c以及超声波束与血流方向之间的夹角θ。
rs because the ultrasound makes a round trip. This quantitative information helps diagnose narrowing of blood vessels and valve dysfunction.
医学超声的多普勒频移方程通常表示为：Δf = (2 f v cosθ) / c。因子2的出现是由于超声波经历了往返路程。这种定量信息有助于诊断血管狭窄和瓣膜功能异常。
11. Doppler Broadening of Spectral Lines | 谱线的多普勒展宽
In a hot gas, atoms and molecules move with a distribution of speeds. The random thermal motions result in a spread of Doppler shifts along the line of sight, causing spectral lines to broaden. This is known as Doppler broadening. The width of a line provides a measurement of the temperature of the emitting gas.
在高温气体中，原子和分子以一定的速度分布运动。随机的热运动导致沿视线方向的多普勒频移有一个分布，从而使谱线变宽，这被称为多普勒展宽。谱线的宽度可以用来测量发射气体的温度。
For a gas at temperature T, the root‑mean‑square velocity of particles leads to a relation between the line width Δλ and T. In fusion plasma diagnostics and astrophysics, Doppler broadening is a key non‑invasive temperature probe.
对于温度为T的气体，粒子的方均根速度导致谱线宽度Δλ与T之间存在一定关系。在聚变等离子体诊断和天体物理学中，多普勒展宽是一种关键的非侵入式温度探测手段。
12. Summary and Key Points | 总结与关键要点
The Doppler effect unifies phenomena from everyday sound to cosmological observations. Remember that for sound, the change depends on whether the source or the observer is moving relative to the medium, whereas for light only relative velocity matters. Memorise the key formulas, know the sign conventions, and be able to convert frequency shifts into velocities and vice versa.
多普勒效应将日常声音现象与宇宙学观测统一了起来。要记住，对声音而言，频移取决于波源或观察者是否相对于介质运动，而对光来说仅相对速度起作用。记住关键公式，理解符号惯例，并能够将频率移动转换为速度，反之亦然。
- Sound Doppler shift general formula / 声波多普勒频移一般公式： f' = f (v ± uₒ)/(v ∓ uₛ)
- Low‑speed EM Doppler / 低速电磁波多普勒： z = Δλ/λ ≈ v/c
- Relativistic Doppler for recession / 相对论退行多普勒： f' = f √((1 − β)/(1 + β)) with β = v/c
- Hubble’s law / 哈勃定律： v = H₀ d
- Medical ultrasound / 医学超声： Δf = (2 f v cosθ)/c
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