The Doppler Effect | 多普勒效应

📚 The Doppler Effect | 多普勒效应

The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave. It is a fundamental concept in physics with significant applications in astronomy, radar technology, and medical imaging. In the context of IB Mathematics, the Doppler effect provides a rich context for mathematical modelling, functional analysis, and the study of rational functions and their derivatives.

多普勒效应描述了当观察者与波源发生相对运动时，波动频率的变化。这是物理学的一个基本概念，在天文学、雷达技术和医学成像等领域有重要应用。在 IB 数学的范畴内，多普勒效应为数学建模、函数分析以及有理函数及其导数的研究提供了丰富的背景。

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1. Introduction to the Doppler Effect | 多普勒效应简介

The perceived pitch of a siren changes as an ambulance passes by: it sounds higher when approaching and lower when receding. This everyday phenomenon is the Doppler effect. Mathematically, we can relate the observed frequency f' to the source frequency f, the speed of sound v, and the velocities of the source vₛ and the observer v₀.

当救护车驶过时，警笛的音调会发生变化：靠近时音调较高，远离时音调较低。这一日常现象就是多普勒效应。从数学上，我们可以将观察到的频率 f' 与声源频率 f、声速 v、声源速度 vₛ 和观察者速度 v₀ 联系起来。

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2. The Classical Doppler Formula for Sound | 声波经典多普勒公式

For sound waves in a stationary medium, the general observed frequency when both the source and the observer are moving along the line joining them is given by a rational expression:

对于在静止介质中的声波，当声源和观察者沿着两者连线运动时，一般观察频率由一个有理表达式给出：

f' = f × (v ± v₀) / (v ∓ vₛ)

Here v is the speed of sound in the medium (typically 340 m·s⁻¹ in air), v₀ is the speed of the observer, and vₛ is the speed of the source. The upper signs in the numerator and lower signs in the denominator correspond to motion that increases the observed frequency (i.e., moving towards each other). The opposite choice reduces the frequency.

其中 v 是介质中的声速（空气中通常为 340 m·s⁻¹），v₀ 是观察者的速度，vₛ 是声源的速度。分子取上符号且分母取下符号对应使观察频率增大的运动（即相向运动）。相反的符号组合则使频率降低。

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3. Derivation for Moving Source | 声源运动的推导

source moves a distance vₛ T, so the wavelength in front of the source is compressed to λ' vₛ T = (v − vₛ)/f. The wave speed in the medium remains v, so the observed frequency is f' = v/λ' = f × v/(v − vₛ).

考虑静止观察者和以速度 vₛ 向观察者运动的声源。在一个周期 T = 1/f 内，声源移动了 vₛ T 的距离，因此声源前方的波长被压缩为 λ' = vT − vₛ T = (v − vₛ)/f。介质中的波速仍为 v，因此观察到的频率为 f' = v/λ' = f × v/(v − vₛ)。

If the source moves away, the wavelength is stretched: λ' = (v + vₛ)/f, giving f' to the source moving at the speed of sound, a non-linear regime where a shock wave forms.

若声源远离，波长被拉伸：λ' = (v + vₛ)/f，得到 f' = f × v/(v + vₛ)。这个关于 vₛ 的有理函数在 vₛ = v 处有一条垂直渐近线，对应声源以声速运动的情况，此时会形成冲击波，属于非线性范围。

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4. Derivation for Moving Observer | 观察者运动的推导

ving observer the effective wave speed becomes v + v₀. The wavelength remains unchanged, so the frequency detected is f' = (v + v₀)/λ = f × (v + v₀)/v.

现在让声源静止，观察者以速度 v₀ 向声源运动。波峰仍以速度 v 传播，但相对于运动的观察者，有效波速变为 v + v₀。波长保持不变，因此检测到的频率为 f' = (v + v₀)/λ = f × (v + v₀)/v。

If the observer moves away, the relative speed is v − v₀, so f' onal forms are different: for the moving source the denominator is modified, while for the moving observer the numerator changes. This asymmetry disappears when both motions are small compared to v.

若观察者远离，相对速度为 v − v₀，因此 f' = f × (v − v₀)/v。注意函数形式有所不同：对运动声源，分母被修正；而对运动观察者，分子发生改变。当两种运动速度远小于 v 时，这种不对称性就会消失。

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5. Combined Formula and Sign Conventions | 合并公式与符号约定

Combining both effects yields the general formula shown in Section 2. It is essential to adopt a consistent sign convention, typically taking the direction from source to observer as positive. The table below summarises the choice of signs.

将两种效应合并就得到了第 2 节中的一般公式。采用一致的符号约定至关重要，通常将声源指向观察者的方向取为正。下表总结了符号的选择。

Relative motion	Numerator (v ± v₀)	Denominator (v ∓ vₛ)
Observer towards source (or source towards observer)	v + v₀	v − vₛ
Observer away from source (or source away from observer)	v − v₀	v + vₛ

相对运动与符号选择：当观察者靠近声源（或声源靠近观察者）时，使用 f' = f (v + v₀)/(v − vₛ)；当观察者远离声源（或声源远离观察者）时，使用 f' = f (v − v₀)/(v + vₛ)。

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6. Mathematical Properties of the Doppler Curve | 多普勒曲线的数学性质

Treating f' as a function of the source speed vₛ for a stationary observer, f' udy its domain, range and asymptotic behaviour. The domain excludes vₛ = v. For vₛ → v⁻, f' → ∞; for vₛ → v⁺, f' → −∞ (physically irrelevant). The derivative is df' for vₛ < v, indicating an increasing frequency shift as the source approaches sonic speed.

将观察者静止时的频率 f' 视为声源速度 vₛ 的函数，f'(vₛ) = f v/(v − vₛ)，我们可以研究其定义域、值域和渐近行为。定义域不包括 vₛ = v。当 vₛ → v⁻ 时，f' → ∞；当 vₛ → v⁺ 时，f' → −∞（物理上无意义）。导数为 df'/dvₛ = f v/(v − vₛ)²，在 vₛ < v 时始终为正，表明当声源接近声速时频移量增加得越来越快。

For the moving observer case, f'(v₀) = f (v + v₀)/v is a simple linear function with a constant rate of change df' y the Doppler effect for sound is not symmetric between moving source and moving observer.

对于运动的观察者，f'(v₀) = f (v + v₀)/v 是一个简单的线性函数，导数为常数 df'/dv₀ = f/v。曲率的差异凸显了为什么声波的多普勒效应在运动声源和运动观察者之间是不对称的。

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7. Application: Frequency Shift and Derivative Analysis | 应用：频率偏移与导数分析

perceived pitch to a small change in speed. For a source moving towards an observer at 10% of the speed of sound (vₛ = 0.1v), the instantaneous rate of change is df' = f v/(v − 0.1v)² = f v/(0.81 v²) = f/(0.81 v). If f = 1000 Hz and v = 340 m·s⁻¹, then df'/dvₛ ≈ (1000)/(0.81×340) ≈ 3.63 Hz per m·s⁻¹.

观察频率的导数可用于估算感知音调对速度微小变化的敏感程度。对于一个以声速 10% 的速度向观察者运动的声源（vₛ = 0.1v），瞬时变化率为 df'/dvₛ = f v/(v − 0.1v)² = f v/(0.81 v²) = f/(0.81 v)。若 f = 1000 Hz，v = 340 m·s⁻¹，则 df'/dvₛ ≈ (1000)/(0.81×340) ≈ 3.63 Hz/(m·s⁻¹)。

y 1 m·s⁻¹ increase in source speed raises the observed frequency by about 3.6 Hz. Such calculations are useful in designing speed detection devices and in understanding the precision required in experimental measurements.

这表明在 vₛ = 0.1v 附近，声源速度每增加 1 m·s⁻¹，观察到的频率上升约 3.6 Hz。这类计算在速度检测设备设计和理解实验测量精度要求方面非常有用。

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8. The Doppler Effect for Electromagnetic Waves | 电磁波的多普勒效应

Electromagnetic waves do not require a medium, so the classical formula cannot be applied directly. Instead, the relativistic Doppler effect must be used, which depends only on the relative velocity between source and observer. For motion along the line of sight, the observed frequency is

电磁波不需要介质，因此经典公式不能直接应用。取而代之的是相对论性多普勒效应，它只依赖于源与观察者之间的相对速度。对于沿视线方向的运动，观察频率为

f' = f √[(1 + β)/(1 − β)]

. The reciprocal square root arises from the Lorentz transformation of the wave four-vector.

其中 β = v/c，v 为相对速度（靠近时取正），c 为光速（≈ 3.00 × 10⁸ m·s⁻¹）。倒数的平方根源于波四维矢量的洛伦兹变换。

When the source recedes, the sign in the denominator changes, giving f' ess than f. This formula is fundamental in cosmology for interpreting redshifts of distant galaxies.

当源远离时，分母中的符号改变，得到 f' = f √[(1 − β)/(1 + β)]，该值始终小于 f。这一公式在天体物理中对于解释遥远星系的红移至关重要。

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9. Approximation at Low Speeds (Binomial Expansion) | 低速近似（二项式展开）

For β ≪ nsion. Writing √[(1+β)/(1−β)] = (1+β)½ (1−β)⁻½ and expanding each factor up to O(β²) gives

当 β ≪ 1 时，相对论性多普勒因子可用二项式展开近似。将 √[(1+β)/(1−β)] = (1+β)½ (1−β)⁻½ 并对每个因子展开至 O(β²)，得到

f' ≈ f (1 + β + ½ β² + …)

The linear term β yields f' classical formula for small speeds. The quadratic term becomes significant when v is a notable fraction of c, and it is responsible for the transverse Doppler effect that has no classical analogue.

线性项 β 给出 f' ≈ f (1 + v/c)，这与低速下的经典公式一致。当 v 占 c 相当比例时，二次项变得重要，并且它导致了在经典物理中没有对应项的横向多普勒效应。

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10. Redshift and Cosmological Considerations | 红移与宇宙学考量

In astronomy, the redshift z is defined as the relative change in wavelength: z = (λ' − λ)/λ. For a receding source, λ' ivistic Doppler formula and the relation c = f λ, we can express z directly in terms of β:

在天文学中，红移 z 定义为波长的相对变化：z = (λ' − λ)/λ。对于退行的源，λ' > λ，因此 z > 0。利用相对论多普勒公式以及 c = f λ，我们可以直接用 β 表达 z：

1 + z = √[(1 + β)/(1 − β)]

ft to distance. These mathematical manipulations bridge observational data with the expansion of the universe.

解出 β 得到 β = [(1+z)² − 1]/[(1+z)² + 1]。对于小的 z，β ≈ z，而线性哈勃定律 v = H₀ d 将红移与距离联系起来。这些数学处理将观测数据与宇宙膨胀联系了起来。

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11. Practical Examples and IB-style Questions | 实例与 IB 风格题目

Example 1: ches a stationary observer at 30 m·s⁻¹. Find the observed frequency, taking v = 340 m·s⁻¹.

例 1： 一列火车的汽笛发出 500 Hz 的频率。它以 30 m·s⁻¹ 的速度驶向静止的观察者。假设声速 v = 340 m·s⁻¹，求观察频率。

Solution: Since only the source moves towards the observer, we use f' = f v/(v − vₛ) = 500 × 340/(340 − 30) = 500 × 340/310 ≈ 548.4 Hz.

解：由于只有声源朝向观察者运动，使用 f' = f v/(v − vₛ) = 500 × 340/(340 − 30) = 500 × 340/310 ≈ 548.4 Hz。

Example 2: its light of frequency 6.0 × 10¹⁴ Hz. Calculate the observed frequency and the redshift z.

例 2： 一颗以 0.05c 退行的恒星发出频率为 6.0 × 10¹⁴ Hz 的光。计算观察频率和红移 z。

Solution: β = 0.05, receding so use f' = f √[(1 − β)/(1 + β)] = 6.0 × 10¹⁴ × √0.95/1.05 ≈ 5.71 × 10¹⁴ Hz. The redshift z = Δλ/λ = f/f' 1 ≈ 0.050. Note that at this speed, the linear approximation z ≈ β is already very close.

解：β = 0.05，退行，故使用 f' = f √[(1 − β)/(1 + β)] = 6.0 × 10¹⁴ × √0.95/1.05 ≈ 5.71 × 10¹⁴ Hz。红移 z = Δλ/λ = f/f' − 1 ≈ 0.050。注意在此速度下，线性近似 z ≈ β 已经非常接近。

These examples illustrate the importance of selecting the correct formula based on the wave type and the state of motion. IB exam questions often require setting up the rational or radical function, finding intercepts with axes, asymptotes, or using differentiation to locate maximum rates of change.

这些例子说明了根据波的类型和运动状态选择正确公式的重要性。IB 考试题目通常要求建立有理函数或根式函数，寻找与坐标轴的交点、渐近线，或利用微分找出最大变化率的位置。

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12. Summary and Key Takeaways | 总结与要点回顾

limits, derivatives, binomial expansions, and radical expressions. The classical formula f' = f (v ± v₀)/(v ∓ vₛ) models sound waves, while the relativistic form f' = f √[(1 ± β)/(1 ∓ ole of differentiation in analysing frequency sensitivity. Mastery of these mathematical models strengthens analytical skills relevant to the IB Mathematics syllabus.

多普勒效应将波动物理与多种数学主题联系在一起：有理函数、极限、导数、二项式展开和根式表达式。经典公式 f' = f (v ± v₀)/(v ∓ vₛ) 用于声波模型，而相对论形式 f' = f √[(1 ± β)/(1 ∓ β)] 则支配光的行为。核心要点包括符号约定的重要性、声波中声源与观察者运动的不对称性，以及微分在分析频率灵敏度中的作用。掌握这些数学模型，有助于强化与 IB 数学大纲相关的分析技能。

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