The Doppler Effect | 多普勒效应

📚 The Doppler Effect | 多普勒效应

The Doppler effect is the apparent change in frequency or wavelength of a wave caused by relative motion between the source and the observer. It is observed for any wave, including sound, water waves and electromagnetic radiation.

多普勒效应是指由于波源与观察者之间的相对运动，导致观察者接收到的波频率或波长发生表观变化的现象。任何波都可能出现多普勒效应，包括声波、水波和电磁辐射。

1. What is the Doppler Effect? | 什么是多普勒效应？

The effect is named after Christian Doppler, who described it in 1842. The source frequency fₛ does not change; only the frequency detected by the observer changes because wavefronts are compressed or stretched by the relative motion.

该效应以克里斯蒂安·多普勒的名字命名，他在 1842 年描述了这一现象。波源发出的频率 fₛ 本身不会改变；改变的是观察者检测到的频率，因为相对运动会压缩或拉伸波前。

A common example is an ambulance siren: as the ambulance approaches, the siren sounds higher in pitch; as it moves away, the pitch becomes lower. The same principle applies to light from stars and galaxies.

一个常见的例子是救护车警笛：当救护车靠近时，警笛音调听起来更高；当它远离时，音调变低。同样的原理也适用于恒星和星系发出的光。

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2. Wavefronts and Observed Wavelength | 波前与观测波长

When a source is stationary, wavefronts spread out evenly in all directions. The wavelength is the same in every direction because each wavefront is emitted from the same point.

当波源静止时，波前向各个方向均匀扩散。由于每个波前都是从同一点发出的，因此各个方向的波长都相同。

When the source moves, the wavefronts in front of the source are pressed closer together. The observed wavelength decreases. Behind the source, the wavefronts are stretched apart, so the observed wavelength increases.

当波源运动时，波源前方的波前被压得更近，观测到的波长减小；波源后方的波前被拉得更开，因此观测到的波长增大。

Since the wave speed v in a given medium is constant, a smaller wavelength means a higher observed frequency. This follows directly from the wave equation:

由于波在给定介质中的速度 v 是恒定的，波长越小，观测到的频率就越高。这直接来自波动方程：

v = fλ

Therefore, a compressed wavefront pattern gives a higher frequency, and a stretched pattern gives a lower frequency.

因此，波前被压缩会产生更高的频率，波前被拉伸则会产生更低的频率。

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3. Moving Source: Approaching the Observer | 波源靠近观察者

Consider a source moving towards a stationary observer with speed vₛ. In one period T = 1/fₛ, the source travels a distance vₛT. The wavefront emitted at the start of the period travels a distance vT.

考虑一个以速度 vₛ 靠近静止观察者的波源。在一个周期 T = 1/fₛ 内，波源移动的距离为 vₛT；周期开始时发出的波前传播的距离为 vT。

The distance between successive wavefronts ahead of the source is therefore reduced to vT − vₛT. The observed wavelength is:

因此，波源前方相邻波前之间的距离减少为 vT − vₛT。观测到的波长为：

λₒ = (v − vₛ) / fₛ

Using v = fₒλₒ, the observed frequency becomes:

利用 v = fₒλₒ，观测到的频率变为：

fₒ = fₛ v / (v − vₛ)

Because the denominator is smaller than v, the observed frequency is higher than the source frequency. This is why an approaching siren sounds higher pitched.

由于分母小于 v，观测到的频率高于波源频率。这就是靠近的警笛听起来音调更高的原因。

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4. Moving Source: Receding from the Observer | 波源远离观察者

When the source moves away from a stationary observer, the wavefronts behind the source are stretched. In one period, the distance between successive wavefronts increases to vT + vₛT.

当波源远离静止观察者时，波源后方的波前被拉伸。在一个周期内，相邻波前之间的距离增加到 vT + vₛT。

The observed wavelength is:

观测到的波长为：

λₒ = (v + vₛ) / fₛ

The observed frequency is therefore:

因此观测到的频率为：

fₒ = fₛ v / (v + vₛ)

Here the denominator is greater than v, so the observed frequency is lower than the source frequency. An ambulance siren sounds lower pitched as it recedes.

此时分母大于 v，因此观测到的频率低于波源频率。救护车警笛在远离时听起来音调较低。

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5. Moving Observer: Approaching the Source | 观察者靠近波源

If the source is stationary but the observer moves towards it with speed vₒ, the observer encounters wavefronts more rapidly. The relative speed between the observer and the wavefronts is v + vₒ.

如果波源静止，但观察者以速度 vₒ 向波源靠近，观察者会更快地遇到波前。观察者与波前之间的相对速度为 v + vₒ。

The wavelength in the medium is still λ = v / fₛ. The observed frequency is the relative speed divided by the wavelength:

介质中的波长仍为 λ = v / fₛ。观测频率等于相对速度除以波长：

fₒ = (v + vₒ) / λ = fₛ (v + vₒ) / v

The observed frequency is higher because the observer is moving into the wavefronts. The denominator remains v, while the numerator increases.

由于观察者迎着波前运动，观测频率会升高。分母保持为 v，而分子增大。

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6. Moving Observer: Receding from the Source | 观察者远离波源

When the observer moves away from a stationary source, the relative speed between the observer and the wavefronts is v − vₒ. The observer takes longer to receive successive wavefronts.

当观察者远离静止波源时，观察者与波前之间的相对速度为 v − vₒ。观察者接收相邻波前所需的时间更长。

The observed frequency is:

观测到的频率为：

fₒ = (v − vₒ) / λ = fₛ (v − vₒ) / v

This gives a lower observed frequency because the observer is moving with the wave direction, effectively reducing the rate at which wavefronts arrive.

这会产生较低的观测频率，因为观察者沿波的传播方向运动，有效降低了波前到达的速率。

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7. Sign Convention and Key Equations | 符号规则与关键公式

For a stationary observer and a moving source, the formula is fₒ = fₛ v / (v ± vₛ). Use the minus sign when the source moves towards the observer, and the plus sign when it moves away.

对于静止观察者和运动波源，公式为 fₒ = fₛ v / (v ± vₛ)。当波源靠近观察者时用减号，远离时用加号。

For a stationary source and a moving observer, the formula is fₒ = fₛ (v ± vₒ) / v. Use the plus sign when the observer moves towards the source, and the minus sign when moving away.

对于静止波源和运动观察者，公式为 fₒ = fₛ (v ± vₒ) / v。当观察者靠近波源时用加号，远离时用减号。

Condition | 条件	Formula | 公式	Result | 结果
Source moving towards stationary observer | 波源靠近静止观察者	fₒ = fₛ v / (v − vₛ)	Higher frequency | 频率升高
Source moving away from stationary observer | 波源远离静止观察者	fₒ = fₛ v / (v + vₛ)	Lower frequency | 频率降低
Observer moving towards stationary source | 观察者靠近静止波源	fₒ = fₛ (v + vₒ) / v	Higher frequency | 频率升高
Observer moving away from stationary source | 观察者远离静止波源	fₒ = fₛ (v − vₒ) / v	Lower frequency | 频率降低

If both source and observer move, the combined form can be written as fₒ = fₛ (v ± vₒ) / (v ∓ vₛ). Use the upper signs when they move towards each other, and the lower signs when they move apart.

如果波源和观察者都在运动，组合形式可以写成 fₒ = fₛ (v ± vₒ) / (v ∓ vₛ)。当二者相互靠近时取上面的符号，彼此远离时取下面的符号。

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8. Electromagnetic Waves and Redshift | 电磁波与红移

For light and other electromagnetic waves, the Doppler effect causes a shift in observed wavelength. When a galaxy or star is moving away from Earth, its spectral lines are shifted towards longer wavelengths. This is called redshift.

对于光和其他电磁波，多普勒效应会引起观测波长的移动。当星系或恒星远离地球运动时，其谱线会向更长波长方向移动，这称为红移。

If the source is moving towards Earth, the wavelengths become shorter and the light is blueshifted. The fractional change in wavelength is given approximately by z = Δλ / λ ≈ v / c for speeds much smaller than the speed of light c.

如果波源朝向地球运动，波长会变短，光发生蓝移。当速度远小于光速 c 时，波长的相对变化近似为 z = Δλ / λ ≈ v / c。

Redshift measurements are used to determine the recessional speeds of galaxies and provide evidence for the expansion of the Universe.

红移测量用于确定星系的退行速度，并为宇宙膨胀提供证据。

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9. Common Applications | 常见应用

The Doppler effect has many practical uses. Radar speed guns emit microwaves and detect the frequency shift when the waves reflect from a moving vehicle. The speed of the vehicle can then be calculated.

多普勒效应有许多实际应用。雷达测速仪发射微波，并检测微波从运动车辆反射后的频率变化，从而计算出车辆的速度。

In medicine, Doppler ultrasound measures the speed of blood flow by detecting the frequency shift of ultrasound waves reflected from red blood cells.

在医学中，多普勒超声通过检测红细胞反射的超声波频率变化来测量血流速度。

Astronomers use the Doppler effect to measure the radial velocity of stars and to detect planets orbiting distant stars through tiny periodic shifts in the star’s spectral lines.

天文学家利用多普勒效应测量恒星的视向速度，并通过恒星谱线的微小周期性移动来探测围绕遥远恒星运行的行星。

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10. Worked Example and Exam Tips | 例题与应试技巧

Example: A police car siren emits sound at a frequency of 800 Hz. The car travels at 30 m/s towards a stationary observer. The speed of sound in air is 340 m/s. Calculate the observed frequency.

例题：一辆警车的警笛发出频率为 800 Hz 的声音。警车以 30 m/s 的速度驶向静止的观察者。空气中的声速为 340 m/s。计算观察者听到的频率。

Since the source is moving towards the observer, use fₒ = fₛ v / (v − vₛ).

由于波源朝向观察者运动，使用 fₒ = fₛ v / (v − vₛ)。

fₒ = 800 × 340 / (340 − 30) = 800 × 340 / 310 ≈ 877 Hz

The observed frequency is about 877 Hz, which is higher than 800 Hz. If the car moved away, the denominator would be 340 + 30, giving about 735 Hz.

观测到的频率约为 877 Hz，高于 800 Hz。如果警车远离，分母将是 340 + 30，结果约为 735 Hz。

Exam tips: identify which object is moving and whether the motion is towards or away. Always check whether the frequency should increase or decrease. Use the sign that gives the physically correct result.

应试技巧：先判断哪个物体在运动，以及运动方向是靠近还是远离。始终检查频率应升高还是降低，选择能给出正确物理结果的符号。

Remember that v is the speed of the wave in the medium, not the speed of the source or observer. Also ensure all speeds are in the same units before substituting into the formula.

记住 v 是波在介质中的传播速度，而不是波源或观察者的速度。同时确保代入公式前所有速度的单位一致。

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