📚 5 The Doppler effect | 多普勒效应
The Doppler effect is the change in observed frequency of a wave when there is relative motion between the source and the observer. Named after Christian Doppler, this phenomenon is commonly experienced with sound waves, such as the changing pitch of a siren as an ambulance passes by. It is a wave property that applies to all types of waves, including sound, light, and water waves, as long as there is relative motion along the line connecting the source and the observer.
多普勒效应是指当波源与观察者之间存在相对运动时，观察到的波的频率发生变化的现象。这一现象由克里斯蒂安·多普勒命名，常见于声波中，例如救护车驶过时警笛音调的变化。多普勒效应是所有波共有的特性，只要波源与观察者之间有沿连线方向的相对运动，无论声波、光波还是水波都会发生。
1. Moving Source with a Stationary Observer | 波源运动、观察者静止
When the source moves towards a stationary observer, the wavefronts ahead of the source are compressed, resulting in a shorter wavelength and a higher observed frequency. Conversely, if the source moves away, the wavefronts are stretched, leading to a longer wavelength and a lower observed frequency. The medium itself is assumed to be stationary.
当波源朝向静止的观察者运动时，波源前方的波阵面被压缩，波长变短，观察到的频率升高。反之，如果波源远离，波阵面被拉伸，波长变长，观察到的频率降低。这里假设介质本身是静止的。
The observed frequency f' can be expressed as:
观察到的频率 f' 可表示为：
f' = f₀ × v / (v ∓ uₛ)
where f₀ is the source frequency, v is the speed of sound in the medium, uₛ is the speed of the source, the minus sign (−) is used when the source moves towards the observer, and the plus sign (+) when the source moves away.
其中 f₀ 为波源频率，v 为介质中的声速，uₛ 为波源速度，减号 (−) 用于波源朝向观察者运动，加号 (+) 用于波源远离观察者运动。
2. Moving Observer with a Stationary Source | 观察者运动、波源静止
If the observer moves towards a stationary source, the relative speed at which wavefronts are intercepted increases, causing a higher observed frequency. If the observer moves away, fewer wavefronts are encountered per unit time, and the observed frequency drops. The wavelength in the medium remains unchanged because the source is at rest.
若观察者朝向静止波源运动，拦截波阵面的相对速率增大，导致观察到的频率升高。若观察者远离，单位时间内遇到的波阵面减少，频率下降。由于波源静止，介质中的波长保持不变。
The relationship is given by:
其关系式为：
f' = f₀ × (v ± uₒ) / v
where uₒ is the speed of the observer. The plus sign (+) applies when the observer moves towards the source, and the minus sign (−) when the observer moves away.
其中 uₒ 为观察者速度，加号 (+) 用于观察者朝向波源运动，减号 (−) 用于观察者远离波源运动。
3. General Doppler Equation for Sound | 声音多普勒效应的一般方程
When both the source and the observer move relative to the medium, the two effects combine into a single general formula:
当波源和观察者都相对于介质运动时，两种效应合并为一个通用公式：
f' = f₀ (v ± uₒ) / (v ∓ uₛ)
The sign convention is crucial:
符号规则至关重要：
- Numerator (v ± uₒ): use '+' when the observer moves towards the source, and '−' when away.
- 分子 (v ± uₒ)：观察者朝向波源运动用 '+'，远离用 '−'。
- Denominator (v ∓ uₛ): use '−' when the source moves towards the observer, and '+' when away.
- 分母 (v ∓ uₛ)：波源朝向观察者运动用 '−'，远离用 '+'。
Note that the signs in the numerator and denominator are independent. Always identify the direction of each motion relative to the line joining source and observer.
注意分子和分母中的符号是独立的。务必根据波源和观察者连线方向判断各自的运动方向。
4. Worked Example for Sound | 声音多普勒效应例题
A car horn emits a steady frequency of 500 Hz. The car travels at 25 m/s directly towards a stationary listener. The speed of sound in air is 340 m/s. Calculate the frequency heard by the listener.
一辆汽车喇叭发出稳定的 500 Hz 频率。汽车以 25 m/s 的速度径直驶向静止的听者。空气中的声速为 340 m/s。计算听者听到的频率。
f' = f₀ × v / (v − uₛ) = 500 × 340 / (340 − 25) ≈ 540 Hz
Since the source is moving towards the observer, we use the minus sign in the denominator. The observed frequency is higher than the emitted frequency, which matches the experience of a rising pitch as a vehicle approaches.
因为波源朝向观察者运动，分母取减号。观察到的频率高于发射频率，这与车辆靠近时音调升高的体验一致。
5. Applications of the Doppler Effect in Sound | 多普勒效应在声波中的应用
The Doppler effect for sound has many practical uses. Police radar guns bounce microwaves off moving vehicles, but ultrasonic Doppler devices measure blood flow by detecting the frequency shift of ultrasound reflected from red blood cells. Bats and dolphins use echolocation based on Doppler shifts to navigate and hunt prey. Weather radar also exploits the Doppler shift to measure wind speeds within storms.
声波的多普勒效应有许多实际用途。警用测速雷达向移动车辆发射微波，而医用超声多普勒设备通过检测红细胞反射的超声频移来测量血流速度。蝙蝠和海豚利用基于多普勒频移的回声定位进行导航和捕食。气象雷达同样利用多普勒频移测量风暴中的风速。
In all these cases, the shift in frequency encodes the relative speed between the instrument and the target. The technique is non-invasive and provides real-time velocity data.
在所有这些情景中，频率的变化编码了仪器与目标之间的相对速度。这种技术无需侵入即可提供实时速度数据。
6. Doppler Effect for Light: Redshift and Blueshift | 光的多普勒效应：红移与蓝移
Light waves also exhibit the Doppler effect, but because light does not require a medium, the shift depends only on the relative velocity between source and observer. For speeds v much less than the speed of light c, the fractional change in wavelength is approximately:
光波同样表现出多普勒效应，但由于光不需要介质，频移仅取决于波源与观察者之间的相对速度。当速度 v 远小于光速 c 时，波长的相对变化近似为：
Δλ / λ₀ ≈ v / c
− λ₀. If the source moves away from the observer, Δλ is positive, resulting in a shift towards longer wavelengths called redshift. If it moves towards, Δλ is negative, producing a blueshift.
其中 λ₀ 是波源发射的原始波长，Δλ = 观察波长 − λ₀。如果波源远离观察者，Δλ 为正，波长变长，称为红移。如果波源靠近，Δλ 为负，波长变短，称为蓝移。
At relativistic speeds, a more accurate formula is used: fobs = fsrc . However, for most astrophysical observations, the low-speed approximation is sufficient.
在接近光速时，需使用更精确的公式：fobs = fsrc √((1+β)/(1−β))，其中 β = v/c。不过，大多数天体物理观测中，低速近似就足够了。
7. Astronomical Doppler Effect and Hubble's Law | 天文多普勒效应与哈勃定律
s of the Doppler effect is in cosmology. Edwin Hubble observed that the light from distant galaxies is redshifted, with the redshift increasing with distance. This led to Hubble' al velocity, d is the distance, and H₀ is the Hubble constant. The redshift provides direct evidence that the universe is expanding.
多普勒效应最深远的应用之一在宇宙学中。埃德温·哈勃观测到遥远星系的光发生红移，且红移量随距离增大，由此得出哈勃定律：v = H₀ d，其中 v 为退行速度，d 为距离，H₀ 为哈勃常数。红移为宇宙膨胀提供了直接证据。
ern astronomy uses this relationship to estimate the distances and speeds of galaxies billions of light-years away.
红移 z 定义为 z = Δλ / λ₀，对于低速情形有 z ≈ v/c。现代天文学利用这一关系估算数十亿光年外星系的距离和速度。
8. Shock Waves and Sonic Booms | 冲击波与声爆
ese wavefronts is a cone, known as the Mach cone, with half-angle given by sin θ = v / uₛ.
当波源速度超过介质中的声速 (uₛ > v) 时，多普勒效应的描述不再适用。波源跑得比自身发出的波阵面快，形成相长干涉，产生冲击波。波前的包络面是一个圆锥，称为马赫锥，半顶角满足 sin θ = v / uₛ。
The sharp pressure change at the shock wave is heard as a sonic boom. The same principle applies to the bow waves produced by boats moving faster than water waves.
冲击波处的剧烈压强变化被听到即为声爆。同样的原理也适用于船速超过水波速度时产生的艏波。
9. Common Pitfalls and Problem-Solving Strategy | 常见错误与解题策略
Students often confuse the signs in the Doppler equations. Follow this systematic approach:
学生常常混淆多普勒公式中的符号，可遵循以下系统步骤：
- Step 1: Draw a diagram, label the velocity directions of source and observer relative to the medium. Clarify who is moving and in which direction along the line of sight.
- 步骤一：画示意图，标出波源和观察者相对于介质的速度方向。明确谁在运动以及沿视线方向如何运动。
- Step 2: Decide which formula applies. If only the source moves, use f' = f₀ v / (v ∓ uₛ). If only the observer moves, use f' = f₀ (v ± uₒ) / v. For both moving, use the combined formula.
- 步骤二：判断适用公式。只有波源运动时用 f' = f₀ v / (v ∓ uₛ)；只有观察者运动时用 f' = f₀ (v ± uₒ) / v；两者都运动时用组合公式。
- Step 3: Apply the sign convention carefully: numerator + if observer moves towards source; denominator − if source moves towards observer. Always think about whether the frequency should increase or decrease; your sign choice must match that physical expectation.
- 步骤三：小心应用符号规则：观察者朝向波源运动时分子取 + ；波源朝向观察者运动时分母取 −。始终思考频率应升高还是降低，符号选择必须符合物理直觉。
- Step 4 ing that a positive Δλ means redshift (receding source). Ensure that speeds are much less than c, or use the relativistic formula.
- 步骤四：对于光，使用 Δλ/λ₀ ≈ v/c，注意正的 Δλ 表示红移（波源远离）。确保速度远小于光速，否则使用相对论公式。
10. Summary and Key Points | 总结与要点
| Key Concept | Equation / Description |
| Sound – moving source | f' = f₀ v / (v ∓ uₛ) |
| Sound – moving observer | f' = f₀ (v ± uₒ) / v |
| Sound – general | f' = f₀ (v ± uₒ) / (v ∓ uₛ) |
| Light (low speed) | Δλ / λ₀ ≈ v / c |
| Redshift (source receding) | λ_obs > λ₀, fobs < f₀ |
| Blueshift (source approaching) | λ_obs < λ₀, fobs > f₀ |
| Hubble's Law | v = H₀ d |
| Shock wave (Mach cone) | sin θ = v / uₛ (uₛ > v) |
The Doppler effect connects wave behaviour with relative motion, revealing shifts in frequency that carry information about velocity. Its applications span from medical diagnostics to the expansion of the universe. Mastery of the sign conventions and the distinction between mechanical and electromagnetic waves is essential for success in IB Physics.
多普勒效应将波动行为与相对运动联系起来，其频移携带着速度信息。从医学诊断到宇宙膨胀，它的应用极其广泛。掌握符号规则，区分机械波与电磁波，是攻克 IB 物理多普勒效应这一主题的关键。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图