Unit C.5: Doppler Effect单元 C.5：多普勒效应

The Doppler shift depends only on the component of relative velocity along the line joining source and observer (the radial component). A source moving in a circle around the observer at constant radius produces no shift, because its radial velocity is always zero even though its speed is large.

多普勒频移只取决于相对速度沿波源与观察者连线方向（径向）的分量。绕观察者做等半径圆周运动的波源不产生频移，因为其径向速度始终为零，尽管速率很大。

This is why an aircraft flying tangentially overhead shows the strongest pitch drop just as it crosses directly above you: that is when the radial velocity swings fastest from $+v$ (approaching) through $0$ to $-v$ (receding).

这就是为何切向飞过头顶的飞机在它恰好越过你正上方时音调降得最明显：此刻径向速度从 $+v$（接近）经 $0$ 最快地摆到 $-v$（远离）。

In one period $T = 1/f$ the source emits one wavefront and itself advances a distance $v_s T$ toward the observer. The next wavefront is emitted from that advanced position. So ahead of the source, successive wavefronts are separated not by $\lambda = vT$ but by

在一个周期 $T = 1/f$ 内，波源发出一个波前，自身朝观察者前进 $v_s T$。下一个波前从前进后的位置发出。因此在波源前方，相继波前的间距不再是 $\lambda = vT$，而是

The observer still measures the wave travelling at speed $v$, so the frequency received is

观察者测得的波速仍为 $v$，故接收频率为

Replacing $v_s \to -v_s$ for a receding source gives the $+$ version. This derivation shows clearly that what changes is the wavelength, not the wave speed in the medium.

对远离的波源用 $v_s \to -v_s$ 即得 $+$ 形式。该推导清楚表明：改变的是波长，而非介质中的波速。

A moving observer does not change the wavelength in the medium — the wavefronts are still spaced by $\lambda = v/f$. What changes is the rate at which the observer sweeps through them. Moving toward the source at $v_o$, the observer meets wavefronts at the relative speed $v + v_o$, so

运动的观察者不改变介质中的波长——波前间距仍为 $\lambda = v/f$。改变的是观察者掠过波前的速率。以 $v_o$ 朝波源运动时，观察者以相对速度 $v + v_o$ 遇到波前，故

A moving source instead changes $\lambda$ itself (Section C5.2), which is why $v_s$ enters in the denominator. Same relative speed, different physics — and the two formulas only coincide to first order in $v/v_{\text{sound}}$.

而运动的波源改变的是 $\lambda$ 本身（见 C5.2），这就是 $v_s$ 进入分母的原因。相同相对速度、不同物理——两个公式仅在 $v/v_{\text{声}}$ 的一阶近似下一致。

For light in vacuum $c = f\lambda$ is fixed, so $f\lambda = \text{const}$. Differentiating, $\lambda\,\Delta f + f\,\Delta\lambda \approx 0$, hence

真空中光满足 $c = f\lambda$ 恒定，即 $f\lambda = \text{常数}$。微分得 $\lambda\,\Delta f + f\,\Delta\lambda \approx 0$，于是

The magnitudes match (both $\approx v/c$), but a redshift means $\Delta\lambda > 0$ and $\Delta f < 0$. The data-booklet line $\frac{\Delta f}{f} \approx \frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$ is a magnitude statement; in problems you decide red vs blue from the physical direction of motion. This formula is the non-relativistic limit — for $v$ comparable to $c$ the full relativistic Doppler formula is needed, but the IB course stays at $v \ll c$.

两者大小相等（都约 $v/c$），但红移意味着 $\Delta\lambda > 0$ 而 $\Delta f < 0$。数据手册的 $\frac{\Delta f}{f} \approx \frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$ 是大小关系式；解题时由运动的物理方向判断红/蓝。该公式是非相对论极限——当 $v$ 接近 $c$ 时需用完整的相对论多普勒公式，但 IB 课程停留在 $v \ll c$。

Not everything is redshifted. The Andromeda galaxy, our nearest large neighbour, is blueshifted: its local gravitational pull toward the Milky Way exceeds the cosmological recession at such close range, so it is actually approaching us. Cosmological redshift only dominates for galaxies far enough that the expansion of space outruns local "peculiar" velocities.

并非一切都红移。仙女座星系是离我们最近的大邻居，呈蓝移：在如此近的距离上，它朝银河系的局部引力吸引超过宇宙学退行，故它实际上正在接近我们。宇宙学红移只在足够远、空间膨胀超过局部"本动"速度的星系上占主导。

In a binary star system, the two stars alternately approach and recede as they orbit, so their spectral lines shift back and forth periodically — a Doppler signature used to detect unseen companions, including exoplanets.

在双星系统中，两颗恒星随轨道运动交替接近与远离，其谱线周期性来回移动——这一多普勒特征被用来探测看不见的伴星，包括系外行星。

In medical Doppler ultrasound, a probe sends sound (typically a few $\mathrm{MHz}$) into tissue; it reflects off red blood cells moving at speed $v$ and returns shifted by $\Delta f \approx \frac{2 f v \cos\theta}{v_{\text{sound}}}$, where $\theta$ is the angle between the beam and the flow direction. The $\cos\theta$ appears because only the component of blood velocity along the beam Doppler-shifts the echo.

在医学多普勒超声中，探头向组织发出声波（通常几 $\mathrm{MHz}$）；它从以速率 $v$ 运动的红细胞反射，返回时频移 $\Delta f \approx \frac{2 f v \cos\theta}{v_{\text{声}}}$，其中 $\theta$ 是声束与血流方向的夹角。出现 $\cos\theta$ 是因为只有沿声束方向的血流速度分量才产生多普勒频移。

Clinicians measure $\Delta f$, know $f$, $v_{\text{sound}}$ in tissue, and $\theta$, and solve for blood-flow speed $v$ — for example to detect a narrowed artery, where flow speeds up. This is the same physics as the radar gun, just with sound in tissue instead of microwaves in air.

临床上测出 $\Delta f$，已知 $f$、组织中 $v_{\text{声}}$ 与 $\theta$，即可解出血流速度 $v$——例如检测变窄的动脉（此处血流加速）。这与雷达枪是同样的物理，只是把空气中的微波换成组织中的声波。