---
title: "The Doppler Effect"
book: "High School Physics"
subject: physics
language: en
chapter: 23
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/23-the-doppler-effect
---

# Chapter 23 — The Doppler Effect

An ambulance closes in, siren blazing, and sweeps past — and at that instant the note slides audibly downward, though the driver touched nothing. The siren is honest; the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) is not: motion squeezes it ahead and stretches it behind. This chapter measures the squeeze, then rides the same idea from sirens to [radar](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#met-g10-signals-and-waves-echo) guns, to blood in an artery, and out to a planet betrayed by a wobble in starlight.

## 23.1 The effect

**Definition 23.1 (Doppler effect).**

The *Doppler effect* is the change of the received [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) of a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) when source and receiver move relative to each other: the received [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f'$ is higher than the emitted [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f$ while they approach, lower while they recede, and equal to $f$ only while their distance momentarily stops changing.

**Remark 23.2 (Pitch, not loudness).**

Two things change as the ambulance passes. The *[loudness](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness)* swells and fades because the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) spreads with distance ([Chapter 21](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#ch-g12-sound-acoustics)); the *[pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness)* shifts because of the speed of approach or recession, and drops abruptly at the passing. [Loudness](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness) says “how far”; [pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness) says “how fast”.

## 23.2 A moving source: crowded wavefronts

The key picture: once a crest leaves the source, the medium owns it. Each crest expands as a circle at the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) speed $v$ ([Chapter 20](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#ch-g12-mechanical-waves)), centered on the point where it was *emitted* — the medium neither knows nor cares that the source has moved on.

**Proposition 23.3 (Moving source).**

A source emits crests with [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $T$ ([frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f = 1/T$) while moving in a straight line at speed $v_s < v$ through a medium in which the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) travels at speed $v$. An observer at rest in the medium receives

$$
\lambda' = (v \mp v_s)\,T, \qquad f' = f\,\frac{v}{v \mp v_s},
$$

the upper signs for a source approaching, the lower for one receding.

**Proof.** Between two crests the first advances $vT$ while the source advances $v_s T$. Ahead, the second crest is therefore emitted $v_s T$ *closer* to the first, $\lambda' = (v - v_s)T$; behind, the same $v_s T$ is added. The crests then sweep past a resting observer at the medium’s speed $v$, one every $\lambda'/v$ seconds, so $f' = v/\lambda' = f v/(v \mp v_s)$. ∎

![Wavefronts of a source moving right: each circle is centered on the point where it was emitted (gray dots), so the crests bunch ahead, ' = (v - v_s)T, and stretch behind, ' = (v + v_s)T.](https://one-course.com/images/onecourse/chapters/physics-2/g12-doppler-effect/fig-9b318ab9e31f.svg)

*[Wavefronts](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavefront) of a source moving right: each circle is centered on the point where it was emitted (gray dots), so the crests bunch ahead, $\lambda' = (v - v_s)T$, and stretch behind, $\lambda' = (v + v_s)T$.*

**Example 23.4 (The ambulance, in numbers).**

A siren at $f = 700\,\mathrm{Hz}$ approaches at $v_s = 25\,\mathrm{m}/\mathrm{s}$; take $v = 340\,\mathrm{m}/\mathrm{s}$ in air ([Chapter 21](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#ch-g12-sound-acoustics)). Approaching: $f' = 700 \times 340/315 \approx 756\,\mathrm{Hz}$; receding: $f' = 700 \times 340/365 \approx 652\,\mathrm{Hz}$. The pass drops the [pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness) by $104\,\mathrm{Hz}$ — a ratio $\approx 1.16$, about two and a half semitones, unmistakable to any ear.

**Remark 23.5 (What motion does not change).**

The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) still travels at $v$: the medium alone sets the speed ([Chapter 20](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#ch-g12-mechanical-waves)). Motion of the source squeezes *[wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength)*, never speeds. And the squeeze has a limit: as $v_s \to v$ the crests ahead pile onto each other and $f' \to \infty$ — the traffic jam of [wavefronts](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavefront) a supersonic aircraft drags as a shock.

## 23.3 A moving observer, and the slow-motion shortcut

**Proposition 23.6 (Moving observer).**

An observer moves at speed $v_o$ straight toward (or away from) a source at rest in the medium. The received [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) is

$$
f' = f\left(1 \pm \frac{v_o}{v}\right) \quad (+\ \text{approaching},\ -\ \text{receding}).
$$

**Proof.** The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) pattern itself is undisturbed: crests of spacing $\lambda = vT$ travel at $v$. An observer running at them meets crests at the relative speed $v + v_o$, one every $\lambda/(v + v_o)$ seconds: $f' = (v + v_o)/\lambda = f(1 + v_o/v)$. Running away, $v - v_o$. ∎

**Definition 23.7 (Radial velocity).**

The *radial velocity* $v_r$ of a source relative to an observer is the component of their relative velocity along the line joining them — the rate at which the distance shrinks or grows. Only this component shifts the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency); motion *across* the line of sight gives no shift at this order.

**Proposition 23.8 (Slow motion: one formula for everybody).**

When the radial speed is small, $v_r \ll v$, it no longer matters who moves:

$$
\frac{\Delta f}{f} = \frac{f' - f}{f} \approx +\frac{v_r}{v} \ \text{(approach)},
\qquad
\frac{\Delta f}{f} \approx -\frac{v_r}{v} \ \text{(recession)}.
$$

**Proof.** The observer formulas are already exactly $1 \pm v_o/v$. For the source, write $x = v_s/v$: the identity $\frac{1}{1 - x} = 1 + x + \frac{x^2}{1 - x}$ shows that $f'/f$ differs from $1 + x$ by a term of order $x^2$ — for $x = 0.1$, a $1\%$ correction on a $10\%$ shift. To first order all four formulas collapse onto $1 \pm v_r/v$. ∎

![Frequency heard as a 440\, Hz siren passes at 15\, m/ s, 8\, m from the microphone: plateaus at fv/(v v_s), and the true f (dotted) crossed essentially at closest approach, where the motion is purely transverse.](https://one-course.com/images/onecourse/chapters/physics-2/g12-doppler-effect/fig-8b4612d7a37a.svg)

*[Frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) heard as a $440\,\mathrm{Hz}$ siren passes at $15\,\mathrm{m}/\mathrm{s}$, $8\,\mathrm{m}$ from the microphone: plateaus at $fv/(v \mp v_s)$, and the true $f$ (dotted) crossed essentially at closest approach, where the motion is purely [transverse](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-transverse).*

**Method 23.9 (Reading a pass).**

A recording of a passing source shows two plateaus: $f_1$ (approach) and $f_2$ (recession). Then

$$
f = \frac{2 f_1 f_2}{f_1 + f_2}, \qquad
v_s = v\,\frac{f_1 - f_2}{f_1 + f_2}.
$$

Indeed $f_1 + f_2 = \frac{2fv^2}{v^2 - v_s^2}$ and $f_1 - f_2 = \frac{2fv\,v_s}{v^2 - v_s^2}$: divide for $v_s$, and compute $2f_1 f_2/(f_1 + f_2)$ for $f$. Two [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) readings yield both the true [pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness) and the speed — no stopwatch, no ruler.

## 23.4 Echoes that measure speed

Bounce a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) off a moving target and the [Doppler effect](#def-g12-doppler-effect-doppler) strikes twice.

**Proposition 23.10 (Reflection doubles the shift).**

A [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) of [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f$ is sent at a reflector approaching head-on at speed $u \ll v$. The [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) returns with

$$
f' = f\,\frac{v + u}{v - u}, \qquad \text{hence} \qquad \Delta f \approx \frac{2u}{v}\,f .
$$

**Proof.** Two shifts in series. As a moving *observer* the reflector receives $f(1 + u/v)$; re-emitting what it receives, it is now a moving *source*, so the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) arrives at $f' = f(1 + u/v)/(1 - u/v) \approx f(1 + 2u/v)$ for $u \ll v$. ∎

**Example 23.11 (The radar gun).**

A traffic [radar](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#met-g10-signals-and-waves-echo) emits microwaves at $f = 24.15\,\mathrm{GHz}$ ($v = c = 3.00 \times 10^{8}\,\mathrm{m}/\mathrm{s}$) and superposes the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) on the outgoing [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave); the beat ([Chapter 21](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#ch-g12-sound-acoustics)) directly [sounds](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) out $\Delta f = 2uf/c \approx 161\,\mathrm{Hz}$ per $\mathrm{m}/\mathrm{s}$ of car speed — a trivially measurable audio [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) carved out of a $24\,\mathrm{GHz}$ carrier. The factor $2$ is not optional: forgetting it flatters every driver by half.

**Example 23.12 (Doppler ultrasound).**

A medical probe sends $f = 5.0\,\mathrm{MHz}$ [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound) into an artery ($v \approx 1540\,\mathrm{m}/\mathrm{s}$ in tissue); red blood cells reflect it. Blood at $u = 0.50\,\mathrm{m}/\mathrm{s}$ returns $\Delta f = 2 \times 0.50 \times 5.0 \times 10^{6}/1540 \approx
3.2\,\mathrm{kHz}$ — an *audible* beat: the cardiologist literally hears the blood accelerate at each heartbeat, then measures $u$ from $\Delta f$.

## 23.5 Doppler in starlight

Light is a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) ([Chapter 22](https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference#ch-g12-light-as-wave)), its [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) striped with sharp lines at [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) fixed by each chemical element ([Chapter 2](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#ch-g10-light-spectra)) — a laboratory-calibrated ruler printed on every star.

**Definition 23.13 (Redshift and blueshift).**

When the [spectral lines](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-emission) of a source appear at [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) *longer* than in the laboratory, the [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) is *redshifted*; when they appear *shorter*, it is *blueshifted* — shifted toward the red or the blue end of the [visible spectrum](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-wavelength).

**Proposition 23.14 (Doppler shift of light).**

For a source receding at radial speed $v_r \ll c$, every [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) is observed stretched by

$$
\frac{\Delta \lambda}{\lambda} = \frac{\lambda' - \lambda}{\lambda} \approx \frac{v_r}{c} \quad \text{(redshift)},
$$

and compressed by the same amount, $\Delta\lambda/\lambda \approx -v_r/c$, for an approaching source ([blueshift](#def-g12-doppler-effect-redshift)).

**Proof.** *Admitted at this level.* ∎

**Remark 23.15 (Where the honest formula lives).**

Light needs no medium, so the [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) derivation cannot be copied — there is no “at rest in the medium”. The exact formula comes from special relativity, later this year ([Chapter 35](https://one-course.com/books/physics/2/en/chapter/35-special-relativity-time-dilation#ch-g12-special-relativity)); the corrections are of order $(v_r/c)^2$, negligible for planes, stars and nearby galaxies. Below we use the slow-motion formula with a clear conscience.

![The same hydrogen line in three spectra: at 656.30\, nm in the laboratory, blueshifted for an approaching star, redshifted for a receding one. The shift, read against the dashed reference, gives the radial velocity.](https://one-course.com/images/onecourse/chapters/physics-2/g12-doppler-effect/fig-f91ad5bdac02.svg)

*The same hydrogen line in three spectra: at $656.30\,\mathrm{nm}$ in the laboratory, [blueshifted](#def-g12-doppler-effect-redshift) for an approaching star, [redshifted](#def-g12-doppler-effect-redshift) for a receding one. The shift, read against the dashed reference, gives the [radial velocity](#def-g12-doppler-effect-radial).*

**Example 23.16 (Weighing a star’s escape).**

In the [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) of the bright star Vega, the hydrogen line of laboratory [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $656.30\,\mathrm{nm}$ is measured at $656.27\,\mathrm{nm}$: $\Delta\lambda = -0.03\,\mathrm{nm}$, a [blueshift](#def-g12-doppler-effect-redshift), so Vega approaches at $v_r = c\,|\Delta\lambda|/\lambda \approx 1.4 \times 10^{4}\,\mathrm{m}/\mathrm{s}$ — $14\,\mathrm{km}/\mathrm{s}$, read off a shift of one part in twenty thousand.

**Example 23.17 (Wobbling stars: binaries and exoplanets).**

When two stars [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) each other, their lines swing [periodically](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) blue and red — a *spectroscopic binary*, [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) measured without ever resolving the pair. A planet does the same to its star, faintly: both circle their common [center of mass](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#def-g11-forces-and-motion-com), so the star’s [radial velocity](#def-g12-doppler-effect-radial) oscillates by a few tens of $\mathrm{m}/\mathrm{s}$ — a shift $\Delta\lambda/\lambda \sim 10^{-7}$. Detecting it is how the first planet around a Sun-like star was found; the weekend problem reruns that discovery, numbers and all.

**Remark 23.18 (Galaxy redshifts).**

The spectra of distant galaxies are all *red*shifted, and the more distant, the more shifted: the universe’s distances are stretching. For nearby galaxies $v_r = c\,\Delta\lambda/\lambda$ reads off the recession speed; for the farthest, shifts outgrow what the slow-motion formula can honestly handle, and the accounting is taken up again from [Chapter 35](https://one-course.com/books/physics/2/en/chapter/35-special-relativity-time-dilation#ch-g12-special-relativity) onward, and in the university volumes.

## 23.6 Exercises

**Exercise 23.1 ★.**

A fire-engine siren emits at $700\,\mathrm{Hz}$ and drives at $25\,\mathrm{m}/\mathrm{s}$ ($v = 340\,\mathrm{m}/\mathrm{s}$). Compute the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) heard by a pedestrian (a) ahead of it; (b) behind it. (c) What does the driver hear?

**Solution of Exercise 23.1.**

(a) $f' = 700 \times 340/315 \approx 756\,\mathrm{Hz}$. (b) $f' = 700 \times 340/365 \approx 652\,\mathrm{Hz}$. (c) $700\,\mathrm{Hz}$: driver and siren move together, their distance never changes.

**Exercise 23.2 ★.**

A friend claims: “the [pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness) drops as the ambulance leaves because the [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) gets weaker with distance.” Untangle the two effects being confused, and say what each depends on.

**Solution of Exercise 23.2.**

The *[loudness](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness)* fades because the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) spreads out — it depends on the distance. The *[pitch](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-pitch-loudness)* drops from approach to recession — the [Doppler effect](#def-g12-doppler-effect-doppler), which depends on how fast the distance changes. Two independent effects.

**Exercise 23.3 ★.**

A cyclist rides at $6.0\,\mathrm{m}/\mathrm{s}$ straight toward a stationary siren emitting at $700\,\mathrm{Hz}$. What [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) do they hear? And riding straight away? Which formula applies, and why not the moving-source one?

**Solution of Exercise 23.3.**

Moving observer: $f' = 700\,(1 + 6.0/340) \approx 712\,\mathrm{Hz}$; riding away, $700\,(1 - 6.0/340) \approx 688\,\mathrm{Hz}$. The siren is at rest in the air, so the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) pattern is undisturbed; only the observer runs through it.

**Exercise 23.4 ★.**

A train horn emits at $400\,\mathrm{Hz}$ while the train runs at $30\,\mathrm{m}/\mathrm{s}$. Compute the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) ahead of the train, behind it, and at rest. Which observer receives which?

**Solution of Exercise 23.4.**

Ahead: $\lambda' = 310/400 = 0.775\,\mathrm{m}$; behind: $370/400 = 0.925\,\mathrm{m}$; at rest: $340/400 = 0.85\,\mathrm{m}$. The short [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) reaches whoever the train approaches, the long one whoever it leaves.

**Exercise 23.5 ★.**

A phone playing a $440\,\mathrm{Hz}$ tone is carried by a runner at $5.0\,\mathrm{m}/\mathrm{s}$. Use the slow-motion formula to estimate the shift heard by someone the runner approaches. Compare it to a semitone (about $6\%$ in [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency)): would a musician notice?

**Solution of Exercise 23.5.**

$\Delta f/f \approx 5.0/340 \approx 1.5\%$: $\Delta f \approx 6.5\,\mathrm{Hz}$, so about $446\,\mathrm{Hz}$ — a quarter of a semitone. A musician would just notice.

**Exercise 23.6 ★★.**

A bat flies at $6.0\,\mathrm{m}/\mathrm{s}$ straight at a wall, emitting at $50.0\,\mathrm{kHz}$. Show that the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) it hears returns at $f' = f(v + v_b)/(v - v_b)$, then compute $f'$ and the shift. Why is this the radar-gun formula in disguise?

**Solution of Exercise 23.6.**

The wall receives $f\,v/(v - v_b)$ (moving source); the bat, a moving observer closing on the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest), hears $(1 + v_b/v)$ times that: $f' = f(v + v_b)/(v - v_b) = 50.0 \times
346/334 \approx 51.8\,\mathrm{kHz}$, a shift of $+1.8\,\mathrm{kHz}$. It is the radar-gun double shift with emitter and reflector roles swapped: only the closing speed matters.

**Exercise 23.7 ★★.**

A [radar](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#met-g10-signals-and-waves-echo) gun at $24.15\,\mathrm{GHz}$ measures a beat of $4.0\,\mathrm{kHz}$ from an approaching car. Find the car’s speed in $\mathrm{m}/\mathrm{s}$ and $\mathrm{km}/\mathrm{h}$. What beat would a car at exactly $50\,\mathrm{km}/\mathrm{h}$ give?

**Solution of Exercise 23.7.**

$u = c\,\Delta f/(2f) = 3.00 \times 10^{8} \times 4000/(2 \times 24.15 \times 10^{9}) \approx
24.8\,\mathrm{m}/\mathrm{s} \approx 89\,\mathrm{km}/\mathrm{h}$. At $50\,\mathrm{km}/\mathrm{h}$ ($13.9\,\mathrm{m}/\mathrm{s}$): $\Delta f = 2 \times 13.9 \times 24.15 \times 10^{9}/3.00 \times 10^{8} \approx 2.2\,\mathrm{kHz}$.

**Exercise 23.8 ★★.**

A $5.0\,\mathrm{MHz}$ Doppler [ultrasound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-ultrasound) probe aimed along an artery ($v = 1540\,\mathrm{m}/\mathrm{s}$ in tissue) hears a beat of $2.6\,\mathrm{kHz}$. Find the blood speed. Why must the probe be angled *along* the flow rather than perpendicular to it?

**Solution of Exercise 23.8.**

$u = v\,\Delta f/(2f) = 1540 \times 2600/1.0 \times 10^{7} \approx 0.40\,\mathrm{m}/\mathrm{s}$. Perpendicular flow has zero [radial velocity](#def-g12-doppler-effect-radial), hence no shift at all.

**Exercise 23.9 ★★.**

Take $f = 1000\,\mathrm{Hz}$, $v = 340\,\mathrm{m}/\mathrm{s}$ and a closing speed of $34\,\mathrm{m}/\mathrm{s}$. Compute $f'$ exactly when (a) the source moves, (b) the observer moves. Compare both to the slow-motion prediction $f(1 + v_r/v)$ and explain the size of the disagreement.

**Solution of Exercise 23.9.**

(a) Source: $f' = 1000 \times 340/306 \approx 1111\,\mathrm{Hz}$. (b) Observer: $f' = 1000 \times 1.1 = 1100\,\mathrm{Hz}$. Slow motion predicts $1100\,\mathrm{Hz}$: exact for the observer, $11\,\mathrm{Hz}$ short for the source — the order-$x^2$ correction, $1\%$ of $f$ for $x = 0.1$.

**Exercise 23.10 ★★.**

The recording of a scooter passing a microphone shows plateaus at $465\,\mathrm{Hz}$ and $415\,\mathrm{Hz}$. Using [Method 23.9](#met-g12-doppler-effect-pass), find the horn’s true [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) and the scooter’s speed in $\mathrm{km}/\mathrm{h}$. Why is the true [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) *not* their average?

**Solution of Exercise 23.10.**

$f = 2 \times 465 \times 415/880 \approx 439\,\mathrm{Hz}$; $v_s = 340 \times 50/880 \approx 19.3\,\mathrm{m}/\mathrm{s} \approx 70\,\mathrm{km}/\mathrm{h}$. The true $f$ is the [harmonic](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-harmonics) mean: approach raises the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) more than recession lowers it, so $f$ sits below the arithmetic average ($440\,\mathrm{Hz}$).

**Exercise 23.11 ★★.**

In a star’s [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre), the hydrogen line of laboratory [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $656.3\,\mathrm{nm}$ is observed at $656.5\,\mathrm{nm}$. [Redshift](#def-g12-doppler-effect-redshift) or [blueshift](#def-g12-doppler-effect-redshift)? Compute the star’s [radial velocity](#def-g12-doppler-effect-radial) and direction.

**Solution of Exercise 23.11.**

Longer [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength): [redshift](#def-g12-doppler-effect-redshift), the star recedes. $v_r = c\,\Delta\lambda/\lambda =
3.00 \times 10^{8} \times 0.2/656.3 \approx 9.1 \times 10^{4}\,\mathrm{m}/\mathrm{s} \approx 91\,\mathrm{km}/\mathrm{s}$.

**Exercise 23.12 ★★★.**

A star wobbles at $55\,\mathrm{m}/\mathrm{s}$ because of an unseen planet. Compute the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude) of the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) swing of a $500\,\mathrm{nm}$ line, as a length and as a fraction of $\lambda$. [Order of magnitude](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-oom): why did planet hunting have to wait for spectrographs stable to one part in $10^{7}$?

**Solution of Exercise 23.12.**

$\Delta\lambda = \lambda K/c = 500 \times 55/3.00 \times 10^{8} \approx 9.2 \times 10^{-5}\,\mathrm{nm}$; $\Delta\lambda/\lambda = 55/3.00 \times 10^{8} \approx 1.8 \times 10^{-7}$. The whole [signal](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-signal) is two parts in ten million: a spectrograph drifting by more than $10^{-7}$ buries it.

**Exercise 23.13 ★★★.**

Every line in a galaxy’s [spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre) is found stretched by $2.4\%$. Compute its recession speed. Astronomers find that such speeds grow in proportion to distance, for every direction of the sky: what picture does this suggest? Is the slow-motion formula still trustworthy here?

**Solution of Exercise 23.13.**

$v_r = 0.024\,c \approx 7.2 \times 10^{6}\,\mathrm{m}/\mathrm{s}$. Speeds proportional to distance, the same in every direction: all distances stretching uniformly — an expanding universe with no privileged center. At $2.4\%$ the corrections are of order $(v_r/c)^2 \approx
6 \times 10^{-4}$: still trustworthy.

**Exercise 23.14 ★★★.**

The Sun rotates: one edge of its disk approaches us at about $2.0\,\mathrm{km}/\mathrm{s}$ while the other recedes equally fast. What does this do to the $656.3\,\mathrm{nm}$ line in light from the whole disk — shift it or broaden it? Compute the effect in $\mathrm{nm}$.

**Solution of Exercise 23.14.**

The two edges shift opposite ways, so the disk-averaged line is *broadened*, not shifted: full width $2\lambda v/c = 2 \times 656.3 \times 2.0 \times 10^{3}/3.00 \times 10^{8}
\approx 8.8 \times 10^{-3}\,\mathrm{nm}$ (about $\pm4.4 \times 10^{-3}\,\mathrm{nm}$).

**Exercise 23.15 ★★★.**

A police car at $40\,\mathrm{m}/\mathrm{s}$, siren at $700\,\mathrm{Hz}$, chases a truck driving at $30\,\mathrm{m}/\mathrm{s}$ in the same direction. (a) Justify $f' = f(v - v_o)/(v - v_s)$ for the truck driver. (b) Compute $f'$. (c) Show that if the truck matched the car’s speed the shift would vanish, and say why that is as it should be.

**Solution of Exercise 23.15.**

(a) Ahead of the car the crests are spaced $(v - v_s)T$; the truck flees at $v_o$ and meets them at relative speed $v - v_o$: $f' = (v - v_o)/\bigl((v - v_s)T\bigr) =
f(v - v_o)/(v - v_s)$. (b) $f' = 700 \times 310/300 \approx 723\,\mathrm{Hz}$. (c) $v_o = v_s$ gives $f' = f$: the separation is constant, and no change of distance means no Doppler shift, by definition.

## 23.7 Problem: The Hunt for an Exoplanet

**Problem 23.1.**

Weekend problem — the hunt for an exoplanet: a train horn calibrates the ear, a radar gun rehearses the double shift, and by Sunday night a star’s spectrum, wobbling by one part in five million, has weighed a planet nobody has ever seen

An astronomy club spends a weekend on one idea — the Doppler shift — rehearsed on Earth, then aimed at a Sun-like star. Data: $v = 340\,\mathrm{m}/\mathrm{s}$ for [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) in air, $c = 3.00 \times 10^{8}\,\mathrm{m}/\mathrm{s}$, $G = 6.67 \times 10^{-11}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{kg}^{2}$; the star has mass $M = 2.0 \times 10^{30}\,\mathrm{kg}$. Admitted, from the mechanics of circular [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) ([Chapter 27](https://one-course.com/books/physics/2/en/chapter/27-satellites-and-planetary-motion#ch-g12-satellites-kepler)): a planet of mass $m$ on a circular [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) of radius $r$ has [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) and speed given by $r^3 = \frac{G M P^2}{4\pi^2}$ and $V = \frac{2\pi r}{P}$, and the star, circling the common [center of mass](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#def-g11-forces-and-motion-com), wobbles at speed $K = \frac{m}{M}\,V$.

**Part I — Friday: the level crossing.** A train passes at constant speed, horn on; a phone records $f_1 = 471\,\mathrm{Hz}$ while it approaches, $f_2 = 411\,\mathrm{Hz}$ after it passes.

1. Explain, with the [crowded-wavefront](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavefront) picture, why $f_1 > f_2$ although the horn never changes.
2. Write the two equations linking $f_1$ , $f_2$ to the true [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f$ and train speed $u$ .
3. Show that $u = v\,\dfrac{f_1 - f_2}{f_1 + f_2}$ and compute it in $\mathrm{m}/\mathrm{s}$ and $\mathrm{km}/\mathrm{h}$ .
4. Show that $f = \dfrac{2 f_1 f_2}{f_1 + f_2}$ and compute it. Why is $f$ slightly *below* the average of $f_1$ and $f_2$ ?
5. Compute the [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) ahead of and behind the train, and the at-rest [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) .

**Part II — Saturday: the [radar](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#met-g10-signals-and-waves-echo) gun.** A traffic patrol demonstrates its [radar](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#met-g10-signals-and-waves-echo): $f = 24.125\,\mathrm{GHz}$, and the gun reads the beat between the outgoing [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) and the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest).

6. Explain why the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) of a moving car is shifted *twice* — name the role the car plays in each shift.
7. Show that for a car at speed $u \ll c$ the beat is $\Delta f \approx \dfrac{2u}{c}\,f$ .
8. A car returns $\Delta f = 3.55\,\mathrm{kHz}$ : compute $u$ .
9. Convert to $\mathrm{km}/\mathrm{h}$ and compare to the $80\,\mathrm{km}/\mathrm{h}$ limit posted there.
10. How many [hertz](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) of beat correspond to $1\,\mathrm{km}/\mathrm{h}$ ? Comment: what makes such a tiny relative shift of a $24\,\mathrm{GHz}$ [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) easy to measure?

**Part III — Sunday: the star that wobbles.** The club downloads the measured [radial velocity](#def-g12-doppler-effect-radial) of a Sun-like star, night after night, plotted below.

![The star’s radial velocity over twelve nights: a clean sinusoid — the signature of a body on a circular orbit, seen edge-on.](https://one-course.com/images/onecourse/chapters/physics-2/g12-doppler-effect/fig-c07bb7f2b23b.svg)

*The star’s [radial velocity](#def-g12-doppler-effect-radial) over twelve nights: a clean sinusoid — the signature of a body on a circular [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite), seen edge-on.*

11. What is actually measured, night after night, in the star’s *[spectrum](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-timbre)* ? State the relation used to turn it into $v_r$ .
12. Read the [amplitude](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-amplitude) $K$ and the [period](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) $P$ of the wobble off the curve.
13. Compute the corresponding swing $\Delta\lambda$ of a $550\,\mathrm{nm}$ line, and $\Delta\lambda/\lambda$ . Compare with Part I: how much harder is this measurement than the train’s?
14. Why does an unseen planet make the star wobble at all? What does the sinusoidal shape indicate about the [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) ?
15. Why does this method measure only the *radial* part of the star’s motion, and what would be seen if the [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) were face-on to us?

**Part IV — Sunday night: weighing the invisible.** Take the [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) circular and edge-on, so $K$ is the star’s full orbital speed.

16. From $P$ and the admitted relation $r^3 = G M P^2/(4\pi^2)$ , compute the planet’s orbital radius $r$ . Compare it to the Earth–Sun distance, $1.5 \times 10^{11}\,\mathrm{m}$ .
17. Compute the planet’s orbital speed $V = 2\pi r/P$ .
18. From $K = (m/M)\,V$ , compute the planet’s mass $m$ .
19. Compare $m$ to Jupiter ( $1.9 \times 10^{27}\,\mathrm{kg}$ ) and to Earth ( $6.0 \times 10^{24}\,\mathrm{kg}$ ), and $r$ to the Sun–Mercury distance ( $5.8 \times 10^{10}\,\mathrm{m}$ ). What sort of world is this?
20. If the [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) were tilted rather than edge-on, would the true mass be larger or smaller than your value? Conclude in one sentence: what did a [periodic](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) shift of one part in five million just deliver?

**Solution of Problem 23.1.**

**1.** Each crest expands from the point where it was emitted: crests bunch ahead of the moving horn and stretch behind it, so the approaching phone meets them faster ($f_1$) than the receding one ($f_2$).

**2.** $f_1 = f\,\dfrac{v}{v - u}$, $f_2 = f\,\dfrac{v}{v + u}$.

**3.** $f_1 - f_2 = \frac{2fv\,u}{v^2 - u^2}$ and $f_1 + f_2 =
\frac{2fv^2}{v^2 - u^2}$; dividing, $u = v\,\frac{f_1 - f_2}{f_1 + f_2} =
340 \times 60/882 \approx 23\,\mathrm{m}/\mathrm{s} \approx 83\,\mathrm{km}/\mathrm{h}$.

**4.** $\dfrac{2f_1 f_2}{f_1 + f_2} = f$ (the common factor cancels): $f = 2 \times 471 \times 411/882 \approx 439\,\mathrm{Hz}$ — the [harmonic](https://one-course.com/books/physics/2/en/chapter/21-sound-and-acoustics#def-g12-sound-acoustics-harmonics) mean, below the average $441\,\mathrm{Hz}$ because approach boosts $f$ more than recession cuts it.

**5.** Ahead: $(v - u)/f = 316.9/439 \approx 0.72\,\mathrm{m}$; behind: $363.1/439 \approx 0.83\,\mathrm{m}$; at rest: $340/439 \approx 0.77\,\mathrm{m}$.

**6.** Once as a moving *observer* receiving the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), once as a moving *source* re-emitting what it received.

**7.** $f' = f\,(1 + u/c)/(1 - u/c) \approx f(1 + 2u/c)$, so $\Delta f = f' - f \approx \dfrac{2u}{c}\,f$.

**8.** $u = c\,\Delta f/(2f) = 3.00 \times 10^{8} \times 3550/(2 \times 24.125 \times 10^{9})
\approx 22\,\mathrm{m}/\mathrm{s}$.

**9.** $22.1 \times 3.6 \approx 79\,\mathrm{km}/\mathrm{h}$: just under the $80\,\mathrm{km}/\mathrm{h}$ limit.

**10.** $\Delta f = 2f/(3.6\,c) \approx 45\,\mathrm{Hz}$ per $\mathrm{km}/\mathrm{h}$. Beating the [echo](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#rem-g10-signals-and-waves-honest) against the emitted [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) leaves only the *difference*: an audio [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency), trivially counted, though it is a $10^{-7}$ fraction of the carrier.

**11.** The [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) of known [spectral lines](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-emission), night after night; then $v_r = c\,\Delta\lambda/\lambda$.

**12.** $K = 55\,\mathrm{m}/\mathrm{s}$, $P = 4.2\,\mathrm{days}$.

**13.** $\Delta\lambda = 550 \times 55/3.00 \times 10^{8} \approx 1.0 \times 10^{-4}\,\mathrm{nm}$, $\Delta\lambda/\lambda \approx 1.8 \times 10^{-7}$ — one part in five million, some $4 \times 10^{5}$ times smaller than the train’s $7\%$ shift.

**14.** Star and planet attract each other equally, so both [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) their common [center of mass](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#def-g11-forces-and-motion-com): the star cannot stand still. A sinusoidal $v_r$ is [uniform circular motion](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#def-g11-forces-and-motion-ucm) seen edge-on — a circular [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite).

**15.** Only the line-of-sight component changes the star–Earth distance, hence the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength). Face-on, $v_r = 0$ at all times: no [signal](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-signal) whatsoever.

**16.** $P = 3.63 \times 10^{5}\,\mathrm{s}$: $r^3 = 6.67 \times 10^{-11} \times 2.0 \times 10^{30} \times
(3.63 \times 10^{5})^2/(4\pi^2) \approx 4.4 \times 10^{29}$, so $r \approx 7.6 \times 10^{9}\,\mathrm{m}$ — about $1/20$ of the Earth–Sun distance.

**17.** $V = 2\pi r/P = 2\pi \times 7.6 \times 10^{9}/3.63 \times 10^{5} \approx
1.3 \times 10^{5}\,\mathrm{m}/\mathrm{s}$.

**18.** $m = M K/V = 2.0 \times 10^{30} \times 55/1.3 \times 10^{5} \approx 8.3 \times 10^{26}\,\mathrm{kg}$.

**19.** About $0.4$ Jupiter masses ($\approx 140$ Earths), orbiting $8$ times closer than Mercury: a gas giant roasting against its star — a “hot Jupiter”.

**20.** A tilt hides part of the motion: the measured $K$ is only the radial share, so the true mass is *larger* — $8.3 \times 10^{26}\,\mathrm{kg}$ is a minimum. A [periodic](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) shift of one part in five million just delivered the [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite), speed and minimum mass of a planet nobody has ever seen.