Christian Doppler

How to use this course

Christian Doppler wrote dozens of papers on mathematics, optics and instruments, but one short lecture made his name. On 25 May 1842 he argued that the frequency we receive from a wave source depends on how the source and the observer move. The idea is simple enough for a first physics course. Its consequences reach from police radar to the discovery of planets around other stars and the expansion of the universe.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Part II moves from Doppler’s own setting (sound in a medium, Lessons 04 to 06) to light (Lessons 07 and 08) and then to the modern uses of the effect in astronomy and atomic physics (Lessons 09 and 10). Doppler himself got several things wrong about light, and the course says where.

Timeline

Salzburg and Vienna, 1803–1835

Christian Andreas Doppler was born on 29 November 1803 in Salzburg, in the family house on what is now Makartplatz. His baptismal record spells the name Christianus Andreas Dopler. The form “Johann Christian Doppler”, still found in some books, is a later error. His father Johann was a master stonemason, and the family firm had made marble work in the region for generations. The boy was physically frail, so the trade was ruled out and his father thought of a career in business for him.

The Salzburg astronomer and mathematician Simon Stampfer noticed his talent and advised further study. From 1822 Doppler attended the young Polytechnic Institute in Vienna, founded in 1815. He found its curriculum narrow, returned to Salzburg in 1825, completed his secondary schooling and the philosophy course at the Lyceum, and supported himself by private tutoring. He then went back to Vienna to study higher mathematics, mechanics and astronomy.

A hard start

In 1829 he became assistant to Adam von Burg, professor of higher mathematics in Vienna. The post was temporary. In that time he published his first papers, on the theory of parallels, the convergence of series and electricity. When it ended in 1833 he applied for teaching posts in Linz, Salzburg, Gorizia, Ljubljana and Prague, and for the chair of higher mathematics at the Vienna Polytechnic, without success. For about eighteen months he worked as a bookkeeper in a cotton-spinning factory.

Discouraged, he decided to emigrate to the United States and travelled to Munich to arrange the journey with the American consul. There the news reached him that he had been appointed to a teaching post in Prague. He took it, and in March 1835 began teaching at the technical secondary school of the Bohemian Estates.

Prague, 1835–1842

Prague gave Doppler a living and, eventually, an intellectual home. In 1836 he married Mathilde Sturm, and all five of their children were born in the city. From 1837 he also taught at the Prague Polytechnic as a substitute lecturer on reduced pay. After a competitive examination held in 1839, he was appointed in 1841 professor of elementary mathematics and practical geometry there.

His most important friendship was with Bernard Bolzano, the philosopher and mathematician who had been removed from his chair at Prague in 1819 for his liberal views. Bolzano proposed Doppler for the Royal Bohemian Society of Sciences, which elected him an associate member in 1840 and an ordinary member in 1843. The Society’s meetings and its Abhandlungen gave Doppler a place to publish.

His papers of these years ranged widely. In 1837 he wrote on observing stars through a comet’s head and on bright spots on the Moon. In 1839 he published an analytical treatment of composite lines, surfaces and bodies. He also designed optical and surveying devices. None of this suggested what came next.

The 1842 paper and its aftermath, 1842–1847

On 25 May 1842 Doppler read a paper to the natural science section of the Bohemian Society, at a small meeting. Its title was Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (On the coloured light of the double stars and certain other stars of the heavens). It appeared in the Society’s Abhandlungen (5th series, vol. 2, pp. 465–482), and Doppler also had it printed separately in Prague that year.

The argument began with a ship. A ship steaming into the waves meets more crests per minute than one at rest, and one running before the waves meets fewer. Sound and light, Doppler said, should behave the same way. He worked out the received frequency for a moving observer and for a moving source, and gave a musical example: an observer approaching a source at one eighth of the speed of sound would hear a C as a D. Then he turned to the sky. Many double stars show a white primary and a coloured companion. He proposed that stars are white by nature and that orbital motion towards or away from us tints them blue or red. Lesson 07 explains why this astronomical application fails, while the principle itself is correct.

Overwork and illness

Doppler’s teaching load in Prague was crushing. Student numbers rose fast, and he examined several hundred students in writing and orally; some surveying exercises began at four in the morning. His strict marking drew complaints and an official reprimand, later withdrawn. In 1844 his health failed and he took sick leave. A lung disease that would kill him nine years later dates from this period.

He kept writing. Papers of 1845 and 1846 dealt with an improved reflecting microscope, a classification of colours and the astronomy of the fixed stars. In 1845 the Dutch meteorologist C. H. D. Buys Ballot tested the acoustic principle on the railway near Utrecht and confirmed it (Lesson 05), while disputing its use for star colours. In 1846 Doppler published in Poggendorff’s Annalen der Physik und Chemie a treatment in which source and observer move at the same time. In 1847 he left Prague for a professorship in the mining town of Schemnitz.

The acoustic Doppler effect

For the general reader. A wave source sends out crests at a steady rate. If you walk towards it, you meet the crests sooner, so you count more of them per second. If the source moves towards you, each crest starts closer to you than the last, so they arrive bunched together. Either way the pitch rises. Moving apart lowers it. This is Doppler’s principle, and for sound the details depend on who moves relative to the air.

Setting

A medium (air) is at rest. Sound travels through it at speed \(c\) in every direction. A source emits a wave of frequency \(f_s\), measured by a clock carried with the source, so its crests leave at intervals \(T_s=1/f_s\). The observer counts crests and reports a frequency \(f_r\). All motion is along the line joining source and observer, and all speeds are below \(c\).

Moving observer, source at rest

With the source at rest, the crests are spaced by the wavelength \(\lambda=c/f_s\) and move at \(c\) through the air. An observer moving towards the source at speed \(v_r\) meets them at relative speed \(c+v_r\), so

Moving source, observer at rest

Now the source moves towards the observer at speed \(v_s\). In one period \(T_s\) a crest travels \(cT_s\) while the source travels \(v_sT_s\) behind it before emitting the next crest. The spacing of crests ahead of the source is therefore \(\lambda'=(c-v_s)T_s\), and they still travel at \(c\):

The general formula follows by applying the two steps in turn: the moving source sets the wavelength in the air to \((c-v_s)/f_s\), and the moving observer then meets crests at relative speed \(c+v_r\). Doppler’s 1842 paper treated the two special cases separately; his 1846 paper combined them.

The asymmetry

For sound, a source approaching at speed \(v\) and an observer approaching at speed \(v\) do not give the same result. Writing \(\beta=v/c\),

The two agree to first order in \(\beta\) and differ at second order. The difference is real: the air defines a preferred frame. Doppler noticed that the two cases gave different results and drew the conclusion that it matters which body moves. For light in vacuum there is no medium, and relativity removes the asymmetry (Lesson 08).

Oblique motion, wind and the medium

For the general reader. When a car passes you, its horn drops in pitch. The drop is gradual, because what matters at each moment is how fast the car is closing or opening the distance to you, and that changes as it goes by. Wind, by contrast, does not change the pitch you hear from a horn standing still. Getting these two facts right settled the first great argument about Doppler’s principle.

Radial velocity

Let the source move with velocity \(\mathbf v_s\) and the observer with velocity \(\mathbf v_r\), relative to still air. Let \(\hat{\mathbf n}\) be the unit vector from the source at the moment of emission to the observer at the moment of reception. Over one period only the components along \(\hat{\mathbf n}\) change the path length, to first order in the period, so

For a source passing a fixed observer at speed \(v\), with \(\theta\) the angle between the velocity and the line to the observer at emission, \(f_r=f_s/(1-(v/c)\cos\theta)\). Far ahead \(\theta\to0\) and far behind \(\theta\to\pi\), so the pitch falls from \(f_s/(1-\beta)\) to \(f_s/(1+\beta)\). Most of the fall happens while the source is within a few times the closest-approach distance.

Wind

Put the source and observer on the \(x\)-axis, source at the origin, observer at positive \(x\). Suppose the air moves with uniform velocity \(w\) along \(+x\), and source and observer have ground velocities \(u_s,u_r\) along \(+x\). In the frame of the air the waves travel at \(c\), the source moves at \(u_s-w\) and the observer at \(u_r-w\). Applying the formula of Lesson 04 in that frame,

If \(u_s=u_r\), the ratio is exactly 1, whatever the wind. When source and observer are at rest on the ground, or move together, wind changes the wavelength and the travel time but not the number of crests received per second.

The tests

In 1845 Buys Ballot arranged a test on the railway near Utrecht. A locomotive pulled an open car carrying horn players who held a steady note, while musicians with trained ears listened by the track and judged the pitch as the train approached and receded. The pitch rose and fell as predicted. In the same year he argued that the astronomical application to star colours did not follow.

Ernst Mach, who had just taken his doctorate in Vienna, took up the question in papers of 1860 and 1861. He demonstrated the effect with a sounding pipe carried round on a rotating tube. An observer in the plane of rotation hears the pitch rise and fall, while one on the axis hears no change. Mach also identified the source of Josef Petzval’s objection (Lesson 11). Petzval had proved, correctly, that a moving medium does not change the frequency between a fixed source and fixed observer. He then treated a moving source as equivalent to a moving medium, which is the step the wind calculation above shows to be false.

Reflection: radar and ultrasound

For the general reader. A police radar gun and a hospital Doppler scanner work the same way. They send out a wave, a moving object reflects it, and the echo returns at a slightly different frequency. The object acts twice: once as a moving observer receiving the wave, and once as a moving source sending it back. So the shift is doubled.

Derivation

A transmitter at rest emits frequency \(f_0\). A reflector approaches it at speed \(v\) along the line of sight. As a moving observer, the reflector receives \(f_1=f_0(1+v/c)\). It re-emits this frequency as a moving source, so the transmitter receives \(f_2=f_1/(1-v/c)\).

The fractional shift \(2v/c\) is tiny for radio waves: a car at 30 m/s shifts a microwave signal by \(2\times10^{-7}\) of its frequency. Measuring \(f_2\) directly would be hopeless, so the receiver mixes the echo with the transmitted wave. The product contains the difference frequency \(|f_2-f_0|\), which falls in the audio range and is easy to count. Medical Doppler ultrasound works the same way; the beat frequency is often played through a loudspeaker.

Light, star colours and spectral lines

For the general reader. Doppler’s title promised to explain the colours of stars. That part was wrong. Stars are coloured mainly because they have different temperatures, and their speeds are far too small to change their colour visibly. What survived was the principle, once others saw that it should be applied to the sharp dark lines in a star’s spectrum rather than to its overall colour.

What Doppler claimed

Doppler argued that a white star moving fast enough towards us would look bluish, and one moving away reddish, because the visible band of its light would be pushed towards one end of the spectrum. He used this to explain the coloured companions of double stars, colour changes reported by earlier observers of double stars, and the red colour of many variable stars. He also treated light waves as longitudinal, like sound, even though Augustin Fresnel had shown they are transverse. Bolzano pointed out in 1847 that the principle applies equally to transverse waves.

Why the colours do not change

Stellar speeds are typically tens of kilometres per second, so \(\beta\sim10^{-4}\), and every frequency is multiplied by roughly \(1+\beta\). A deeper reason is that a star’s light is close to a blackbody spectrum, and the whole spectrum shifts together: infrared light is shifted into the visible as red light is shifted out.

Sketch. The quantity \(I_\nu/\nu^3\) (specific intensity divided by frequency cubed) is invariant under changes of inertial frame; it is proportional to the photon occupation number, which is a count and does not depend on the observer. For a blackbody, \(I_\nu/\nu^3\propto 1/(e^{h\nu/k_BT}-1)\). Substituting \(\nu=\nu'/D\) gives \(1/(e^{h\nu'/k_B(DT)}-1)\), the same form at temperature \(DT\). A star moving at 50 km/s changes its apparent temperature by under 0.02 per cent. The 2.7 K cosmic microwave background shows exactly this effect: our motion makes the sky slightly hotter in one direction and cooler in the other.

Spectral lines

In 1848 Hippolyte Fizeau, in Paris, made the decisive point independently: the measurable effect is a displacement of the narrow lines of a spectrum, whose rest wavelengths can be known from laboratory sources. For this reason the effect is called the Doppler–Fizeau effect in France. Astronomers write the shift as a redshift

with \(v_r\) the radial velocity, positive for recession. William Huggins in London attempted the first such measurement in 1868, on Sirius; his value was later revised, but the method was established. Aristarkh Belopolsky confirmed the optical Doppler effect in the laboratory in 1901, using light reflected from rapidly moving mirrors.

The relativistic Doppler effect

For the general reader. Light needs no medium, so there is no air to say whether the source or the observer is “really” moving. Einstein’s relativity of 1905 gives a single formula that depends only on their relative velocity. It contains a new effect that Doppler could not have predicted: a moving clock runs slow, so even a source moving sideways appears redshifted.

Derivation

A plane wave \(\cos(\mathbf k\cdot\mathbf x-\omega t)\) has a phase that every observer agrees on, because it counts crests. Hence \(k^\mu=(\omega/c,\mathbf k)\) is a four-vector. Let the source move with velocity \(v\) along the \(x\)-axis of the observer’s frame \(S\), and let \(S'\) be the source’s rest frame, where the light has angular frequency \(\omega_0\). In \(S\) the light reaches the observer travelling at angle \(\theta\) to the \(x\)-axis, so \(k_x=(\omega/c)\cos\theta\). The Lorentz transformation of the time component gives

Longitudinal case. For \(\theta=0\) (head-on approach),

and for recession \(\beta\to-\beta\). The formula is symmetric: an observer moving towards a source at rest gives the same result. It is the geometric mean of the two classical answers \(1+\beta\) and \(1/(1-\beta)\), as Exercise 8.1 shows.

Transverse case. For \(\theta=\pi/2\) the classical shift vanishes, but \(f=f_0/\gamma\): a redshift by the time-dilation factor. If instead the angle is measured in the source frame, so that the light is emitted at right angles to the motion in that frame, the observer sees a blueshift \(f=\gamma f_0\). The two statements describe different rays and do not conflict.

Testing time dilation

Woldemar Voigt derived the longitudinal formula in 1887 from a transformation that anticipated Lorentz’s. Einstein proposed in 1907 to test the second-order term with fast-moving ions. Herbert Ives and G. R. Stilwell did so in 1938. They observed light emitted by fast hydrogen ions forwards and, via a mirror, backwards. The two lines sit at \(\lambda_\pm=\gamma\lambda_0(1\mp\beta)\), so the first-order shifts cancel in their mean, which is displaced by

They found the predicted displacement. Later experiments with Mössbauer absorbers on rotors and with lasers on fast ion beams confirmed time dilation to a few parts in a million.

Radial velocities in astronomy

For the general reader. A star that wobbles towards and away from us shows spectral lines that shift back and forth. Doppler hoped to see orbital motion in double stars; astronomers eventually did, but in the lines, not the colours. The same wobble, made far smaller, revealed the first planet around a Sun-like star in 1995. On a larger scale, the redshifts of galaxies showed that the universe is expanding.

Spectroscopic binaries

In 1889 Edward Pickering at Harvard and Hermann Carl Vogel at Potsdam found the first binary stars detected only through periodic shifts of their lines. Consider a star of mass \(M_1\) in a circular orbit with a companion of mass \(M_2\), period \(P\), and orbital plane inclined at angle \(i\) to the sky. The star’s radial velocity varies sinusoidally with semi-amplitude \(K_1=v_1\sin i\), where \(v_1=2\pi a_1/P\) and \(a_1=aM_2/(M_1+M_2)\).

Derivation. Kepler’s third law gives \(G(M_1+M_2)P^2=4\pi^2a^3\). Then \(K_1^3=(2\pi/P)^3a^3\,M_2^3\sin^3 i/(M_1+M_2)^3\). Substitute \(a^3=G(M_1+M_2)P^2/4\pi^2\) and simplify to get \(K_1^3=2\pi G\,(M_2\sin i)^3/\big(P(M_1+M_2)^2\big)\).

Exoplanets

For a planet with \(M_2\ll M_1\), \(K_1\approx v_2\,(M_2/M_1)\sin i\), where \(v_2\) is the planet’s orbital speed. In 1995 Michel Mayor and Didier Queloz found a periodic radial-velocity signal of about 59 m/s with a period of 4.23 days in the star 51 Pegasi: a planet of roughly half Jupiter’s mass, orbiting closer to its star than Mercury does to the Sun. They shared the 2019 Nobel Prize in Physics for the discovery.

Galaxies and Hubble’s law

From 1912 Vesto Slipher measured large redshifts in the spectra of spiral nebulae. In 1929 Edwin Hubble combined such velocities with distance estimates and found \(v\approx H_0d\). For small redshifts \(v\approx cz\) and the Doppler reading is adequate. In general relativistic cosmology the redshift is not a Doppler shift of a velocity through space: it measures the expansion of space between emission and reception, \(1+z=a(t_{\text{obs}})/a(t_{\text{em}})\), where \(a\) is the cosmic scale factor.

Doppler broadening and laser cooling

For the general reader. In a hot gas, atoms fly in all directions at different speeds. Each one’s light is Doppler-shifted by a different amount, so a sharp spectral line is smeared out, and the width of the smear acts as a thermometer. Turned around, the same effect lets lasers slow atoms down: an atom moving towards a laser beam sees the light shifted into resonance, absorbs it, and is pushed back.

Doppler broadening

In a gas at temperature \(T\), the velocity component \(v_z\) along the line of sight of an atom of mass \(m\) is distributed as \(p(v_z)\propto\exp(-mv_z^2/2k_BT)\). To first order the atom emits at \(\nu=\nu_0(1+v_z/c)\). Changing variables,

The width grows as \(\sqrt{T/m}\). Light atoms in hot gases give the broadest lines. Astronomers use the width to measure temperatures and turbulent motions in stellar atmospheres and interstellar clouds.

Doppler cooling

In 1975 Theodor Hänsch and Arthur Schawlow, and independently David Wineland and Hans Dehmelt for trapped ions, proposed cooling atoms with light. Tune a laser slightly below an atomic resonance (red detuning). An atom moving towards the beam sees the light Doppler-shifted up towards resonance and absorbs strongly, each absorption giving it a momentum kick \(\hbar k\) opposite to its motion. Spontaneous emission goes in random directions and averages to no net kick. With beams from all six directions, the atom feels a friction force, called optical molasses.

Cooling is limited by the randomness of the kicks, which heats the atoms. Balancing friction against this heating gives a minimum temperature when the detuning equals half the natural linewidth \(\Gamma\) (in angular frequency units).

In 1988 William Phillips’s group measured sodium atoms colder than this limit, which led to the discovery of sub-Doppler cooling mechanisms. Steven Chu, Claude Cohen-Tannoudji and Phillips shared the 1997 Nobel Prize in Physics for laser cooling and trapping.

Schemnitz, Vienna and the Petzval dispute, 1847–1852

In 1847 Doppler accepted the professorship of mathematics, physics and mechanics at the Mining and Forestry Academy in Schemnitz, now Banská Štiavnica in Slovakia, with the title of Bergrat. The revolutions of 1848 and the Hungarian uprising made the town unsafe, and he returned to Vienna. In 1848 he was elected to the new Imperial Academy of Sciences in Vienna, and the University of Prague gave him an honorary doctorate.

In 1849 he succeeded his old mentor Simon Stampfer as professor of practical geometry at the Vienna Polytechnic. On 17 January 1850 an imperial decree named him the first director of the Physical Institute of the University of Vienna, founded to train physics teachers, and professor of experimental physics. Among his students was the Augustinian friar Gregor Mendel, sent to Vienna by his abbot from 1851 to 1853.

The dispute

In 1852 the mathematician Josef Petzval, famous for the portrait lens he had designed in 1840, attacked Doppler’s principle in a series of lectures to the Academy. Petzval proved a correct theorem: a wave equation in a uniformly moving medium gives no change of frequency between a fixed source and a fixed receiver. He then argued that Doppler’s result was wrong, and he dismissed Doppler’s formula as a mere fraction. Doppler replied in May 1852 before a large audience. The exchange ended with Petzval’s third lecture in October, when Doppler was already too ill to respond. As Lesson 05 shows, the two men were describing different situations. Ernst Mach made this clear in 1860 and 1861.

Venice, and the long road to acceptance

By late 1852 Doppler’s lung disease was severe. In November he travelled to Venice, then part of the Austrian Empire, hoping that the milder climate would help. It did not. His wife joined him as his condition worsened, and he died in Venice on 17 March 1853, aged 49. He is buried in the cemetery on the island of San Michele. Anton Schrötter read an obituary to the Vienna Academy on 30 May 1853.

Acceptance

The principle won general acceptance only in the 1870s, after spectroscopy gave it a practical use. Fizeau’s 1848 remark about spectral lines, Huggins’s stellar measurements from 1868, the detection of the Sun’s rotation from the shifts of lines at opposite edges of its disc, and Belopolsky’s 1901 laboratory test turned a disputed proposal into a standard tool. Relativity then changed its form at second order (Lesson 08).

Legacy

• Astronomy: radial velocities of stars, spectroscopic binaries and stellar masses, exoplanets, galactic rotation curves and cosmological redshifts all rest on the principle.
• Technology: radar, weather radar, medical Doppler ultrasound, laser Doppler velocimetry and satellite navigation all measure Doppler shifts.
• Atomic physics: Doppler broadening, Doppler-free spectroscopy and laser cooling.
• Honours: a bust by G. Leiseck was unveiled in the arcaded courtyard of the University of Vienna in 1901. His collected papers appeared in Ostwald’s Klassiker in 1907, edited by Hendrik Lorentz. A lunar crater bears his name, and Austria’s Christian Doppler Research Association funds laboratories linking universities and industry.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Christian Andreas Doppler, MacTutor History of Mathematics.
2. Christian Doppler, Wikipedia.
3. F. Klemm, Doppler, Christian, Neue Deutsche Biographie 4 (1959), and E. Lommel, Allgemeine Deutsche Biographie 5 (1877), Deutsche Biographie.
4. Dictionary of Scientific Biography, Doppler, Christian.
5. Christian Doppler’s Prague years, Acta Universitatis Carolinae 46 (2005).
6. Christian Doppler Fonds, Josef Maximilian Petzval (on the 1852 dispute).
7. D. D. Nolte, A commotion in the stars: the history of the Doppler effect, Galileo Unbound.

Physics

1. C. Doppler, Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels, Abh. k. böhm. Ges. Wiss. (5) 2 (1842), 465–482. Text on Wikisource.
2. Abhandlungen von Christian Doppler, ed. H. A. Lorentz, Ostwalds Klassiker 161, Leipzig, 1907.
3. Doppler effect and Relativistic Doppler effect, Wikipedia.
4. A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 17 (1905), §7.
5. H. E. Ives and G. R. Stilwell, An experimental study of the rate of a moving atomic clock, J. Opt. Soc. Am. 28 (1938).
6. M. Mayor and D. Queloz, A Jupiter-mass companion to a solar-type star, Nature 378 (1995).
7. G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics, Wiley, 1979 (invariance of \(I_\nu/\nu^3\), line broadening).
8. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer, 1999.