Doppler effect

Experiment by Buys Ballot (1845) depicted on a wall in Utrecht (2019)

Doppler first proposed this effect in 1842 in his treatise "Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels" (On the coloured light of the binary stars and some other stars of the heavens).[6] The hypothesis was tested for sound waves by Buys Ballot in 1845.[p 1] He confirmed that the sound's pitch was higher than the emitted frequency when the sound source approached him, and lower than the emitted frequency when the sound source receded from him. Hippolyte Fizeau discovered independently the same phenomenon on electromagnetic waves in 1848 (in France, the effect is sometimes called "effet Doppler-Fizeau" but that name was not adopted by the rest of the world as Fizeau's discovery was six years after Doppler's proposal).[p 2][7] In Britain, John Scott Russell made an experimental study of the Doppler effect (1848).[p 3]

In classical physics, where the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the relationship between observed frequency ${\displaystyle f}$ and emitted frequency ${\displaystyle f_{\text{0}}}$ is given by:[8]

$${\displaystyle f=\left({\frac {c\pm v_{\text{r}}}{c\pm v_{\text{s}}}}\right)f_{0}}$$

where

• ${\displaystyle c}$ is the propagation speed of waves in the medium;
• ${\displaystyle v_{\text{r}}}$ is the speed of the receiver relative to the medium, added to ${\displaystyle c}$ if the receiver is moving towards the source, subtracted if the receiver is moving away from the source;
• ${\displaystyle v_{\text{s}}}$ is the speed of the source relative to the medium, added to ${\displaystyle c}$ if the source is moving away from the receiver, subtracted if the source is moving towards the receiver.

Note this relationship predicts that the frequency will decrease if either source or receiver is moving away from the other.

Equivalently, under the assumption that the source is either directly approaching or receding from the observer:

$${\displaystyle {\frac {f}{v_{wr}}}={\frac {f_{0}}{v_{ws}}}={\frac {1}{\lambda }}}$$

where

• ${\displaystyle v_{wr}}$ is the wave's velocity relative to the receiver;
• ${\displaystyle v_{ws}}$ is the wave's velocity relative to the source;
• ${\displaystyle \lambda }$ is the wavelength.

If the source approaches the observer at an angle (but still with a constant velocity), the observed frequency that is first heard is higher than the object's emitted frequency. Thereafter, there is a monotonic decrease in the observed frequency as it gets closer to the observer, through equality when it is coming from a direction perpendicular to the relative motion (and was emitted at the point of closest approach; but when the wave is received, the source and observer will no longer be at their closest), and a continued monotonic decrease as it recedes from the observer. When the observer is very close to the path of the object, the transition from high to low frequency is very abrupt. When the observer is far from the path of the object, the transition from high to low frequency is gradual.

If the speeds ${\displaystyle v_{\text{s}}}$ and ${\displaystyle v_{\text{r}}\,}$ are small compared to the speed of the wave, the relationship between observed frequency ${\displaystyle f}$ and emitted frequency ${\displaystyle f_{\text{0}}}$ is approximately[8]

Observed frequency	Change in frequency
${\displaystyle f=\left(1+{\frac {\Delta v}{c}}\right)f_{0}}$	${\displaystyle \Delta f={\frac {\Delta v}{c}}f_{0}}$

where

• ${\displaystyle \Delta f=f-f_{0}}$
• ${\displaystyle \Delta v=-(v_{\text{r}}-v_{\text{s}})}$ is the opposite of the velocity of the receiver relative to the source: it is positive when the source and the receiver are moving towards each other.

Proof

Given ${\displaystyle f=\left({\frac {c+v_{\text{r}}}{c+v_{\text{s}}}}\right)f_{0}}$

we divide for ${\displaystyle c}$

$${\displaystyle f=\left({\frac {1+{\frac {v_{\text{r}}}{c}}}{1+{\frac {v_{\text{s}}}{c}}}}\right)f_{0}=\left(1+{\frac {v_{\text{r}}}{c}}\right)\left({\frac {1}{1+{\frac {v_{\text{s}}}{c}}}}\right)f_{0}}$$

Since ${\displaystyle {\frac {v_{\text{s}}}{c}}\ll 1}$ we can substitute the geometric expansion:

$${\displaystyle {\frac {1}{1+{\frac {v_{\text{s}}}{c}}}}\approx 1-{\frac {v_{\text{s}}}{c}}}$$

• 
  Stationary sound source produces sound waves at a constant frequency f, and the wave-fronts propagate symmetrically away from the source at a constant speed c. The distance between wave-fronts is the wavelength. All observers will hear the same frequency, which will be equal to the actual frequency of the source where f = f₀.
• 
  The same sound source is radiating sound waves at a constant frequency in the same medium. However, now the sound source is moving with a speed υ_s = 0.7 c. Since the source is moving, the centre of each new wavefront is now slightly displaced to the right. As a result, the wave-fronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source. An observer in front of the source will hear a higher frequency f = c + 0/c – 0.7c f₀ = 3.33 f₀ and an observer behind the source will hear a lower frequency f = c − 0/c + 0.7c f₀ = 0.59 f₀.
• 
  Now the source is moving at the speed of sound in the medium (υ_s = c). The wave fronts in front of the source are now all bunched up at the same point. As a result, an observer in front of the source will detect nothing until the source arrives and an observer behind the source will hear a lower frequency f = c – 0/c + c f₀ = 0.5 f₀.
• 
  The sound source has now surpassed the speed of sound in the medium, and is traveling at 1.4 c. Since the source is moving faster than the sound waves it creates, it actually leads the advancing wavefront. The sound source will pass by a stationary observer before the observer hears the sound. As a result, an observer in front of the source will detect nothing and an observer behind the source will hear a lower frequency f = c – 0/c + 1.4c f₀ = 0.42 f₀.

With an observer stationary relative to the medium, if a moving source is emitting waves with an actual frequency ${\displaystyle f_{0}}$ (in this case, the wavelength is changed, the transmission velocity of the wave keeps constant; note that the transmission velocity of the wave does not depend on the velocity of the source), then the observer detects waves with a frequency ${\displaystyle f}$ given by

$${\displaystyle f=\left({\frac {c}{c\pm v_{\text{s}}}}\right)f_{0}}$$

A similar analysis for a moving observer and a stationary source (in this case, the wavelength keeps constant, but due to the motion, the rate at which the observer receives waves and hence the transmission velocity of the wave [with respect to the observer] is changed) yields the observed frequency:

$${\displaystyle f=\left({\frac {c\pm v_{\text{r}}}{c}}\right)f_{0}}$$

Assuming a stationary observer and a source moving at the speed of sound, the Doppler equation predicts a perceived momentary infinite frequency by an observer in front of a source that is traveling at the speed of sound. All the peaks are at the same place, so the wavelength is zero and the frequency is infinite. This overlay of all the waves produces a shock wave which for sound waves is known as a sonic boom.

When the source moves faster than the wave speed the source outruns the wave. The equation gives negative frequency values, which have no physical sense in this context (no sound at all will be heard by the observer until the source passes past them).

Lord Rayleigh predicted the following effect in his classic book on sound: if the observer were moving from the (stationary) source at twice the speed of sound, a musical piece previously emitted by that source would be heard in correct tempo and pitch, but as if played backwards.[9]

Since 1968 scientists such as Victor Veselago have speculated about the possibility of an inverse Doppler effect. The size of the Doppler shift depends on the refractive index of the medium a wave is traveling through. But some materials are capable of negative refraction, which should lead to a Doppler shift that works in a direction opposite that of a conventional Doppler shift.[22] First experiment that detected this effect was conducted by Nigel Seddon and Trevor Bearpark in Bristol, United Kingdom in 2003.[p 6] Later inverse Doppler effect was observed in some inhomogeneous materials and predicted inside Vavilov–Cherenkov cone.[23]

• Doppler, C. (1842). Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens). Publisher: Abhandlungen der Königl. Böhm. Gesellschaft der Wissenschaften (V. Folge, Bd. 2, S. 465–482) [Proceedings of the Royal Bohemian Society of Sciences (Part V, Vol 2)]; Prague: 1842 (Reissued 1903). Some sources mention 1843 as year of publication because in that year the article was published in the Proceedings of the Bohemian Society of Sciences. Doppler himself referred to the publication as "Prag 1842 bei Borrosch und André", because in 1842 he had a preliminary edition printed that he distributed independently.
• "Doppler and the Doppler effect", E. N. da C. Andrade, Endeavour Vol. XVIII No. 69, January 1959 (published by ICI London). Historical account of Doppler's original paper and subsequent developments.
• David Nolte (2020). The fall and rise of the Doppler effect. Physics Today, v. 73, pgs. 31 - 35. DOI: 10.1063/PT.3.4429
• Adrian, Eleni (24 June 1995). "Doppler Effect". NCSA. Archived from the original on 12 May 2009. Retrieved 2008-07-13.

• Media related to Doppler effect at Wikimedia Commons
• Doppler Effect, ScienceWorld