# Doppler effect

Corresponding Wikipedia article: Doppler effect

## Definition

The Doppler effect emerges as a result of the different speeds of the source and the receiver of beams (waves). This effect relates to the mechanical (acoustic, the magnetic) waves, and to the light beams. This effect manifests as a variation in one of three parameters of the received waves (beams) with respect to their initial values:

• Frequency and wavelength.
• Velocity of wave or beam.
• Rate of time.

These variations are equivalent and mutually exclusive. For any medium (including aether) the actual variating parameters are the frequency and wavelength of the received wave. The relative variation in a wavelength $$\lambda$$, a time period $$T$$, a frequency $$f$$ and a speed of wave (beam) $$c$$, are interrelated by the equation: $\frac{\lambda+\Delta\lambda}{\lambda}=\frac{T+\Delta T}{T}=\frac{f}{f+\Delta f}=\frac{c}{c+\Delta c}\tag{1}$

Let the source of waves (beams) moves with respect to the receiver (observer) at a velocity $$v$$ and radiates the waves (beams) at a speed of $$c$$ (regardless of the medium anisotropy). At some point, the wave (beam) reaches the receiver in a time $$T$$. The source displacement from this point within a time period $$T$$ causes a wave (beam) delay by $$\Delta T$$. These quantities are interrelated according to the law of cosines: $c(T+\Delta T)=T\sqrt{c^2+v^2-2cv\;\cos\Theta}\tag{2}$ This general equation (2) has two special cases: the longitudinal and the transverse Doppler effects.

## Longitudinal effect ($$\Theta$$ = 0° или $$\Theta$$ = 180°)

$\frac{T+\Delta T}{T}=\frac{c\pm v}{c}\tag{3}$ When $$v\ll c$$, the following approximate equality is valid with high accuracy: $\frac{c\pm v}{c}\approx\frac{c}{c\mp v}\tag{4}$ Thus, the longitudinal Doppler effect at $$v\ll c$$ can be represented by a sum of the source and the beam (wave) velocities.

The beam sources and receivers are the elementary particles (mostly electrons), which are also the rotating beams and are varying their own speeds when passing from one medium to another (see "Radiation and photons"). In this regard, for the optical Doppler effect the actual value of the particle velocity $$v$$ at its real speed $$V$$ is determined by the law with a refractive index $$n$$: $v=\frac{V}{n}\tag{5}$ This law causes the entanglement of light by the moving aether in the Fizeau experiment (see "Interference").

## Transverse effect ($$\Theta$$ = 90°)

$\frac{T+\Delta T}{T}=\sqrt{1+\frac{v^2}{c^2}}\tag{6}$ The transverse optical Doppler effect is called the relative velocity time dilation. In fact, the frequency and wavelength of the relativistic particle is varying for an observer. This formula is practically applicable for a range of angles $$\Theta$$ in a neighborhood of 90°, in which this effect appears in the particle accelerators. The Ives–Stilwell experiment and other [1] confirmed the transverse Doppler effect for a velocity of about 106...107 m/s.

## References

1. Л.А. Победоносцев, Я.М. Крамаровский, П.Ф. Паршин, Б.К. Селезнёв, А.Б. Берёзин Экспериментальное определение доплеровского смещения линий водорода на пучках ионов H2+ в диапазоне энергий 150-2000 кэВ. Журнал технической физики, Том 59, в. 3, 1989 г., с. 84.