Doppler effect

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DOPPER effect
L. G. Aslamazov

In a famous anecdote about the sign of physicist Robert Wood, it is said that when Bud’s car passed an intersection on a red light and was stopped, Bud explained his mistake to the effective Doppler: for a moving person, the frequency of the light source changes and the traffic light signal seemed green to him. They say that the policeman turned out to be no layman. He knew that a noticeable change in frequency could only occur at a very high speed of the car, close to the speed

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Sveta. Pozny the policeman did not argue with the meaning of the physicist and simply fined him for speeding. And although this venerable story may discourage us from ever trying to explain anything in our lives by the Doppler effect, nevertheless, apparently everyone has encountered this effect. When a buzzing train passes us or an airplane flies over us, we hear a sudden change in the tone of the sound. This is also Doppler effective.

Why does the frequency of the sound of a moving source change? Let’s try to figure it out. Let’s draw a picture of waves emitted by a moving source. The speed of sound propagation in the medium does not depend on the speed of the source, and the wave pattern will be a set of circles, the centers of which will be mixed in the direction of movement of the source (picture at the beginning of the article). Let now the observed stand, for example, behind the source, so that the source moves away from it with speed $v$. Let us assume that initially the distance between the source and the observed was equal to $u$, ($u$ is the speed of sound). then after 1 second the distance between them will be $u + v$ and at this distance all $\omega$ maxima created by the source per unit of time decrease ($\omega$ - sound frequency). Before observing for 1 second only those maxima that were initially at a distance of $u$ from it. Therefore, the frequency of the sound $\omega_1$ perceived by the observable will be less, and

$$\frac{\omega_1}{\omega} = \frac{u}{u + v},$$

from where $\omega_1 = \omega \frac{1}{1 + \frac{v}{u}}$

If the source approaches the application, then similarly we obtain

$$\omega_1 = \omega \frac{1}{1 - \frac{v}{u}}$$

If it is not the source, but the receiver of the sound that moves, the same change in frequency occurs qualitatively, but quantitatively the result will be somewhat different. If the applicator moves towards the source with a speed of $v$, then in 1 second it will pass not mismo $\omega$, but a mismo of a larger number of maxima $\omega_2$, and $\frac{\omega_2}{\omega} = \frac{u + v}{u}$, hence $\omega_2 = \omega \left(1 + \frac{v}{u}\right)$.

If the receiver moves away from the source, it will note a lower frequency

$$\omega_2 = \omega \left(1 - \frac{v}{u}\right).$$

This change in frequency when a source or application moves is called the Doppler effect. Considering $v$ positive in both cases when the distance increases, we can write

$$\omega_1 = \frac{1 \omega}{1 + \frac{v}{u}}, \quad \omega_2 = \omega \left(1 - \frac{v}{u}\right).$$

If the speed of the source is small compared to the speed of sound, then, using formulas for approximate calculations, $\frac{1}{1 + \frac{v}{u}}$ can be approximately replaced by $1 - \frac{v}{u}^*$, and then $\omega_1 \approx \omega_2$. However, in modern

*) Check this for example for $\frac{u}{v} = 0.01$. 0.05: 0.1; 0.2

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The technical speed of the source or receiver is not at all small compared to the speed of sound (for example, the speed of aircraft), and then the Doppler efficiency in both cases is significantly different.

Now that we have understood the Doppler efficiency for sound waves, it would seem that there is nothing simpler than writing similar formulas for light - it is enough to put the speed of sound $u$ together with the speed of light $c$ everywhere, and the answer is ready. However, here we are faced with an amazing obstruction: such formulas cannot be correct, since they contradict Galileo’s principle of relative.

According to this principle, it is impossible to distinguish uniform rectilinear motion from a state of rest, in other words, any physical phenomena must proceed in the same way and any physical experiments must give the same result in all coordinate systems moving relative to each other at constant speeds (that is, in inertial systems). We can consider, for example, the case of movement of a source and a stationary observable in a reporting system moving with the source, and then, according to our formulas, the change in the frequency of light in both systems would be different.

In the propagation of sound, this generality should not confuse us. After all, here, in addition to the source and receiver, the medium in which the sound propagates plays a role. The movement of the source to the receiver and the receiver to the source gives different results precisely because in the first case the relative movement