The Doppler effect. If a sound source approaches or moves away from the listener's ear, the sound waves emanating from the sound source will reach the listener's ear shortened or elongated. Since the wavelength determines the pitch of the sound, the sound will be perceived as higher or lower in pitch.

A person stands next to a straight railway track, and a train passes by at a speed of $c=60 \frac{\mathrm{km}}{\text { hour }}$. The train's whistle emits a sound of $N=2048$ Hz when the train is stationary. What pitch of sound will the observer hear as the train approaches and then moves away? The speed of sound in air is $V=340 \frac{\mathrm{m}}{\mathrm{sec}}$.
Let the original wavelength be $\lambda$. This means that at the moment when the wave motion center emits a compression, the previously emitted compression is $\lambda$ meters away from the wave motion center. However, if this center itself is moving at a speed of $c$ and thus travels a distance of $c T$ in $T$ seconds, then the previously emitted wave is not $\lambda$, but $\lambda - c T$ or $\lambda + c T$ away from the wave motion center, depending on whether it is in front of or behind the direction of motion. The new wavelengths are therefore

$$
\lambda^{\prime} = \lambda - c T \quad \text{and} \quad \lambda^{\prime \prime} = \lambda + c T
$$

Since

$$
\lambda^{\prime} = \frac{V}{N^{\prime}} \quad \text{and} \quad \lambda^{\prime \prime} = \frac{V}{N^{\prime \prime}}
$$

it follows that

$$
\begin{aligned}
& N^{\prime} = \frac{V}{\lambda - c T} = V : \left( \frac{V}{N} - \frac{c}{N} \right) = N \cdot \frac{V}{V - c} = N \cdot \frac{1}{1 - \frac{c}{V}} \\
& N^{\prime \prime} = \frac{V}{\lambda + c T} = V : \left( \frac{V}{N} + \frac{c}{N} \right) = N \cdot \frac{V}{V + c} = N \cdot \frac{1}{1 + \frac{c}{V}}
\end{aligned}
$$

Substituting the values, we get $N^{\prime} = 2153$ and $N^{\prime \prime} = 1952$. We note that textbooks typically provide the following formula:

$$
N^{\prime} = N \left(1 + \frac{c}{V}\right) \quad \text{and} \quad N^{\prime \prime} = N \left(1 - \frac{c}{V}\right)
$$

This is only a first approximation, which we obtain by writing the previous exact formula as $N^{\prime} = N \left(1 - \frac{c}{V}\right)^{-1}$ and expanding it according to the binomial theorem, keeping only the first two terms. However, the formula given in textbooks is correct when the observer is moving towards the sound source.

The problem was solved (according to the textbook formula) by: Bánó L., Bayer N., Blum J., Erdélyi I., Ehrenstein P., Esztó P., Fekete M., Frank A., Freund E., Fodor H., Heimlich P., Kiss E., V. ker. áll. főgim. math. köre, Szilas O., Sztrokay K.