Answer to Question #164229 in Optics for enam

Question

   Two automobiles are equipped with the same single-frequency horn. When one is at rest and the

      other is moving toward an observer at 15ms^-1, a beat frequency of 5.5Hz is heard. What is the

      frequency the horns emit? Assume T = 20°C.

Answer 1

The beat frequency can be found as follows:

$$f_{beat}=f'-f,$$

here, $f$ is the original frequency and $f'$ is the Doppler shift frequency.

The Doppler shift frequency can be written as follows:

$$f'=f(\dfrac{v}{v-v_s}).$$

Substituting $f'$ into the previous equation, we get:

$$f_{beat}=f(\dfrac{v}{v-v_s})-f=f[(\dfrac{v}{v-v_s})-1],$$ $$f=\dfrac{f_{beat}}{[(\dfrac{v}{v-v_s})-1]}=\dfrac{5.5\ Hz}{[(\dfrac{343\ \dfrac{m}{s}}{343\ \dfrac{m}{s}-15\ \dfrac{m}{s}})-1]}=120.3\ Hz.$$