Question

Two factories are sounding their sirens at $800Hz$. A man goes from one factory to another at a speed of $2m{{s}^{-1}}$. The velocity of sound is $320m{{s}^{-1}}$. The number of beats heard by the person in one second will be
$\begin{align}
  & A)10 \\
 & B)2 \\
 & C)8 \\
 & D)4 \\
\end{align}$

Answer 1

Hint: Using the concept of Doppler effect, both maximum apparent frequency as well as minimum apparent frequency is calculated. Maximum apparent frequency is related to the sum of velocity of the siren and the velocity of the listener whereas minimum apparent frequency is related to the difference between velocity of the siren and the velocity of listener. Number of beats heard by the person in one second is equal to the difference between maximum apparent frequency and minimum apparent frequency.
Formula used:
\[\begin{align}
  & {{n}_{\max }}=\left( \dfrac{V+{{V}_{L}}}{V} \right)n \\
 & {{n}_{\min }}=\left( \dfrac{V-{{V}_{L}}}{V} \right)n \\
\end{align}\]

Complete answer:
Using the concept of Doppler effect, we know that maximum apparent frequency is related to the sum of velocity of the siren and the velocity of the listener whereas minimum apparent frequency is related to the difference between velocity of the siren and the velocity of listener. If ${{n}_{\max }}$ and ${{n}_{\min }}$ represent these apparent frequencies respectively, they are given by
${{n}_{\max }}=\left( \dfrac{V+{{V}_{L}}}{V} \right)n$
${{n}_{\min }}=\left( \dfrac{V-{{V}_{L}}}{V} \right)n$
where
$n$ is the frequency of source (siren)
$V$ is the velocity of sound
${{V}_{L}}$ is the velocity of the listener
Let this set of equations be denoted by M.
From the question, we have
$\begin{align}
  & n=800Hz \\
 & V=320m{{s}^{-1}} \\
 & {{V}_{L}}=2m{{s}^{-1}} \\
\end{align}$
Substituting these values in the set of equations denoted by M, we have
${{n}_{\max }}=\left( \dfrac{320+2}{320} \right)\times 800=805Hz$
${{n}_{\min }}=\left( \dfrac{320-2}{320} \right)\times 800=795Hz$
Let this set of equations be denoted by N.
Now, we know that the number of beats heard by the person in one second is equal to the difference between maximum apparent frequency and minimum apparent frequency. If $n{}_{b}$ denotes the number of beats per second, then, we have
${{n}_{b}}={{n}_{\max }}-{{n}_{\min }}$
Let this be equation 1.
Now, substituting the set of equations denoted by N in equation 1, we have
${{n}_{b}}={{n}_{\max }}-{{n}_{\min }}\Rightarrow 805-795=10$

Therefore, the correct answer is option $A$.

Note:
When there is a relative motion between a source of sound and a listener, the apparent frequency of sound heard by the listener is different from the actual frequency of sound produced by the source. This phenomenon is termed as Doppler’s effect in sound. Also, the change in frequency depends on whether the source is moving towards the observer or the observer is moving towards the source.