Question

Two whistles A and B produce notes of frequencies 600 Hz and 596 Hz respectively. There is a listener at the midpoint of the line joining them. Both the whistles and the listener start moving with the speed of 30m/s in the same direction. If the speed of the sound is 330m/s, the number of beats that will be heard by the listener are:
A) 2
B) 4
C) 6
D) 8

Answer 1

Here two sources of sound are given which are whistle A and whistle B and a single observer i.e. listener. After a while both sources and the observer start moving with the same speed, now by considering the Doppler effect we can find the frequencies which are observed by the observer due to both whistles. By using the frequencies observed by the listener we can find the number of beats.
Formula used:

& {{f}_{Beat}}={{f}_{1}}-{{f}_{2}} \\\ & f={{f}_{S}}\left( \dfrac{v\pm {{v}_{O}}}{v\pm {{v}_{S}}} \right) \\\ \end{aligned}$$ **Complete answer:** Let us first draw the diagram for better understanding  In the above diagram A and B are the sources i.e. whistles and L represents the listener. $${{v}_{A}},{{v}_{B}}\text{ and }{{v}_{L}}$$are the speed of whistle A, whistle B and the listener respectively. As both sources and the listener are moving in same direction with same velocity, therefore we can write $${{v}_{A}}={{v}_{B}}={{v}_{L}}={{v}_{C}}$$ Here $${{v}_{C}}$$denotes common speed of A, B and L. Now the Doppler Effect gives us a relation between the frequency observed by the observer and the frequency of the source. If f is the frequency observed by observer and $${{f}_{S}}$$is the frequency of the sources and both are moving i.e. both are in motion, then according to Doppler Effect $$f={{f}_{S}}\left( \dfrac{v\pm {{v}_{O}}}{v\pm {{v}_{S}}} \right)$$ $${{v}_{O}}$$ is the speed of the observer, it will be negative if the observer is moving away from the source and it will be positive in case it is moving towards the source. $${{v}_{S}}$$is the speed of the source, it will be positive in case it is moving away from the observer and negative in case it is moving towards the observer. v is the speed of the sound. Now let us calculate the frequency which will be observed by the listener due to whistle A. As we can see from the diagram above that A is moving towards the listener, therefore its speed will be taken negative. On the other hand, the listener is moving away from the source therefore its speed will be negative too. If $${{f}_{1}}$$ is the frequency observed by listener due to A and $${{f}_{A}}$$is the frequency of A, then from Doppler Effect $$\begin{aligned} & {{f}_{1}}={{f}_{A}}\left( \dfrac{v-{{v}_{C}}}{v-{{v}_{C}}} \right) \\\ & \Rightarrow {{f}_{1}}={{f}_{A}}\text{ }...............\text{(i)} \\\ \end{aligned}$$ Now considering the second case of Whistle B and the listener, as B is moving away from L therefore it will be positive and L is moving towards B therefore it will be Positive too. If $${{f}_{2}}$$is the frequency observed by the listener due to B and $${{f}_{B}}$$is the frequency of the whistle B, then from Doppler Effect we have $$\begin{aligned} & {{f}_{2}}={{f}_{B}}\left( \dfrac{v+{{v}_{C}}}{v+{{v}_{C}}} \right) \\\ & \Rightarrow {{f}_{2}}={{f}_{B}}\text{ }..................\text{(ii)} \\\ \end{aligned}$$ The number of beats is given as the difference in the frequencies producing it. $${{f}_{Beat}}={{f}_{1}}-{{f}_{2}}$$ Using equations (i) and (ii), we get $${{f}_{Beat}}={{f}_{A}}-{{f}_{B}}$$ We have given the values of frequencies $${{f}_{A}}\text{ and }{{f}_{B}}$$which are 600 Hz and 596 Hz respectively. Therefore above equation becomes $$\begin{aligned} & {{f}_{Beat}}=600-596 \\\ & {{f}_{Beat}}=4Hz \\\ \end{aligned}$$ Hence 4 beats will be heard by the listener per second. **Option B is the correct answer.** **Note:** Beats are produced by the interference of sound waves having nearly the same frequency and travelling in the same direction. In case the sources, whistles A and B travelled in opposite directions then no beats would have been produced. Here we didn’t substitute the values of speed of sound or speed of listener and the sources in the formula because terms in denominator and numerator cancelled each other giving us value 1.