question_answer

Sound waves are passing through two routes-one in straight path and the other along a semicircular path of radius r and are again combined into one pipe and superposed as shown in the figure. If the velocity of sound waves in the pipe is \[\upsilon ,\]then frequencies of resultant waves of maximum amplitude will be integral multiples of

A)  \[\frac{v}{r(\pi -2)}\]

B)  \[\frac{v}{r(\pi -1)}\]

C)  \[\frac{2v}{r(\pi -1)}\]

D)  \[\frac{v}{r(\pi +1)}\]

Correct Answer: A

Solution :

 Path difference \[=(\pi r-2r)\] \[=(\pi -2)r=n\lambda \](where\[f=\]frequency) \[v=f\times \lambda \] \[\Rightarrow \] \[\frac{v}{\lambda }=f\Rightarrow f=\left[ \frac{v}{(\pi -2)r} \right]n\] So, multiples integral\[=\left[ \frac{v}{(\pi -2)r} \right]\]