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Physics Question on Waves

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their PthP^{th} overtone is

A

p+12p\frac{p+1}{2p}

B

p+12p+1\frac{p+1}{2p+1}

C

2(p+1)2p+1\frac{2\left(p+1\right)}{2p+1}

D

p2p+1\frac{p}{2p+1}

Answer

2(p+1)2p+1\frac{2\left(p+1\right)}{2p+1}

Explanation

Solution

Frequency of pp th overtone of open organ pipe
=(p+1)v2L=(p+1) \frac{v}{2 L}...(i)
where, L=L= length of organ pipe and that of closed organ pipe is
=(2p+1)V4L=(2 p+1) \frac{V}{4 L}...(ii)
So, the ratio of pp th overtone of open to closed
=(p+1)v2L(2p+1)v4L=\frac{(p+1) \frac{v}{2 L}}{(2 p+1) \frac{v}{4 L}} (using Eqs. (i) and (ii)
=2(p+1)(2p+1)=\frac{2(p+1)}{(2 p+1)}