A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically i

Question

A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to

Professor PC Thomas Classes · Answer

(d) : The fundamental frequency $(f)$ is related to length of the pipe $(L)$ and speed of sound in air $(v)$ as $ f=\frac{v}{\lambda} $ \ For a pipe open at both ends, the fundamental frequency corresponding to a wavelength $\lambda$ that is twice the length of the pipe is \ $f=\frac{v}{2 L}$ \ When the pipe is dipped vertically in water so that half of its length is submerged, the new length of the air column becomes \ $L^{\prime}=\frac{L}{2}$ \ For the new length of the air column, the wavelength $\lambda^{\prime}$ is given by 
$\lambda^{\prime}=4 L^{\prime}$
$\lambda^{\prime}=4 	imes \frac{L}{2}$  (ii)
$\lambda^{\prime}=2 L$

The new fundamental frequency, 
$ f^{\prime}=\frac{v}{\lambda^{\prime}}=\frac{v}{2 L} $
 Comparing eq. (i) and (iii), we get $ f^{\prime}=f $