Question

A cylindrical tube open at both ends has a fundamental frequency$f$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now

• A. $\frac{f}{2}$
• B. $f$
• C. $\frac{3f}{4}$
• D. $2f$

Answer 1

The correct option is B

$f$

Step 1. Finding frequency before tube being dipped in water

The tube is with two open ends, So an integer number of half wavelength have to fit into the tube of length $L$.

$L=\frac{n\lambda }{2}$

where $\lambda$ is the wavelength of wave and $n$ is integer value

$\Rightarrow f=\frac{\nu }{\lambda }=\frac{n\nu }{2L}$

$\Rightarrow f=\frac{\nu }{2L}$, is the fundamental frequency

Step 2. Finding frequency when half tube is in water:

This can be considered as tube with one open and one close end,

for such a case, An odd-integer number of quarter wavelength have to fit into the tube of length $\frac{L}{2}$.

$\frac{L}{2}=\frac{n\lambda }{4}$

$\Rightarrow f\text{'}=\frac{\nu }{\lambda }=\frac{n\nu }{2L}=nf{\text{'}}_0$

where $f_o\text{'}$ is the new fundamental frequency

$\Rightarrow f{\text{'}}_0=\frac{\nu }{2L}=f$

So, the option B is correct