Question

A sound consists of four frequencies : $200\text{Hz},400\text{Hz},1125\text{Hz}$ and $1600\text{Hz}$. A sound filter is made by passing this sound through a bifurcated pipe as shown. The sound wave has to travel a distance $40\text{cm}$ more in the right branch than in the straight pipe. The speed of sound in air is $300\text{m/s}$. which of the given frequencies will almost completely muffle or silence the outlet?

• A. $200\text{Hz}$
• B. $400\text{Hz}$
• C. $1125\text{Hz}$
• D. $1600\text{Hz}$

Answer 1

The correct option is C $1125\text{Hz}$

For silence zone, destructive interference should occur at the outlet.
Which means,
Path difference, $\Delta x=(2n+1)\frac{\lambda }{2}$ [$n$ is an integer]
since, $\lambda =\frac{v}{f}$, we can rewrite the above formula as,
$\Delta x=\frac{(2n+1)v}{2f}$
$\Rightarrow f=\frac{(2n+1)v}{2\Delta x}$

Here $v=300\text{m/s},\Delta x=40\text{cm}\text{or}0.4\text{m}$
So,
$f=\frac{(2n+1)}{2}\times \frac{300}{0.4}=375(2n+1)$
Among the given options, $1125\text{Hz}$ is the only odd integer multiple of $375$

Therefore, the outlet will be silenced for $1125\text{Hz}$