Question 277: Waves

Question

In fundamental mode, the time required for the sound wave to reach up to closed end of a pipe filled with air is ' $t$ ' second. The frequency of vibration of air column is (Neglect end correction)

Accepted Answer

Given:
Time taken for sound to reach from open end to closed end = $t$ seconds.
The pipe is closed at one end and open at the other (closed pipe).
Neglect end correction.
Step 1: Express length of pipe in terms of $t$ and speed of sound
Let the length of pipe be $L$.
Let speed of sound in air = $v$.
Time to travel from open to closed end:
$$ L = v t $$
Step 2: Write the fundamental frequency for closed pipe
For a closed pipe, the fundamental (first harmonic) forms a node at closed end and antinode at open end:
Length of air column ($L$) in fundamental mode:
$$ L = \frac{\lambda}{4} $$
where $ \lambda $ = wavelength.
So,
$$ \lambda = 4L $$
Step 3: Express frequency in terms of $t$
Frequency,
$$ f = \frac{v}{\lambda} $$
Substitute $\lambda$ and $L$:
$$ f = \frac{v}{4L} $$
Since $L = v t$, we substitute:
$$ f = \frac{v}{4 \cdot v t} $$
$$ f = \frac{1}{4 t} $$
So, the correct option is:
$ \boxed{(4 t)^{-1}} $
That is,
Option A
.

Answer

$(4 t)^{-1}$

Answer

$(2 t)^{-1}$

Answer

$4 t$

Answer

$2 t$

Waves

In fundamental mode, the time required for the sound wave to reach up to closed end of a pipe filled with air is ' t ' second. The frequency of vibration of air column is (Neglect end correction)

Choose the correct answer: