Question 287: Waves

Question

The fundamental frequency of a closed pipe of length $L$ is equal to the second overtone of a pipe open at both the ends of length (XL). The value of X is (Neglect end correction)

Accepted Answer

Step 1: Find the fundamental frequency of closed pipe
A closed pipe (one end closed, one end open) of length $L$ has fundamental frequency:
$ f_1^{	ext{closed}} = \frac{v}{4L} $
where $v$ = speed of sound.
Step 2: Find the second overtone of an open pipe (both ends open)
An open pipe has all harmonics. The $n^{	ext{th}}$ overtone is the $(n+1)^{	ext{th}}$ harmonic.
So,
second overtone
means
third harmonic
.
For an open pipe of length $XL$, frequency of third harmonic:
$ f_3^{	ext{open}} = 3 \left(\frac{v}{2 \cdot XL}ight) = \frac{3v}{2XL} $
Step 3: Equate the two frequencies
Given: Fundamental frequency of closed pipe equals the second overtone (third harmonic) of open pipe:
$ \frac{v}{4L} = \frac{3v}{2XL} $
Step 4: Solve for $X$
Cancel $v$ from both sides:
$ \frac{1}{4L} = \frac{3}{2XL} $
Multiply both sides by $4L \cdot 2XL$:
$ 2XL = 12L $
Divide both sides by $2L$:
$ X = 6 $
Step 5: Final Answer
The value of $X$ is
6
.
Correct Option: C (6)

Answer

4

Answer

5

Answer

6

Answer

7

Waves

The fundamental frequency of a closed pipe of length L is equal to the second overtone of a pipe open at both the ends of length (XL). The value of X is (Neglect end correction)

Choose the correct answer: