Question

The fundamental frequencies of vibrations of air column in pipe open at both ends and in pipe closed at one end are $n_1$ and $n_2$ respectively, then \dots

• A. $n_1 = n_2$
• B. $n_1 = 2n_2$
• C. $2n_1 = n_2$
• D. $3n_1 = 4n_2$

Hint:

Closing one end of a previously open tube immediately drops its fundamental pitch by exactly one full octave (halving its frequency). This is why instruments like the clarinet (effectively closed) can play much lower notes than flutes of similar size!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to mathematically relate the fundamental acoustic frequency of an open organ pipe ($n_1$) to the fundamental frequency of a closed organ pipe ($n_2$) assuming they have identical physical lengths.

Step 2: Key Formula or Approach:
We must recall the fundamental frequency derivations for both types of organ pipes:
1. Open Pipe (open at both ends): Antinodes form at both ends. The pipe length $L$ equals half a wavelength ($\lambda/2$).
$$n_{open} = \frac{v}{\lambda} = \frac{v}{2L}$$
2. Closed Pipe (closed at one end): A node forms at the closed end, and an antinode at the open end. The pipe length $L$ equals a quarter of a wavelength ($\lambda/4$).
$$n_{closed} = \frac{v}{\lambda} = \frac{v}{4L}$$

Step 3: Detailed Explanation:
Let the fundamental frequency of the open pipe be $n_1$:
$$n_1 = \frac{v}{2L}$$
Let the fundamental frequency of the closed pipe be $n_2$:
$$n_2 = \frac{v}{4L}$$
To find the relationship, let's manipulate the equation for $n_1$ so it resembles $n_2$:
Multiply the numerator and denominator of $n_1$ by 2:
$$n_1 = \frac{2v}{4L}$$
Pull the constant 2 out to the front:
$$n_1 = 2 \times \left( \frac{v}{4L} \right)$$
Notice that the term in the parenthesis is exactly the formula for the closed pipe ($n_2$):
$$n_1 = 2 \times n_2$$
This shows that an open pipe vibrates exactly one octave higher (twice the frequency) than a closed pipe of the exact same length.

Step 4: Final Answer:
The relation is $n_1 = 2n_2$, matching option (b).