Question

In an open end organ pipe of length 'L', if the velocity of sound is 'V', then the fundamental frequency will be (Neglect end correction) ______.

• A. $\frac{V}{2L}$ and all harmonics are present.
• B. $\frac{V}{4L}$ and all harmonics are present.
• C. $\frac{V}{2L}$ and even harmonics are present.
• D. $\frac{V}{4L}$ and even harmonics are present.

Hint:

Open pipes sound richer and more musical than closed pipes specifically because they produce all harmonics, providing a fuller overtone spectrum.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question is testing fundamental acoustic theory regarding vibrating air columns inside an organ pipe that is open at both ends.

Step 2: Detailed Explanation:
1. Fundamental Frequency:
In an open pipe, antinodes form at both open ends. In its simplest mode of vibration (fundamental mode), there is exactly one node located perfectly in the middle.
This means the length of the pipe $L$ accommodates exactly half a wavelength: $L = \frac{\lambda}{2} \implies \lambda = 2L$.
Using the wave equation $V = n\lambda$, we find the fundamental frequency $n$:
$$n = \frac{V}{\lambda} = \frac{V}{2L}$$
2. Harmonics Present:
Because the pipe is open at both ends, it can support boundary conditions for any whole number of half-wavelengths ($L = \frac{\lambda}{2}, L = \frac{2\lambda}{2}, L = \frac{3\lambda}{2}$, etc.).
Therefore, the possible frequencies are $1n, 2n, 3n, 4n \dots$
This means all harmonics (both even and odd) are present.
(In contrast, a pipe closed at one end has a fundamental frequency of $V/4L$ and produces only odd harmonics).

Step 3: Final Answer:
The frequency is $\frac{V}{2L}$ and all harmonics are present, matching option (a).