Question

Two closed organ pipes A and B have the same length and A is wider than B. They resonate in the fundamental mode at frequencies \(n_{A}\) and \(n_{B}\) respectively, then

• A. \(n_{A}=n_{B}\)
• B. \(n_{A}>n_{B}\)
• C. \(n_{A}\lt n_{B}\)
• D. either (2) or (3) depending on the ratio of their diameters

Hint:

A wider pipe has a larger end correction, so its effective length is longer and its frequency lower.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For a closed pipe the fundamental frequency is \(n=\dfrac{v}{4(L+e)}\), where \(e\approx0.3d\) is the end correction and \(d\) the diameter.

Step 2: Apply the Condition
Both pipes have the same length \(L\), but A is wider so \(e_{A}>e_{B}\).

Step 3: Compare
Effective length of A is larger, so \(n_{A}\lt n_{B}\).

Final Answer:
The wider pipe A has the lower frequency, so nA is less than nB. \[ \boxed{\text{(C) }n_{A}\lt n_{B}} \]