Question

5th harmonic of a closed organ pipe matches with 1st harmonic of an open organ pipe. Find ratio of their lengths.

• A. $5$
• B. $2$
• C. $\dfrac{5}{2}$
• D. $\dfrac{2}{5}$

Hint:

Closed organ pipes support only odd harmonics, while open organ pipes support all harmonics.

Answer 1

The Correct Option is C

Step 1: Use concept of wavelength instead of frequency formulae

Sound waves produced in organ pipes correspond to standing waves. Hence, the condition of resonance can be written using wavelengths.

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Step 2: Wavelength in a closed organ pipe

In a closed organ pipe, the fifth harmonic corresponds to:

L_closed = 5λ / 4

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Step 3: Wavelength in an open organ pipe

In an open organ pipe, the fundamental mode corresponds to:

L_open = λ / 2

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Step 4: Apply the condition of same frequency

If both pipes produce sound of the same frequency, their wavelengths must be equal.

From the expressions:

λ = 4L_closed / 5
λ = 2L_open

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Step 5: Compare the lengths

Equating the two expressions for wavelength:

4L_closed / 5 = 2L_open

L_closed / L_open = 5 / 2

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Final Answer:

The ratio of lengths of the closed pipe to the open pipe is
L_closed : L_open = 5 : 2