Question
A pipe closed at one end has length . The diagram shows the second overtone standing wave inside the pipe. Find the frequency of this overtone if the speed of sound in air is .
Solution — Step by Step
A pipe closed at one end supports only odd harmonics: fundamental , third harmonic , fifth harmonic , and so on. The second overtone is the third allowed mode after the fundamental.
For closed pipes, the overtones are . So the second overtone is the fifth harmonic:
The second overtone has frequency .
Why This Works
In a closed pipe, the closed end is a displacement node and the open end is an antinode. This boundary condition forces an odd number of quarter wavelengths inside. The first overtone is the third harmonic (skipping the even ones), and the second overtone is the fifth harmonic — not the fourth.
Diagram-based questions test whether you can count nodes correctly. The second overtone diagram shows three nodes (including the closed end) and three antinodes (including the open end).
Alternative Method
Count the wavelengths directly from the diagram. The second overtone fits of a wavelength in the pipe, so . Then . Same answer.
The big confusion: “second overtone” “second harmonic”. In a closed pipe, the second harmonic doesn’t even exist. Always list the allowed harmonics first, then count overtones from the fundamental.
Common Mistake
Computing and picking that option. Closed pipes skip even harmonics. Open pipes have all integer harmonics. Memorise this distinction — it shows up in CBSE Class 11 and NEET every couple of years.