Closed-Pipe Odd Harmonics
For an ideal pipe closed at one end and open at the other, a displacement node and antinode bound the column, so allowed frequencies are f_m=(2m-1)v/(4L_eff), with m=1,2,3,... and even harmonics absent.
Why this shows up in the exam
Closed-organ-pipe frequencies · Water-filled pipe length changes · Open-pipe versus closed-pipe resonance
Learn the idea
A pipe closed at one end fits only odd quarter-wave modes in the ideal model. The closed end cannot let air move, while the open end can; mismatched boundary types require one, three, five, and successive odd quarter-wavelengths.
🧠 Memory hook: Closed-open pipes keep odd harmonics only.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- f_m = (2m-1)v/(4L_eff) — successive closed-pipe natural frequencies
- lambda_m = 4L_eff/(2m-1) — allowed wavelengths of odd quarter-wave modes
How to approach it
- 1Mark displacement node at closed and antinode at open end
- 2Write the odd sequence explicitly
- 3Translate overtone wording into the correct odd harmonic
Common slip-ups that cost marks
- •Calling the first overtone twice the fundamental
- •Using half-wave boundary conditions
- •Confusing harmonic number with overtone number
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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