Open-Pipe Harmonics
For an ideal uniform air column of effective length L_eff open at both ends, displacement antinodes occur at both boundaries and allowed frequencies are f_n=n v/(2L_eff) for every positive integer n.
Why this shows up in the exam
Open-organ-pipe harmonics · Pipe-length and gas comparisons · Frequency differences between adjacent modes
Learn the idea
An ideal pipe open at both ends supports every integer harmonic with displacement antinodes at both ends. Air at an open end moves freely, so the open-open column fits an integer number of half-wavelengths, much like a string with matching boundary type at both ends.
🧠 Memory hook: Open-open pipes allow all integer harmonics.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- f_n = n v/(2L_eff) — nth open-pipe harmonic
- lambda_n = 2L_eff/n — allowed wavelength in an open-open column
How to approach it
- 1Identify both ends as open
- 2Write the all-integer harmonic series
- 3Apply end correction only when the problem supplies or implies it
Common slip-ups that cost marks
- •Using physical rather than effective length when end correction matters
- •Keeping only odd harmonics
- •Confusing pressure nodes with displacement nodes
🌟 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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