Organ Pipes Numericals
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Direct answer
A closed pipe (one end stopped) sounds an octave lower and only odd harmonics: the sealed end forces a displacement node, the open mouth a displacement antinode, giving fundamental f1 = v/4L and overtones at 3f1, 5f1, 7f1... An open pipe, antinodes at both ends, sounds f1 = v/2L with all harmonics (2f1, 3f1...) — its fundamental is double the closed pipe's on equal length. With v ≈ 340 m/s, a half-metre closed pipe gives 170 Hz (then 510, 850 Hz) while a half-metre open pipe gives 340 Hz (then 680, 1020 Hz). Real mouths add an end correction of roughly 0.6 times the radius per open end, so a pipe plays slightly lower than its measured length suggests.
What you must remember
- Closed pipe modes: f_n = (2n − 1)v/4L — odd harmonics only; the first overtone is the third harmonic, the second the fifth.
- Open pipe modes: f_n = nv/2L — all harmonics present; the first overtone is the second harmonic (the octave).
- Equal-length ratio: open fundamental : closed fundamental = 2 : 1; conversely, a closed pipe needs half the length to match an open pipe's note.
- End correction: about 0.6 r per open end; effective length L_eff = L + 0.6r (closed) or L + 1.2r (open), lowering all frequencies slightly.
- Speed of sound: about 340 m/s at room temperature in NEET conventions, rising roughly 0.6 m/s per °C; v ∝ √T in kelvin.
- Resonance tube pairing: a fork of frequency f over a closing water column resonates first at L = v/4f, next at 3v/4f; the separation of successive resonances is λ/2 — the standard way to measure v.
- Numbering trap: "fifth harmonic" and "fifth overtone" differ by one; a closed pipe has no second harmonic at all.
Walking down a half-metre pipe
Close one end of a 0.5 m pipe: f1 = 340/(4 × 0.5) = 170 Hz, and the only overtones are 3 × 170 = 510 Hz and 5 × 170 = 850 Hz — the even numbers 340 and 680 are forbidden, because the closed node cannot support them. Open both ends: f1 = 340/(2 × 0.5) = 340 Hz with 680 and 1020 Hz — the octave and the third harmonic return. Now invert to the resonance tube: hold a 340 Hz fork over a water column; the shortest resonant length is L = v/4f = 340/1360 = 0.25 m, the next comes at 0.75 m, their difference being λ/2 = 0.5 m (λ = 1 m). Add the end correction for a tube of radius 2 cm: each open end contributes about 1.2 cm, so the true resonance sits at 26.2 cm — and a student who ignores it will compute v about 5% too low.
Where NEET sets the trap
The even-harmonic ban is the chapter's assertion-reason favourite: "a closed pipe can never sound the second harmonic" is true, and the reason is the permanent node, not the pipe's material. Length-versus-frequency inversions punish proportional reasoning: halving the length doubles every frequency, and questions ask it backwards ("a pipe gives 512 Hz; what after halving?"). Diagram questions show a pipe with nodes and antinodes marked and ask for the mode number — count the quarter-wavelengths. The end correction's direction is worth one mark alone: real pipes sound flatter (lower) than v/4L predicts because the effective length is longer. Finally, temperature items note that a warmer day raises v and sharpens every pipe's pitch — a fact that pairs naturally with a musician's complaint.
Frequently asked questions
What is the fundamental of a closed pipe 50 cm long at 340 m/s?
f1 = v/4L = 340/(4 × 0.5) = 170 Hz, with overtones possible only at 510 Hz and 850 Hz — odd multiples.
Which harmonics are absent from a closed organ pipe?
All even harmonics; the closed end must remain a displacement node, so only odd multiples of the fundamental fit the pipe.
An open and a closed pipe of equal length have fundamentals in what ratio?
2 : 1 — v/2L against v/4L; the open pipe sounds an octave higher on the same length.
What is end correction and how large is it?
The antinode forms slightly outside the mouth, adding about 0.6 times the radius per open end to the effective length — making real pipes sound marginally flatter than geometry suggests.
A 340 Hz fork resonates with a closed tube at 25 cm. What is the next resonance length?
75 cm: successive closed-tube resonances sit half a wavelength apart, here λ = 1 m from λ = 4 × 0.25 m.