Physics · Ch 14 — Waves

Stationary Waves in Organ Pipes: Fundamental Mode and Harmonics

14.10

Stationary Waves in Organ Pipes: Fundamental Mode and Harmonics

An organ pipe is a tube of air, open at one or both ends, inside which a longitudinal stationary wave can be set up by blowing air across an opening (or over a reed) at one end, exciting the enclosed air column into resonance. Because the wave here is the longitudinal displacement of air particles along the pipe's own length, the boundary condition at each end is fixed by whether that end is open or closed to the atmosphere. At an OPEN end, the air is free to move (it connects directly to the large body of atmospheric air outside), so the displacement amplitude is a maximum there -- an open end is always a displacement ANTINODE. At a CLOSED end (a rigid cap), the enclosed air cannot move at all at that point, so the displacement there is always zero -- a closed end is always a displacement NODE. (Note that a closed end is a node of DISPLACEMENT but simultaneously an antinode of pressure variation, and vice versa for an open end, though this syllabus's calculations use only the displacement description above.) The two possible configurations of a pipe -- open at both ends, or open at one end with the other closed -- lead to two structurally different harmo …

Figure 1Stationary-wave (displacement) patterns in an open organ pipe versus a closed organ pipe

What this figure shows. Two side-by-side vertical panels, each drawn as a narrow rectangular tube of the same length $L$ with the air-column displacement envelope sketched as a pair of mirror-image dashed curves bulging left-right out of the tube's central axis, representing the two extreme instants of the oscillation. The LEFT panel shows the OPEN PIPE'S FUNDAMENTAL MODE: both ends of the tube are drawn open (no end cap, with small outward arrows suggesting free air movement) and both are marked as displacement antinodes ('A'), with exactly one node ('N') at the exact midpoint of the tube; a brace along the tube marks its length as $L=\lambda_1/2$ , matching the string's fundamental in form. The RIGHT panel shows the CLOSED PIPE'S FUNDAMENTAL MODE: the bottom end of the tube is drawn capped/sealed (a solid end wall) and marked as a displacement node ('N'), the top end is drawn open and marked as an antinode ('A'), with no other nodes or antinodes in between (just the one quarter-wavelength loop); a brace along the tube marks its length as $L=\lambda_1/4$ . A caption beneath both panels notes that, for tubes of the identical length $L$ and the same speed of sound $v$ , the open pipe's fundamental $f_1=v/2L$ is e …

Table 2Comparison of the open organ pipe and the closed organ pipe: boundary conditions, allowed wavelengths, and harmonics present

	End conditions	Allowed wavelengths	Frequency of $n$ -th mode	Harmonics present	Fundamental
Open pipe	Antinode -- Antinode	$\lambda_n=2L/n$	$f_n=nv/2L$	All integers: $1,2,3,4,\dots$	$f_1=v/2L$

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