Physics · Waves · NEET
An open end connects to the free outside air, so the air molecules there are free to vibrate with maximum displacement. Maximum displacement means a displacement antinode. Since both ends are open, both ends must be antinodes. The lowest mode that fits an antinode at each end has exactly one node in the middle, giving length L = lambda/2, so lambda = 2L and f = v/2L.
Yes. An open pipe produces ALL harmonics: f, 2f, 3f, 4f, 5f... Both odd and even harmonics are present. The nth harmonic is f_n = n(v/2L) for n = 1, 2, 3, 4... This is the key difference from a closed pipe, which gives only odd harmonics.
Harmonic counts from the fundamental: 1st harmonic = f, 2nd harmonic = 2f, 3rd = 3f. Overtone counts the tones ABOVE the fundamental: 1st overtone = 2nd harmonic = 2f, 2nd overtone = 3rd harmonic = 3f. So for an open pipe: nth overtone = (n+1)th harmonic. Mixing these up is the most common exam mistake.
v/4L is the closed-pipe fundamental (node at closed end, antinode at open end, length = lambda/4). An open pipe has antinodes at both ends, so the shortest fitting wave is length = lambda/2, giving f = v/2L. That is why an open pipe of the same length sounds an octave higher (twice the frequency) than a closed pipe.
For the nth harmonic of an open pipe, the number of nodes equals n. Fundamental (1st harmonic) has 1 node, 2nd harmonic has 2 nodes, 5th harmonic has 5 nodes. The number of antinodes is n+1 (because both ends are antinodes).
The ratio of the fundamental frequency of an open pipe to that of a closed pipe of the same length is:
The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe of length L. The length of the open pipe is:
The 4th overtone of a closed organ pipe has the same frequency as the 3rd overtone of an open pipe. The ratio of the length of the closed pipe to the length of the open pipe is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
f = v/2L, where v is the speed of sound in the air column and L is the length of the pipe. This is the lowest note the pipe can produce.
All harmonics are present: f, 2f, 3f, 4f, 5f... The general formula is f_n = n(v/2L) for n = 1, 2, 3, 4...
For the same length, an open pipe has double the fundamental frequency of a closed pipe (v/2L vs v/4L), so it sounds one octave higher in pitch.
The speed of sound v increases with temperature (v is proportional to the square root of absolute temperature). Since f = v/2L, higher temperature raises every harmonic frequency of the pipe.
An open pipe contains both even and odd harmonics, while a closed pipe has only odd harmonics. More harmonics give a fuller, richer tone, which is why open pipes are common in musical instruments like the flute.
