End Corrections for Resonance Tubes

by Dr. John Askill

If a tuning fork is set in vibration and held over the open end of a tube closed at one end, sound waves which travel down the tube, will be reflected back from the closed end. If the returning wave is exactly in phase, a condition of constructive interference exists which we call resonance and the loudness of the sound will be greatly increased. The displacement of the air will be greatest at the open end of the tube, whereas the pressure difference will be greatest at the closed end, and vice versa.
The lowest frequency where a condition of resonance exists in the tube occurs when the wavelength is four times the length of the tube. This is called the fundamental mode of vibration. Higher modes of resonance exist when the wavelength is 4/3(third harmonic), 4/5(fifth harmonic),... of the length of the tube. The first four resonance modes for tubes open at both ends, and closed at one end are shown below:



The antinodes (A) are displacement antinodes (pressure nodes) and the nodes (N) are displacement nodes (pressure antinodes). If the frequency is fixed, as in the case of the tuning fork held over the open end of a tube, the resonance lengths are v/4f, 3v/4f, 5v/4f and so on for a tube closed at one end, and v/2f, 2v/2f, 3v/2f for a tube open at both ends.

When resonance exists, the displacement is a minimum (node) at the closed end, but the antinode is not exactly at the open end. It is actually a small distance beyond it. This extra distance beyond the end of the tube is called the end correction . The acoustic lengthof the tube is equal to its physical length plus the end correction.
In a typical laboratory experiment with a water reservoir-resonance tube apparatus about 1m long, and tuning forks of frequency about 500 Hz, three positions of resonance can usually be found. If the positions of resonance from the open end of the tube are L1, L2, and L3, the wavelength of the sound wave is equal to
4(L1 + e) or 4/3(L2 + e) or 4/5(L3 + e), where e is the end correction. From these relations the wavelength may be calculated from 2(L2 - L1) or 2(L3 - L2) or (L3 - L1), and the end correction may be calculated from (L2 - 3L1) or (3L3 - 5L2) or (L3 - 5L1). Since the frequency of the tuning fork is known, both the velocity of sound may be determined using the relation
velocity = frequency x wavelength
and the value of the end correction may be determined from the relations above.
The results of over a thousand measurements of the end correction for a closed tube 1 m long by students in classes of Physics for Life Sciences and Physics of Music class are that for a cylindrical tube of inside diameter 2.80 0.01 cm, and for tuning forks of 480 Hz (B4) and 512 Hz (C5), the value of the end correction for the open end is 1.28 0.05 cm. This gives a value of the end correction coefficient, e/d as 0.46+/- 0.02 for the open end of a cylindrical pipe. Very few references to this value are available in the open literature. For comparison, the theoretical value given by Rossing for the end correction coefficient for the open end of a cylindrical pipe is 0.305.
If the tube is open at both ends, the end correction coefficient is then twice as much or 0.92, so that to a close approximation, the the acoustical length for an open pipe is equal to its physical length plus its diameter, and that for a closed pipe is equal to its physical length plus its diameter
The velocity of sound in air at room temperature can also be determined with this procedure from the product of frequency x wavelength to an accuracy of typically %.


John Askill, 1998,1999.