Say that we have a tube of length $L$. In the tube, there is a standing wave of wavelength $\lambda$. Then, $L=\lambda$.


In the above diagram, the wave's amplitude is highest at the ends and center of the tube. The wave has two nodes of with zero amplitude on either side of the middle of the tube.

I would expect these nodes to be at exactly 1/4 of the tube's length from each end. Beginning at the left side of the tube: If the tube contains one whole wavelength, then the first $90{}^{\circ}$ of the wave should bring the wave from the maximum down to zero. The next $90{}^{\circ}$ should take the wave to the minimum at the center of the tube, and so on. Dividing the wavelength into four equal parts should make these locations occur at even intervals, or at even quarters of the tube's length, $L$. Correct?

However, I have built tubular bells in the past. In all of the literature (and in experience), the correct locations for these node points is $0.224L$, rather than $0.25$. Would someone mind explaining this?


the open ends of the tube are surrounded by air outside the tube which is coupled to the air inside the tube. this coupling effect draws energy from the resonant air inside of the tube and radiates it outside the tube. this energy loss represents a damping term in the resonance equation and replaces the undamped resonant frequency with the damped resonant frequency, which is slightly lower. this increases the effective length of the resonant tube slightly and shifts the positions of the nodes.


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