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I am looking for a more precise calculation for determining an accurate end correction for an open-ended pipe of a relatively high fundamental frequency and am hoping someone might be able to steer me in the right direction.

Researching online I have found the general calculation of adding 0.6 times the radius (or 0.3*d) to either end but I would like to know where this "constant" is derived from as I am skeptical of it's ability to render accurate results based on the possible variations relating to a given frequency and it's resulting acoustic impedance, etc...

I was able to find a simple formula where the volume of a half-sphere derived from the diameter of the pipe is converted to a cylinder of equal volume w/ a given length which then determines the supposed end correction value, however this value is considerably less than the general 0.6*r mentioned above.

To give an idea of my experiment; I am working with a cylindrical pipe that has an outer diameter of .5", an inner diameter of .425" and am trying to determine a length that will produce the desired frequency of 17.424 kHz.

Any help would be greatly appreciated!