Here's what I'm aware of regarding the background of the end correction calculation. Beranek's Acoustics
, published in 1954, has the math behind this. The derivation is complex, so I won't get into that. But the end result is this:
leff = lact + lf + luf
leff = effective length of tube (in meters)
lact = actual length of tube (in meters)
lf = end correction for flanged end of tube (in meters)
luf = end correction for unflanged end of tube (in meters)
The actual length is self-explanatory. Formulas for the two end corrections are as follows:
lf = 0.8488 * a
luf = 0.6132 * a
where "a" is the radius of the tube in meters
Beranek further says:
Originally Posted by Beranek
"If the tube is not round, we may replace a by sqrt(S / pi), where S is the cross-sectional area of the tube.
Of course, this is the same as replacing the diameter d by 2 * sqrt(S / pi).
This comes from section 5.6 of Beranek, pages 131-133. For lf and luf, Beranek actually uses 0.85 and 0.613 respectively, but I've replaced these by a slightly more accurate calculation from the electroacoustics book by Leach.