End Correction

When an air column vibrates -- whether in a pipe closed at one end or open at both ends -- the boundary conditions require an ANTINODE to sit at every OPEN end (since the air there is comparatively free to move) and a NODE to sit at every CLOSED end (since the air there has almost no freedom to move). In practice, though, the antinode at an open end does not form EXACTLY at the physical opening of the pipe, but slightly BEYOND it, because the air just outside the tube is even more free to vibrate than the air still confined just inside it; since the air particles right at the plane of the opening are also not entirely free to move in every direction, some reflection genuinely occurs at a small distance outside the tube rather than exactly at its mouth. This small extra distance, between the pipe's physical open end and the true position of the antinode, is called the END CORRECTION, e. Reynold's first-order approximation gives $e=0.3d$, where d is the pipe's inner diameter. …