Tone generation in an open-end organ pipe: How a resonating sphere of air stops the pipe*

According to the classical Helmholtz picture, an organ pipe while generating its eigentone has two anti-nodes at the two open ends of a cylinder, the anti-nodes being taken as boundary condition for the corresponding sound. Since 1860 it is also known that the pipe actually sounds lower, which is to say the pipe "sounds longer," than it is, a long-standing enigma. As for the pipe end, we have resolved this acoustic enigma by detailing the physics of the airflow at the pipe's open end and showing that the actual boundary condition is the pipe's acoustically resonating vortical sphere (PARVS). The PARVS geometry entails a sound-radiating hemisphere based on the pipe's open end and enclosing a vortex ring. In this way we obtain a physical understanding of the dependence of the end correction upon the pipe's radius and also of the sound of the flute, mankind's oldest musical instrument.