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

TL;DR

This paper explains the long-standing enigma of why organ pipes sound longer than their physical length by revealing that the boundary condition at the open end is a resonating vortical sphere, not a simple boundary.

Contribution

It introduces the concept of the acoustically resonating vortical sphere (PARVS) as the boundary condition at the pipe's open end, providing a new physical explanation for sound radiation.

Findings

Resolves the enigma of pipe length perception

Introduces PARVS as a boundary condition

Explains dependence of sound on pipe radius

Abstract

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 according to the classical picture the pipe actually sounds lower, which is to say that the pipe so-to-speak sounds longer than it is, a long-standing enigma. As for the pipe's end, we have resolved this acoustic enigma by detailing the physics of the airflow at the pipe's open end and showing that the boundary condition is actually 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 not only a physical explanation of sound radiation from the organ-pipe's open end, in particular, of its puzzling dependence upon the pipe's radius, but also an appreciation of it as realization of the sound of the flute, mankind's oldest musical instrument.