End correction

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In acoustics, end correction is a short distance applied or added to the actual length of a resonance pipe, in order to calculate the precise resonance frequency of the pipe.

A simple notion is that the fundamental resonance of a pipe occurs when the resonator length is half or a quarter of the sound wavelength. It is however well recognized that the practical frequency comes out lower than this, you have to apply an end correction, the pipe appears to be acoustically somewhat longer than its physical length.[1]

A theoretical basis for computation of the end correction is the radiation acoustic impedance of a circular piston. This impedance represents the ratio of acoustic pressure at the piston, divided by the flow rate induced by it. The air speed is typically assumed to be uniform across the tube end. This is a good approximation, but not exactly true in reality, since air viscosity reduces the flow rate in the boundary layer very close to the tube surface. Thus, air column inside the tube is loaded by the external fluid due to sound energy radiation. This requires an additional length to be added to the regular length for calculating the natural frequency of the pipe system.

It is denoted by  \Delta L and sometimes by  e . In organ pipes, an antinode is not formed exactly at the open end. Rather, antinode is formed a little distance  \Delta L away from open end outside it.

This  \Delta L is known as end correction, which can be calculated as,

  • for a closed pipe (with one opening):
 \Delta L = 0.6 \cdot r = 0.3 \cdot D ,

where r is the hydraulic radius of the neck and D is the hydraulic diameter of the neck;

  • and for an open pipe (with two openings):
 \Delta L = 1.2 \cdot r = 0.6 \cdot D .

There is no scientifically proven and accepted value for the end correction of a resonant tube, various values ranging from 0.3r to 0.6r have been suggested from numerous disparate experiments. Lord Rayleigh was the first experimenter to publish a figure, in 1871: it was 0.3r.