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Multi-Way Conventional loudspeakers with crossovers

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Old 16th December 2017, 10:33 AM   #14871
Ro808 is offline Ro808  Netherlands
diyAudio Member
Join Date: Mar 2010
A very concise definition is:

"The “end correction” is the distance from the open end of a pipe to the antinode just above it".
This applies to standing waves in a pipe.

From Wiki:
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.

The concept of End Correction is widely used in Structural and Fluid Dynamics, Antenna Theory and notably the Acoustics of musical instruments (pipe and horn).

The point of reflection for air pressure fronts in brass instruments is not at the end of the tube, but some distance outside of it. End correction is the difference between the end of the physical tube and the actual point of reflection. This distance differs according to frequency.

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 scientist to publish a figure, in 1871: it was 0.3r.

Consider a pipe closed at one end and open at the other. A simple model tells that resonance will occur when the pipe length (plus end correction) is 1/4,3/4,5/4 etc. of the sound wavelength by assumption that antinode occurs at the open end. Due to the pipe wall waves can't expand freely, so there is not a perfect antinode. By addition of an end correction, the pipe appears to be acoustically somewhat longer than its physical length.

The most accurate value appears to be 0.58r (It is independent of the wavelength but does depend on the shape of the tube. Value is quoted for a circular cross section). Normally in such experiments, you eliminate (and can calculate) the end correction by finding two consecutive resonating lengths L1 and L2 at the same frequency, for example two cases where the L1+c is 1/4 wavelength and L2+c is 3/4 wavelength. Subtracting gives L2 - L1 = half wavelength. In most experiments, the speed of sound in the tube is calculated. However, it must be remembered this is not the same as the speed in free air.

Martin J. King discussed end correction in his articles on Transmission Line and Back Loaded Horn Physics.

If I'm not mistaken, Danley Soundlabs has implemented (a form of) end correction in some of their Synergy horns.
The Equivalent Throat system is one of the few theories that includes end correction in the combined application of a compression driver + horn.

The science behind the end correction is thoroughly adressed by Neville H. Fletcher and Thomas D. Rossing in "The Physics of Musical Instruments".

The topic is also discussed in the works of:
Howard and Craig
Kurze and Riedel
Junger and Feit
Morse and Feshbach, Morse and Ingard
and numerous others.

Last edited by Ro808; 16th December 2017 at 10:57 AM.
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