I learn more about waveguides every day. From what I understand today, An "optimal" waveguide shape is specific to the crossover frequency for the driver it is mounted in front of AND the dispersion of the lower frequency driver it is paired with. An off-the-shelf waveguide may provide better response from a driver, though not as good if it were tailored to the system it is included in.

A waveguide also induces backpressure forces on the driver diaphragm. This may induce distortion. For this reason metal domes seem to be favored.

This is timely. I'm encouraged by the interest expressed here. I've copied some posts by Pooge and others from the AA forum that provide the formulas needed to plot "optimal" shapes. I plan to slash my way through Excel this week to see what I can come up with.

The following are excerpts from some of those posts and are shared to whet your appetite. For the “full” story, go to Audio Asylum and search “waveguide’

Waveguide shape

Pooge ( A ) on March 29, 2005 From "Sound Radiation in Acoustic Apertures" by Geddes, JAES, April 1993:

"A short waveguide may not yield the best directivity control at the low frequencies, while a long waveguide will yeild poor high-frequency results (on axis) due to the aperture diffraction since the first "hole" [in the axial frequency response] is found to be located at ..."

It was found that "diffraction limit" for a circular mouth is

kR = 15.0/sin^2(angle);

where R = the length of the horn

K is the wavenumber, or (2*pi*f)/c,

where c=the speed of sound

"This frequency will be called the aperture diffraction limit for a waveguide. Above this value the frequency response and the directivity will be strongly affected by the mouth diffraction. Below this limit the waveguide will behave as a directive point source. [The formula] can be used as a rule of thumb for deciding the correct length of a waveguide." Thus:

"kR" should be less than "15.0/sin^2(angle)".

What is clear,… is that a baffle and a radiused termination of the waveguide into the baffle will reduce the diffraction effect that causes the condition for which the hole in the freqency response was found. The final curve at the mouth is not part of the 'horn flare', which is what the waveguide equations determine. It's simply a radius to reduce diffraction (and probably reflections back down the horn) from the mouth. There probably is some theoretical basis for this, but I would guess that empirical is pretty good. I think Pooge is right that Geddes uses a ~2" radius - simply blend that into the curve for the flare as smoothly as possible and you should be in good shape.

For anyone that wants to play with other throat sizes and waveguide angles, here are the formulas:

[(y^2)/{((a/2)^2)*sin^2(ang)}] - [(x^2)/{((a/2)^2)*cos^2(ang)}] = 1

where a= 2r/sin(ang)

where r= the radius of the throat or y at x=0

and ang=the angle from the x-axis to one wall of the waveguide.