If a horn is too small for the frequency it is operating at (an acoustic
mouth circumference much less than about 1 wl), then placing it next to
an identical unit will normally extend the low cutoff.
When the area ratio between the mouth and throat is not large, the low
cutoff of the horn is set by the ¼ wavelength resonance and in this
situation the resonance is roughly set by the length of the horn path.
While a horn is not at maximum efficiency until it is ½ wl long or
longer, it can provide usable output (significant gain over the same
driver not horn loaded) down to about the ¼ wl resonance and at much
lower frequencies, only has the output of a sealed box (the driver, its
compliance and back volume).
The quarter wavelength resonance is the lowest mode a pipe closed at one
end and open at the other resonates at, for a pipe to resonate at the
same frequency with both ends open (½ wave length resonance) it has to
be about twice as long.
In a horn, as the mouth area becomes larger, some of the air out in
front of the horn is still governed by (highly coupled to) the pressure
gradient at the mouth and so is an acoustic length added to the horn
path length. How much it adds to the length depends on its shape with
round or square (including the mirror image) being the best and
typically adding about .6 to .8 times the horn mouth radius. A large
mouth can add significantly to the path length and can have a
significant effect on the low cutoff.
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