[Paul Spencer]
work in progress - help/comments anyone?
- If I have a target length and mouth area, what do I do about expansion contours?
- What are the forumulas for other types of expansion contours?
ACOUSTIC LENGTH
1/4 frequency wavelength determines the low frequency limit of horn loading
wavelength in metres = 344 / frequency (Hz)
20 Hz ~ 4.3m axial length required
30 Hz ~ 2.8m
40 Hz ~ 2.2m
50 Hz ~ 1.7m
Please note: this is a minimum; it is preferable to make the horn twice this length, as it results in the horn working in a velocity controlled manner
Acoustic length = physical length + (mouth diameter x 0.6)
The second part of the equation (mouth diameter x 0.6) accounts for the fact that the effective length extends a little longer than the axial length.
MOUTH SIZE
This relates to whether it radiates into free space,
half space (outdoors on the ground),
quarter space (on the ground against a wall)
or one eigth space (corner).
Each subsequent step requires half the mouth size of the previous
1. free space - requires mouth circumference equal to the wavelength of the LF cutoff
2. half space (outdoors) - as for #1 /2
3. quarter space - as for #1 /4
4. eigth space (corner) - as for #1 /8
For free space horns
Mouth Area = Pi * [c / (2 * Pi * f )]^2
FOLDINGS
<This section gleaned from John Sheerin -
https://ldsg.snippets.org/HORNS/design.html)
Parallel walls creates resonances which absorb power
from the horn's output if it falls inside the horn's
operating band, creating a notch in the frequency response.
This resonance would occur at the frequency whose half
wavelength is equal to the distance between the parallel surfaces
( f = 344 / x / 2 ).
Additional resonances would occur at odd multiples of this
frequency (1, 3, 5, etc.). This would indicate that to go higher
in frequency using a horn with parallel side walls, one would
need a narrower horn (consider the popular Lowther-style rear-loaded horns).
Folding can also introduce anomalies into the response.
EXPANSION CONTOUR
- Conical
- Exponential / Hyperbolic (longest)
- Tractrix (shortest)
Exponential contours are considered best for bass horns.
EXPANSION CONTOUR FORMULAS
EXPONENTIAL
The hyperbolic / exponential formula
(source: John Sheerin -
https://ldsg.snippets.org/HORNS/design.html)
Area = Throat Area [ cosh ( x*2*Pi*f / c) + M * sinh ( x*2*Pi*f / c) ] ^2
where
x = distance from throat
f = the cutoff frequency of the horn
M = the flare constant - M = 1 is exponential, 0 < M < 1 is hyperbolic
c = the speed of sound, approximately 13538 inches per second
or 344 m/s (depends on temperature, etc.)
EXAMPLE
- 40 Hz target cutoff
- corner loaded design
1. Target acoustic length
1/2 wavelength = 4.3m
1/4 wavelength = 2.15m
2. Mouth area
area = 1/8 x Pi x [c / (2 x Pi x f )]^2
f = 40
hence, area = 1/8 x Pi x [c / (2 x Pi x 40 )]^2
area = 0.7 sqm
3. Length correction
correction = 0.6 x mouth diameter
= 0.6 x 0.77
= 0.46m
4. Physical length
Physical length = 4.3 - 0.46
= 3.84m
or 1.7m for quarter wavelength version
This is correct assuming both 1/2 and 1/4 wave version have the same mouth area.
COMMENTS
This would make a bass horn with a similar mouth size to the Lab12
which does not require a stack of 4. The length would also be similar
to the Lab horn, although it is actually a little longer.
This is by no means a design one would build, but a preliminary
starting point to give an idea of some numbers to simulate.
