How can I calculate a 'tractrix revistid' ?

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alujoe2

Member
Joined 2009
#1
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I would like to know the formula to calculate the 'tractrix revisited' curve, thank you for your help.
 
#5
Hi 4real,

thank you for the link. It is obvious to see that 'Kugelwellen' and 'tractrix revisited' are similar. I was hoping for some magical numbers, but seems to be as simple as radius * 0.5. In other words 'tractrix revisited' sounds much more interesting than 'Kugelwellen with half radius' or 'Kugelwellen using same radius as Tractrix' 🙂.
 
#7
Quoted from the link above:

Hello,

The nearest conventional ( = old) profile to the Le Cléac'h profile is the Kugelwellen horn. (here compared among with other profiles to the Tractrix and T = 1 Le Cléac'h horn ).

Notice that the Kugelwellen follows the same logic as the Tractrix ("constant speed" translation of spherical cap wavefronts of a constant radius along the axis. The radius used for the Kugelwellen being 2 times the radius used for the Tractrix).

A main difference between the Tractrix and the Kugelwellen is that the Kugelwellen expansion is exponential all along but the expansion of the Tractrix (as I calculated it) departs from an exponential law near the mouth. If we recalculate the tractrix with a pure exponential expansion of spherical cap wavefronts, then we find a horn the mouth of which curves back as for the Kugelwellen. I called this a "revisited Tractrix" horn.

Best regards from Paris, France

Jean-MIchel Le Cléac'h
Click to expand...

and for me the best information:

To say that the cut off frequency of a horn is related to the (mean) diameter of its mouth is a common mistake.

In fact whatever the length of a horn having a known expansion, its cut-off is the same , BUT what differs is the ripple in the response curve which is due to the interferences between normal waves emitted by the diaphragm and propagating inside the horn with waves backreflected from the mouth to the throat.

We are still wrongly using a semiempirical rule originating from Keele who said that the ripple due to those reflections passes through a minimum when the mouth diameter is equal to the wavelength at cut-off. This is based on a model using the false assumption that the wavefronts are plane. Further studies demonstrated that such minimum doesn't exist.

The domain of application of this rule of thumb is only for cut horns (all Salmon horns must be considered as cut horns). If you use a minimum diameter equal to the wavelength at cutoff then you obtain an "acceptable" amount of ripple in the response of the horn. (acceptable for whom?)

Doing that, we link that diameter to the cut-off frequency but we cannot reverse that rule so: it is not true that the cut-off frequency is related to the diameter of the mouth.

Now, IMHO,to encouter this rippling, it is highly desireable to use quasi infinite horns like complete (= uncut) Le Cléac'h horn, complete Kugelwellen or complete Tractrix. But only the Le Cléac'h horn doesn't do any asumption on the shape of the wavefronts. In this sense it is a better design than the Kugelwellen and the Tractrix, the design of which is eroneously based on the asumption that wavefronts are spherical caps.

Best regards from Paris,

Jean-Michel Le Cléac'h
Click to expand...

The length of the horn sets the minimum cut off frequency, the size and shape of the mouth determinates the ripple and therefore the usable cut off frequency.
Considering size, for high frequency the JMMLC horns are probably the best compromise and for lower frequency stuff a 'tractrix revisited' using factor 0.4 - 0.5 could be the choice or whatever else floats your boat...
 
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