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Re: Mouth reflections (reposted from Thomas Dunker)

194.214.158.221

Dear Thomas.

OT: You are right I am in long vacations from Joenet (after a many years lurking period, I was asked to come back on Joenet. As I saw that ancient names were around -and yours BTW- it was a pleasure for me to come back. Rapidly my messages were treated as “non-sense”, my name feminized, and because no real discussion was possible… I have no more fun to post on Joenet anymore…too bad !).

Your message need few answers from my part.

You are right saying that my opinion is that, for any horn , a curvature at the mouth with some rolling back is a good way to reduce reflected waves from the mouth to the membrane (It exists other means, absorbing materials at the mouth, slots...). As I said this is perfectly reflected by the positive changes in the smoothness of the electrical impedance curve. But for a more direct measurement, I’ll plan to use a method similar to the one described in the document:
http://www.microflown.com/rd/books/book/c.pdf
to see the influence on the acoustical reactance and resistance curves.

You are wrong thinking that the profile one can calculate with my method is intended to both reduce the axial length of the horn while providing a curved mouth . It was not originally conceived in that goal and I never said that a curved mouth allows to reduce the length of one horn. In my method, no limit for the axial length is imposed by the user, the horn is calculated from the throat to the “mouth” according to the expansion law you want to use (choosing the T value, T around 0.7 for an hypex and 1 for an exponential) and it can be continued until wavefronts (the “wavefront” not the “cross section”) at a very long axial distance from the throat, more important: its shape is only a result from the geometrical calculation, no initial assumption of the shape of the horn is done. The only hypothesis is that the dimension of the minute discrete elements used and their simple shape (with parallelism between the upper and lower surfaces and orthogonality between those upper and lower surfaces and the sides of the elements) are a warranty that the Webster’s equation is correctly applied (and solved as hyperbolic expansion) at the microscopical scale (that’s not exactly how I should say the idea because anyone could argue that whatever the shape of the horn the Webster’s equation applies…).

In other messages I yet said what I think of horns designed with a cross section expansion (exponential,…). One more time i have to say: Webster’s equation is a differential equation and describe what happens inside an “infinitesimal volume = at the microscopic scale, not at the macroscopic scale. The plane wave hypothesis at that infinitesimal scale is valid but cannot be translated at the macroscopical scale = on the whole horn. Horns calculated with a (planar) cross sectional evolution following exponential law, hypex law or every kind of law belonging to the hyperbolic family are a mistake! At the macroscopical scale it is the area of equipressure and equiphase surfaces propagating inside the horn that should follow the expansion law solution of Webster’s equation. (Theoricaly within a perfect horn equipressures surfaces should be also equiphase surfaces).

It could seem very presumptuous from me to say that the profile calculated by my method is a more accurate way to calculate a more exact geometry of a horn belonging to the hyperbolic family but in fact as I yet said most of the merit belongs to Voigt .

While the results are similar your own view of what is a tractrix horn is different than mine. We should not give such importance to the fact that the tractrix horn is calculated from the mouth to the throat. I am sure that initially Voigt used a method a bit similar to mine (or preferably mine is quite similar to Voigt’s one) and begins to build the geometry of the horn he was studying from the throat. If you use my spreadsheet using T = 1 (expansion according to an exponential law) you’ll obtain a profile more than 98% similar to a tractrix profile. Furthermore if you modify the value of T only the axial length will varies, the diameter of the cross-section at the max axial distance will not change. This means that the diameter of the “mouth” of the horn (undependantly on how we define the mouth) is only dependant on the cut-off frequency, not of the chosen expansion law. As a consequence, it is similar to calculate a horn from the mouth to the throat than from the throat to the mouth.

Nor the Tractrix nor the horn calculated with my method are short horns (in the sense that noone decide to cut them or to increase their curvature at the mouth at a given distance from the throat). They don’t result from an empirical choice for the shape, they result only from pure physics and pure geometry (even if the number of discrete elements used is a limitation). An infinite horn having an exponential variation of the wavefront area cannot possess an infinite axial length (again horns having an exponential cross section variation are mistakes!) . The true distance to throat for a given point on the profile is the length measured along the sides of the horn, not the axial distance. I don’t know why Voigt choose to stop the expansion of the horn he worked on. As the legend said (see what Edgar said about the subject in Positive Feedback I think )it’s Voigt’s graphical assistant who said “it looks like a tractrix curve!” and yes a tractrix has an asymptote and cannot curves back… but I am pretty sure that’s Voigt knew that the wavefronts exponential expansion at a further distance imposes that the horn curves back after the 180° opening angle.

About the Western Electric horns , specially the WE15 and the WE66A, I worked long ago on their expansion and I was convinced that they were calculated using curved wavefronts (double circular curvature like N-S and E-W to give an idea). They are made of 3 parts, at the throat there is an adaptation element. Then there is a part having a nearly linear (slow) flare in one direction (N-S to give an idea) a kind of exponential (rapid) flare in the other direction (E-W to give an idea). If W1a is the width of the wavefront along the first direction and W2a is the width in the other direction the area Aa = W1a x W2a follows a perfect exponential expansion corresponding to a frequency cut-off Fc. Then there is a 3rd part terminated at the mouth having a kind of exponential (rapid)flare in the first direction (reference N-S ) and a nearly linear (slow) flare in the second direction (reference E-W ). If W1b is the width of the wavefront along the first direction and W2b is the width in the other direction the area Ab = W1b x W2b follows a perfect exponential expansion corresponding to the same frequency cut-off Fc as the second part.
For sure the axial length of the 2 last parts play an important role in the way such horns perform.
I am also certain that the choice of the change from a rapid flare in one direction for the second part to a slow flare in the 3rd part in the E-W direction (and inversely: from a slow flare in one direction for the second part to a rapid flare in the 3rd part in the N-S direction)is very important in the way the WE horn performs.
We can see that such change in flare was then used in other kind of horns like the Mantaray horn.

Well have to stop for the moment…

Best regards.

Jean-Michel Le Cléac’h , Paris, France


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