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In Reply to: Mouth reflections (reposted from Thomas Dunker) posted by JMLC on June 08, 2004 at 00:10:16:
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:
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…
Hello Jean Michel,
There is not much I can add to the discussion of the early W.E. theatre horns, as I have not studied these in depth. I have read the articles and patents concerning the Fletcher Horn System of 1933, however. I regard this system as pivotal in the evolution of the technology, as it inspired the Shearer Two Way Horn System, which in turn established the two way system as a standard which lasted for many decades.
The Fletcher System contained the first use of a compression driver with a complex phasing plug to minimize path length differences from different parts of the diaphragm. The driver was designed to create a plane wave in the horn throat. The Fletcher also utilized the first multicellular horn. The design of the horn is noteworthy as it also assumes a plane wave in the initial throat section. The individual cells are (nearly) parallel where they begin, then flare out to establish the pattern of the horn. This pattern describes a portion of a sphere, the exact shape determined by the number and arrangement of cells. This pattern is quite uniform up to the frequency where the individual cells begin beaming, as Thomas pointed out in a previous post. The onset of this is about 7kHz., about the upper limit of the response desired from the theatre systems of the 1930s.
The design of this equipment was, like the earlier W.E. horns, absolutely brilliant. These designs of the 1920s and 1930s still provide a healthy challenge, and a standard of comparison for those seeking advancements in the 21st century.
You are right, the 2-ways Shearer system
can be considered as the very true father of all modern 2 ways horn systems used in the proworld.
It is clear that the bass was the poorest part of the older systems based on the WE555 + WE15A horn (or equivalent). Dynamics was not the same for the bass and the mid-high. The directivity also in the high frequencies was a bit high and the sound not evenly spread inside the auditorium. But another bad characteristics was the severe delay of the mid-high due to the length of the "snail-horn".
With the Shearer, the bass is more dynamical due to the horn load and the bass is just a little delayed compared to the mid-high and this is less audible than with the older system. The Shearer also have less directivity in the mid-high than the snail-horn.
But for it seems there was persons to regreat the natural sound of the WE555W + WE15A when at the best listening point...the Shearer had many advantages compared to the older systems but we loose on few point when leaving (not all auditoriums did) the snail-horn...
We are now 70 years later and digital delay is here. The offset between the different loudspeaker in a group is no more a problem. If we had digital delays in the 1930, may be the WE15A will still be there.
I used to listen few month ago to the most extrem audiophile system here in France:
this is a full horn 5 ways system using Ale and Goto compressions drivers and 5 digital amplifiers (previously it used Cary 1810 in place of the digital amplifiers).
Low-mid reproduced by the Goto driver + long exponential horn is the most natural that I ever listened and medium is very excellent too.
That's really interesting to realize that those Goto compression drivers are not inspired by the father of all modern compression driver the famous WE594 that used a multi slots phase plug but they are totally inspired by the WE555 with the old phase plug!
Here again we can question if the changes done by the pro gear within the years is transposable in the audiophile world.
Every audiophile should have the chance to listen one time to such very high level Goto or Ale systems that use WE555 type compression driver to see that modern type compression drivers have very few if no advantages over them...
Hello Jean-Michel and Steve,
Thank you for your elucidative reply to my post, Jean-Michel. I shall have to make a printout and reread it and give it some thought.
I quite follow your reasoning when it comes to the computation of horn flare based on an exponential expansion of increasingly curved wavefronts. A mathematical method for doing this has always been missing from the horn theory, and your contribution is of great value. I am also getting increasingly interested in the study of the mouth discontinuity of impedance at frequencies approaching the horn cutoff. Last week I was visited by Bjorn Kolbrek, and together we located some more literature and discussed horn ideas at length.
I was aware that a copy of N.W. McLachlan's book "Loudspeakers" existed at the University's Acoustics Institute, but I had not had a better look at it until last week, when Bjorn and I made copies of the chapters on horns, drivers and magnet system. McLachlan's style and thoroughness is impressive by any standard, and much of the theory found in this book goes virtually unmentioned in later theory on the subjects. Bjorn brought with him the copies on the horn material for scanning, but I spent some time reading through the chapters we'd copied and some very interesting points were brought to my attention for the first time.
We were also able to leaf through many volumes of Bell Laboratory Record and Bell System Technical Journal from the 1920s and 1930s in the magazine of the main library, but while we found the original documentation on the development of the 555 driver, we could not find any references to the horn research underlying the WE horns of interest. Whatever may once have existed seems lost to history.
To me, reading McLachlan's treatment on the subject of mouth reflections added further depth to the complexity of the phenomenon in the cutoff region. He brings attention to the point (in my interpretation) that for low frequencies, the average pressure of the wave has not yet reached atmospheric pressure at the point of the mouth, and is therefore not "ready to launch" and the wave expands/decompresses more violently, causing a kind of shockwave that ripples back through the horn. It is understandable that this will not happen for high frequencies, where the length of the horn and the mouth dimensions is a high multiple of the wavelength, and the horn can be regarded as infinite.
Experimental evidence reveals that wavefronts having a considerable curvature before "launching" from the horn tend to undergo a flattening at/around the mouth plane, such that the radiation of an imagined hemispherically shaped wave at the mouth is impossible.
McLachlan also suggests that depending on the position along the horn axis, the wave fronts may assume intermediate curvatures, between plane and spherical. That is, perpendicular to the horn walls at the boundary, but not necessarily describing a perfect arc near the center of the horn. This especially seems to be prominent at the mouth, where the wavefront is considerably flatter than the opening angle between the horn walls might suggest. This suggests that the mouth and its termination deserves special attention in order to have the most suitable geometry
and provide the most "gentle release" of the wave from the horn. This would not seem to be a simple matter of truncating the horn profile at some point having the appropriate circumference or area. I have not studied this aspect of Jean-Michel's work very well, but I am very intrigued by the mouth's "beyond 180 degrees" opening angle and its claimed suitability for reducing reflections. I don't question the possibility of this, I just need to fit some theory around it.
Assuming that the wavefront is able to rip away from the horn walls at a point some way before the plane of the mouth perimeter, we would like to see it expand in a smooth, controlled fashion as it radiates into the space in front of the horn. This calls for an analysis of the impedance at the mouth, but as Jean-Michel has pointed out, there is no single point along the axis that can be defined as "the mouth", and as McLachlan points out, the effective mouth impedance would have to be based on knowledge of the curvature of the wave, as impedance calculations based on plane waves certainly don't apply near the mouth as they do at the throat. While classical theory states that the the impedance at any point in the horn is po*c*A, A, the area must be the area of the wavefront, and this must be corrected for curvature.
If the wave propagation velocity through the horn is not constant with frequency, further complexity is added to the computation of effective mouth impedance in the cutoff region.
McLachlan's horn theory is very advanced and tends to continue where the subsequent and current theory tends to stop, and it has got me very uncertain about many assumptions that are frequently made in the better known theory.
For instance, I am thinking about horns like the Iwata horns, which are designed with a lot of thought given to the curvature of wave fronts, and just looking at it gives both Torbjoern Lien and me an intuition that the horn shape is "just right for the sound waves". At least it *seems* very right for a horn having a rectangular cross section. Part of the horn design tradition calls for making the "mouth" plane curved in one or more axes, calling for double curved surfaces in one or more planes. The idea would seem to be making the "boundary" at the horn mouth (where the wave faces expansion unrestricted by the horn walls) correspond to the assumed curvature of the "departing" wave. It is obvious that an axisymmetric horn can not have a "curved mouth" in the same way as the Iwata horn - except if the effective "mouth plane" is regarded as curved surface defined by the wavefront shape "at the mouth", and it is most interesting if special geometry at the "perimeter" can shape this imaginary "mouth plane" suitably for reduced reflections near the cut-off frequency. As Jean-Michel has also suggested, the "effective" horn may extend to a point outside the physical horn structure making the effective length of the horn greater than its physical length. This again would have to assume considerable curvature of wavefronts very close to the physical horn perimeter, for the wavefront to protrude "out of the mouth" while boundaries more or less still follow the last section of the inner horn walls.
If the wavefronts do not have an easily predictable curvature at any point along the horn and near the mouth, the area expansion profile along the axis of the horn can not be computed based on equal increments of distance from the throat. If the waves "balloon out" towards the mouth, the distance between wavefronts/pressure maxima
on the horn axis will increase as the curvature of the wavefront increases and once again another uncertainty is added in terms of finding the optimal physical horn profile near the mouth. I am curious to know how Jean-Michel considers these questions, and what assumptions regarding wavefront curvature (vs. frequency) have been used in the calculation of the geometry near the mouth.
My somewhat limited understanding of the mathematics involved leaves me struggling to get a handle on all these factors, which all seem inevitable consequences of going from an infinite to a finite horn, which in a sense is what happens gradually as the cutoff frequency is approached from above.
As Beranek states, for wavelengths where the horn length and mouth circumference is a a multiple of wavelengths for the wave transmitted, the horn may be regarded as infinite. This is a simplification, but arguably a lot of the challenge of horn design is taming the mechanisms responsible for mouth reflections, and these happen in a loosely defined "cutoff region".
I am thinking, as a more general remark, that the reflections and
disturbances taking place at frequencies falling within this region cause many unpredictable conditions for the propagation of higher frequency wave components that would be present in music signals.
At some point it seems inevitable that we close the theory books for a while and put the theory to the test by constructing and listening to music through actual horns.
It is suggested by Hiraga and many others, and it is becoming very clear to me, that the theory of horns can never be "complete", since all existing models make some fundamental assumptions. The theory must be complemented by empirical work. But horns are time consuming to construct "just for testing" so mathematical models are very useful, and possibly at some level of understanding conjoined aspects of the theory a certain intuition for good design can be developed.
It is very frustrating that extraordinary horns like the old WE horns and Iwata's horns do not have precisely known expansion formulae. For all horns, a correct representation of the expansion profile would have to include a full knowledge of the curvature of the wavefronts and the resulting area at all points along the horn. In the study of the WE15A and similar curved horns, an analysis is made extremely difficult by the curve of the horn axis and unequal outer and inner wall lengths making the wavefront "tilt" as it propagates. If it is tilted ("plane" not perpendicular to horn axis) AND more or less curved, the horn can not be mathematically described with simple traditional horn expansion formulae. Perhaps such a horn would be best designed by working out the area expansion in separate segments to be finally joined together. Bjorn has been testing ideas that the 15A horn may have a "plane wave" expansion somewhere between
exponential and hyperbolical, but due our only having a few points along the axis where approximate areas are known, only approximations can be found. It is clear that the horn does not have a single "flare constant" (as an exponential would) but that the flare constant increases as the mouth is approached. One must wonder with what kind of precision the wavefront shapes in a curved horn like the 15A could be anticipated in the late 1920s, but we know this was considered important, so it must have been given some deliberation. I believe that mechanical computers for the "numerical" solution of differential equations may have been available at the time (when were such computers first used? I have seen these in museums) so it is possible that the design process was quite advanced.
Bjorn has found a patent for the actual 15A horn, which is in the name of DeHart G. Scrantom. This patent shows a nice side view drawing of the 15A horn, and some cross sections, but gives no dimensions or information about the foundation for its expansion profile. Assuming that the side view is approximately to scale, some cross sectional areas can be extrapolated from known dimensions and sectional drawings. In the patent text only the woodworking construction method is described, so the patent really applies only to the method of fabrication. Other patents by Scrantom are also in the field of manufacturing methods. It seems unlikely that he designed the actual horn profile, but he was apparently commissioned by WE for this patent on the aspects of manufacturing procedure and construction materials. Somebody at Bell Labs would have had to give Scrantom precise guidelines for the actual design of the horn, for which a suitable method of mechanical construction could be found. Several WE horns from this period seem to be designed with a common "style", suggesting some method had ben developed. Oddly, Edward Wente, the biggest theorist of the Wente/Thuras/Osmer team, has not written anything specific on horn design/theory that I am aware of, apart from some thoughtful remarks in conjunction with the design of the 555 driver.
Hiraga seems to suggest (can you confirm this, J-M? I am struggling very hard to read the French text in "Les Haut-Parleurs") that aspects of the design of the 15A serves to reduce 3rd harmonic distortion (in the driver? In the horn?). Understanding how that might happen would be very interesting.
I agree with Jean-Michel about the similarity of "flare reversal" in the 15A family of horns and in modern CD horns, and it is known that the 15A has wider vertical dispersion than horizontal. It is very tempting to divide the 15A into two sections; a "throat" section extending to the point where the direction of dominating expansion shifts to vertical, and a "mouth section" with mostly vertical expansion. Most of the length of the horn is made up by the "throat section", which is long enough to contain a little more than a whole wavelength at about 100Hz. It seems to me that this flattened and curved "throat section" has a function in establishing good "waveguide operation" in a wide frequency band, and constricting the wave expansion so that at the point of "flare reversal", all wavefronts have a moderate curvature, somewhat like a cylindrical surface with a small horizontal plane curvature and a more prominent vertical plane curvature. I have been thinking that this might have the effect of confining the "mouth zone" (where the wave breaks away from the horn) to a more limited region than would have been the case if the horn were axisymmetric or square.
The tilted wavefronts due to a curved horn axis makes the analysis of this horn, particularly the mouth, very complicated.
I have been thinking myself, somewhat inspired by the WE13A, that since I will be placing the midrange horns on top of the bass systems in any case, I can make use of the space behind the bass system down to the floor and and make a curved portion of the horn extend downward akin to the throats of the Klangfilm Europa Junior horns, but with more of an S- or question mark shape (like some old radio horn speakers), where the first part of the horn is curved in the opposite direction of the second bend to counteract the tilting of the wavefronts. I have to learn more about the behavior of the wavefronts in a curved duct and how to best construct this curve, but it might make the design of the mouth easier if the tilt angle of the wavefront approaching the mouth can be minimized this way.
Ideally I would have liked to make such a horn with curved mouth geometry more like that of the Iwata horn, but the introduction of double curve surfaces very much complicates the construction. I have therefore been thinking that I could try integrating Iwata's "progressive coupling" and "anti-shock" rib/slit system
into the final portion of the horn near the mouth to reduce mouth reflections in the situation where the mouth has a geometry not as closely corresponding to the presumed curvature of the wavefronts as is the case with the Iwata horns...
Jean-Michel, you mentioned that you have made a spreadsheet for calculating "WE15A style horns" with a spiral curve axis, using your expansion profile. How do you go about correcting the areas for tilted wavefronts in this program, and how do you go about computing the correct expansion near the mouth with this in mind? I would be very interested in trying this spreadsheet and knowing how a rectangular "Le Cleac'h" horn would look. It sounds very interesting to me.
Oh, well - this is all very complex and confusing, and the deeper I dig into the theory, the more questions I have...
Among the things to do in the near future is to visit some professors at the university who have experience with horn,
waveguide and wave propagation theory and look for some further clues. In 1991, Tonni F. Johansen published his doctoral thesis on a numerical method for the prediction of wave propagation from horns, which I have borrowed from the library. It contains a lot of interesting information and references, but will take me some time to go through, and I can only make limited use of the mathematics, but the hope is that some of the computer programs developed during his work on the method are still available and can have some useful purpose.
Well, too much horn theory in my head for now... I will check back in here soon. My computer needs some work before I can read and send e-mail as usual, so please reply here if you have comments. (I am a bit disappointed with the environment for fruitful discussions on the JoeNet these days, so I much prefer this forum. We have also created a Yahoo-group for "Norwegian" horn discussions, and I only have so much time to spend writing. Theory this complex is better discussed
face to face or on the phone, but some times writing it down helps
structure one's thoughts and further understanding...
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