So if I understand correctly, if WG angle < 90 the directivity is less than a baffled disk. (makes sense) And at 90 the polar response is the same as a baffled disk. (I believe you) At 180 the polar response is the same as a baffled disk. (by definition) So where does the directivity widen? My first impression is that since all the modes have "cut-in" by 90, that there will be no further effect and the directivity will be essentially that of a piston in a baffle from 90 to 180. Hence no "pull out".
I had in mind to see the individual OS harmonics. It now occurs to me that it would be more helpful to see the solution itself. The solution to a piston/planar wavefront in an infinite baffle is well known, there are sites that show the polars at different frequencies. The polars of an OS WG at different frequencies can also be calculated exactly. If these were to be shown for different angles of OS WG then we could compare them, answer all my questions. Hopefully answer James' questions too.
Best wishes David
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Indeed! Very impressive. But what makes it so good? Just the OS throat, or is there more going on?
HI David
Your statements are confusing, probably because of the way they are stated.
When the waveguide is < 90 the polar response is wider than that of a rigid disk in a baffle, i.e. the polar response is "pulled-out" by the waveguide. At 90 degrees they become equal as they are exactly the same situation, just analyzed differently. But one cannot take this analogy beyond 90 degrees because it does not hold - in that case, one cannot enforce boundary conditions that are rational.
There are very few closed form solutions done in the OS and PS coordinates owing to the severe difficulty of doing them. Time has not made this any easier since these days one would use numerical FEA methods instead, which has meant that in the last 30-40 years nothing has really been done on these coordinate solutions to further develop them. As I said, I was surprised to understand that Mathmatica had them, but that package is expensive and has a long learning curve.
I do not have the solutions that you desire and I know of nowhere that they have ever been done. They could be done (Mathmatica?), but that would be a monumental task in numerical calculations, one that I personally would not wish on anyone.
I have more experience with these solutions than anyone and what I have done is all in the public domain. I suggest you go there for what answers are available.
The easiest way to think about the OS modes is to recognize that as the frequency falls they have to become identical to the spherical ones. Hence think of the spherical case but remember that as the frequency goes up they change and at high frequencies, like those that we are discussing, they become quite different. However, it is precisely the HFs that are the most difficult to calculate because of convergence of the solutions and the greater number of modes required - they converge very slowly and, in what I found, often will not converge at all even with double precision calculations.
My hunch is that you need to look at these three things as a single unit:
1) the compression driver
2) the waveguide
3) the baffle
Here's a couple of measurements.
The first measurement is an 18Sound XT1086 with a Celestion CDX1-1445 and no baffle whatsoever.
The second measurement is the same waveguide witha BMS 4552 and a big fat roundover on the baffle : 4.5" on each side!
From what I can see, the use of five dollars worth of PVC pipe to create a roundover is making a bigger difference that using a $200 compression driver. For $45, the Celestion performs quite well, the main issue with the first measurement is the dip at 2500hz and the peak at 1800hz. It's true that both can be "EQ'd" away, but the PVC pipe fixes things elegantly.
More importanly, and the big "x factor" is whether the diffraction produced off of the baffle edge in measurement one is audible.
Personally, I'm inclined to say "yes."
For instance, I had these Yamahas in my living room up until two weeks ago. I banished them to my den because I had family coming over for Thanksgiving.
I put these Behringers in their place.
At first, I preferred the Behringers. They look a million times better, and they get plenty loud. But there's just something OFF about the treble, and my hunch is that it's higher order modes. I think the sharp edges of the Behringer cabinet, and the ports adjacent to the tweeter, are generating diffraction. The Yamaha just sounds more transparent in the treble.
side note: I see a lot of people auditioning a million different tweeters, looking for the best. And IMHO, a lot of what they're hearing and attributing to the tweeter is the BAFFLE not the tweeter. The lowly SB19, for $20, exceeds the sound of a lot of expensive domes if the SB19 is placed on a waveguide.
It would be nice if there was something like the Yamaha but in a smaller and more attractive cabinet. The Yamaha is utter overkill for a home theater. The Behringer can do about 110dB, which is plenty for me. The Yamaha can hit about 120dB, which is just unnecessary. Plus, it's ugly.
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There is an 8" version of yamaha (dxr8) though not cheap at $550 each.
Yamaha DXR8 8" 2-way powered PA speaker — 1,100W peak at Crutchfield.com
I think the nice fat rollover where horn mouth to baffle lessens reflections on the yammi.
The dsr112 has a 1.7khz crossover point.
I know i don't shut up about it but get foam or felt around the behringers baffle to catch those reflections.
Norman
Yamaha DXR8 8" 2-way powered PA speaker — 1,100W peak at Crutchfield.com
I think the nice fat rollover where horn mouth to baffle lessens reflections on the yammi.
The dsr112 has a 1.7khz crossover point.
I know i don't shut up about it but get foam or felt around the behringers baffle to catch those reflections.
Norman
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The Behringer is so ridiculously capable and affordable, I'm half tempted to just 3D print an entire new baffle, this time with a roundover on it. Some EQ would be required to fix the response, because it would change with a new baffle, but MiniDSP is only $80....
Picture something like the Revel Salon, and you get the general idea. I've heard the $20K Revels back-to-back with the $2000 Revels, and the baffle is definitely making a difference. And I'll bet most listeners attribute the improvement to the use of a $300 tweeter instead of a $30 tweeter, when the difference is mostly the baffle IMHO. (The $2000 Revel uses an aluminum dome, the $20,000 Revel uses beryllium.)
Picture something like the Revel Salon, and you get the general idea. I've heard the $20K Revels back-to-back with the $2000 Revels, and the baffle is definitely making a difference. And I'll bet most listeners attribute the improvement to the use of a $300 tweeter instead of a $30 tweeter, when the difference is mostly the baffle IMHO. (The $2000 Revel uses an aluminum dome, the $20,000 Revel uses beryllium.)
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Freehafer is the only closed form solution I know. I have seen you cite him but I don't know if you have studied his PhD thesis, it's available on line from MIT. I don't find it an easy read, there's so many variable transformations that I lose track and the scrawly hand written equations don't help. Do you have any comments that would illuminate?
Yes, I have read your book and just reread Ch. 6, I am very appreciative that it's available. I do have a few questions but will think them over a little more before I reply.
Best wishes David
I did not read Freehafer's thesis because by the time that I knew about it I had already done my own calculations. I did read his JASA paper and referenced that in my papers.
To me, what you need to understand is how the solutions of "horns" in OS (or any separable coordinate system) allow one to know the precise shape of the wavefront, which is critical for polar response calculations. This is not possible with any solution derived from Webster's Equation. People make all kinds of assumptions about what the shape is, but they have no way of actually knowing what it is for sure. The thing that is missing from Freehafer's work is this simply fact. He basically never even mentions it. He and even one else were, and still are, obsessed with the impedance of horns and no one, except myself, were too concerned about the shape of the wavefront. I was drawn to this solution for the simple reason that it is the only way to calculate the exact shape of the wavefront and hence the directivity.
After 25 years, I have come to this: the main thing to understand is the zeroth order mode because for the purposes of directivity this is the only one that counts. The higher order modes are important from a subjective point of view, but almost irrelevant from a directivity point of view (except for wide angle devices and fairly high frequencies where they can significantly affect the directivity.) So it's good to suppress the HOMs because they add nothing but bad sound. It turns out that all of the useful coordinate systems result in an almost perfectly spherical wavefront at the mouth - even more so if the HOM are suppressed. Basically, all finite origin coordinate systems look spherical for large r.
From the physics, it is know that if the wavelength are much shorter than the aperture size then the sound basically radiates directly forward. This means that a spherical wavefront in the aperture will radiate directly forward propagating as if it were still restrained by the waveguide walls (except for diffraction at the aperture if not handled properly.) Hence, as long as the zerth mode dominates and the shape of the zeroth mode is nearly spherical, as it is for an OS waveguide below about 10 kHz, the sound radiation polar angle will be just about the same as the angle of the walls. This turns out to be pretty true for any waveguide that I have ever built, so as a rule-of-thumb it is fairly solid.
I would not encourage anyone to go through the calculations of an OS waveguide as they are far too tedious. It's important to understand what the solutions mean, but not much is to be gained by doing them oneself.
(I can only imagine the complexity of the Freehafers thesis since the math at that time was quite different in notation from what we are used to today. It's like when I try and read a modern physics paper in Quantum Field Theory. It's the unfamiliar notion that gets me. When I dig past this, I can follow the math, but if you don't know the modern notation you just get lost. Einstein created his own notation for General Relativity, which is now the standard, but at the time, no one had any idea what he was writing.)
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Yes, I didn't know if you had read it subsequently, hence my comment.
I suspected that may be the case, based on your comments about his work.
I have only read the JASA abstract, didn't follow it up because it seems to be only a partial summary of the full thesis.
Not only is the thesis more detailed, as one would expect, but it covers aspects the JASA paper does not include at all, most importantly the directivity.
Thesis is >HERE< if you, or others, are interested.
Still at work on a more detailed response with further questions.
Best wishes
David
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Patrick, you are blowing my mind man!
I have been a huge constant directivity fan for years, but didn't know why until I read Earl's paper years ago. Thanks Earl!
Currently, I have modified JBL 4722's with upgraded 2453h-SL drivers - JBL's best bang for the buck for large format 4" compression driver on a 2384 waveguide - super smooth with low distortion + dual subs = huge dynamic sound for reasonable $'s.
Every once and a while, I consider "upgrading" to JBL M2 or Revel Salon 2 that you reference above. What is interesting is that both off axis response is excellent. There is a huge shootout thread on AVSForum and here are the results: Speaker Shootout - two of the most accurate and well reviewed speakers ever made - Page 12 - AVS Forum | Home Theater Discussions And Reviews
If you go through the thread, you will find Harman's spinorama's for each and they are amazingly close between the M2 waveguide and Revel dome waveguide. For the M2, there appears be a directivity blip at the XO of 750 or 800 Hz. You will see it once you find the spins. But otherwise, both are excellent speakers, yet the Revel wins on both days... Why?
I read posts like yours and makes me wonder, would I be disappointed to drop $20K on the Revel's only to find out I prefer the dynamic sound of the 4722's for a fraction of the cost.... Or the Revels can be had for a fraction of the cost based on your research...
All the best,
Mitch
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I have the Behringer Truth B2030A and JBL LSR306P MKII neck to neck here. Great for the price but quite far from being great speakers IMO.
Based on what I've measured so far, there doesn't seem to be anything revolutionary about JBLs image control wave guide. But the wave guide looks really trick, which very well may be why JBL chose it.
Below are on-axis measurements of both in sweetspot at a distance of approximately 2 m. They are placed directly on front of a large speaker with much distance to front wall, hence part of the reason for the low level in the bass.
JBL LSR306P MKII:
Behringer Truth B2030A:
Based on what I've measured so far, there doesn't seem to be anything revolutionary about JBLs image control wave guide. But the wave guide looks really trick, which very well may be why JBL chose it.
Below are on-axis measurements of both in sweetspot at a distance of approximately 2 m. They are placed directly on front of a large speaker with much distance to front wall, hence part of the reason for the low level in the bass.
JBL LSR306P MKII:
Behringer Truth B2030A:
HI David
Thanks for the link. I read through it and recognized most of it, but that was certainly not the way I did the problem. I had a computer! He didn't.
I used Numerical Recipes "shooting" algorithm to find the angular functions, which is pretty efficient if you start with a low C value since the solution must be very close to the Legendre polynomials. Then you can increase C using the old functions as starting values. This yields the angular functions and, most importantly, the eigenvalues for any given value of C and theta.
With the Eigenvalues known you can find the radial functions by integrating backwards from a large KR value where we know that the functions must be simple sines and cosines.
Once you have the real part of the radial functions you can use the Wronskian to find the imaginary part, but this is tricky, because the imaginary parts have a logarithmic singularity at the origin and you must find that value at the origin that yields a solution that is orthogonal to the real part at large KR. This works fine for the n=0 mode, pretty well for the n=1 mode, somewhat slow for the n=2 mode, but fails to converge for any mode >2.
There are other techniques that I discovered later would allow for the calculation of modes > 2, but I never tried those.
These calculation would have to be done for many values of C and for many values of theta to get the type of comparison that you were looking for before. And, that is a lot of work!!
I did not see any discussion in the thesis on directivity, since this requires some assumptions of how the mouth is treated. Flat baffles and spherical enclosures give different results. I use the spherical enclosure and treated the mouth as I describe in my text. I assumed no mouth reflection, which is a good assumption in my case, but not so good in the general case.
Thanks for the link. I read through it and recognized most of it, but that was certainly not the way I did the problem. I had a computer! He didn't.
I used Numerical Recipes "shooting" algorithm to find the angular functions, which is pretty efficient if you start with a low C value since the solution must be very close to the Legendre polynomials. Then you can increase C using the old functions as starting values. This yields the angular functions and, most importantly, the eigenvalues for any given value of C and theta.
With the Eigenvalues known you can find the radial functions by integrating backwards from a large KR value where we know that the functions must be simple sines and cosines.
Once you have the real part of the radial functions you can use the Wronskian to find the imaginary part, but this is tricky, because the imaginary parts have a logarithmic singularity at the origin and you must find that value at the origin that yields a solution that is orthogonal to the real part at large KR. This works fine for the n=0 mode, pretty well for the n=1 mode, somewhat slow for the n=2 mode, but fails to converge for any mode >2.
There are other techniques that I discovered later would allow for the calculation of modes > 2, but I never tried those.
These calculation would have to be done for many values of C and for many values of theta to get the type of comparison that you were looking for before. And, that is a lot of work!!
I did not see any discussion in the thesis on directivity, since this requires some assumptions of how the mouth is treated. Flat baffles and spherical enclosures give different results. I use the spherical enclosure and treated the mouth as I describe in my text. I assumed no mouth reflection, which is a good assumption in my case, but not so good in the general case.
Yes, I can see how that works. They all interact.
I certainly see the effect of the rounded baffle, but also see good off-axis consistency even in the ragged plot. That most be the driver+horn combo.
Sure does make a difference. I've used soft termination on the mouths of horns, everything from beach towels to wool to fur. The differences are audible and measurable. And I've seen cardboard Sonotube used as baffle edge - worked well.
FWIW, Mr Iwata used a progressively slotted mouth on the horns he built for his home system. I don't think I've seen anyone else use that. The story was that he designed tunnel entrances and exits for the Shinkansen high speed train. The slots reduced the shockwave from the train rapidly going into or out of a tunnel.
I think it's prettier than the Behringer, but beauty is in the eye of the beholder, right?
Of course I'm used to that style of speaker from work, the Yamaha, Mackie, QSC, JBL, etc.I had in mind your point that the OS solution includes the variation across the wavefront that the Webster Horn Equation lacks. Freehafer solves for this variation, within the horn. This doesn't include mouth effects of course, but Freehafer's solution is only exact for an infinite horn anyway. The coordinate involved becomes asymptotically close to an angular coordinate, so the solution for within an infinite horn should asymptotically approach the polar response. Even idealized angular variation would be useful compared to the total omission inherent in Webster's Equation. I expect it should be a reasonable approximation for well rounded mouth of adequate size.* Make sense?
Best wishes David
* IIRC you observed that the angle of the waveguide is practically equal to the directivity of the speaker, which is consistent.
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Yes, to solve the OS waveguide, you have no choice but to find the curvature of the wavefront. But the solution is still exact even if the horn is not infinite as long as 1) there is no significant mouth reflection or 2) you take the mouth reflection into account (possible, but very difficult.)
With a "well rounded mouth of adequate size" it is a very "reasonable approximation", as my work has shown.
Makes perfect sense.
With a "well rounded mouth of adequate size" it is a very "reasonable approximation", as my work has shown.
Makes perfect sense.
The OS/hyperbolic horn problem is not easy, for a start the differential equation is 2nd order. It's not 1 dimensional like school calculus but 3, and not static but time variable. We can transform the Wave equation to drop the time variation but then it's Helmholtz problem in complex phasors rather than simple real numbers. And finally, it's not in simple, square Cartesian coordinates but in curved (oblate spheroidal) coordinates.
So not an embarrassment to admit it's not obvious, it's not to me
The upside is that the maths is about as hard as you will see, any problem much worse just can't be done analytically and will be solved for specific cases with numerical techniques on computer. It's also a kind of heroic achievement, the peak of more than 2000 years of work by countless people to build the foundations and structure of all the mathematics. Best wishes David
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Hello, All
Looking at the available data for the bulk of the crop of waveguides it looks like the “Constant Directivity” stays close to on track for several octaves and gets more directive above 10K. I also notice that the bulk of the GedLee style polar maps stop at 10K.
Is the Constant Directivity performance above 10K not so critical or is that just the current state of the art?
Thanks DT
I am trying out a pair of JBL 2451J drivers and PT H1010HF-1 (1-1/2 inch throat) waveguides crossed over at 2K, above JBL 2123H’s. So far the imaging is working out fine and the 100 X 100 waveguide coverage provides remarkable space. This wave guide also is not so Constant Directivity above 10K.
Looking at the available data for the bulk of the crop of waveguides it looks like the “Constant Directivity” stays close to on track for several octaves and gets more directive above 10K. I also notice that the bulk of the GedLee style polar maps stop at 10K.
Is the Constant Directivity performance above 10K not so critical or is that just the current state of the art?
Thanks DT
I am trying out a pair of JBL 2451J drivers and PT H1010HF-1 (1-1/2 inch throat) waveguides crossed over at 2K, above JBL 2123H’s. So far the imaging is working out fine and the 100 X 100 waveguide coverage provides remarkable space. This wave guide also is not so Constant Directivity above 10K.