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MultiWay Conventional loudspeakers with crossovers 

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10th May 2012, 05:34 PM  #11 
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Join Date: Apr 2004
Location: Texas

Thank you for the very interesting links, Dave. I don't believe that we are in disagreement. I think that the only questions that arise are which radius to use for the spreadsheet and, at least for line arrays, your PDF suggests it is more of a longest dimension for line arrays. For individual drivers, it may be different, but this is good ground for a practical experiment.
Some time back, I measured a loudspeaker (2 driver with xover in place) at steadily increasing distances and watched as the high frequency did not appear to reach equilibrium until further back while the lower frequency driver showed a more rapid equilibrium (not quite the right word but it gets the point across). This would also be consistent with your PDF as the higher frequency reproduction would otherwise not really require longer measurement distances unless the baffle, and not the driver size, were being considered. Putting this all together (both your work and mine as well as the balloon paper referenced by Tom), I think that the spreadsheet is a useful tool (biased as I might be) in helping to understand the approximate error in measurement for measurements taken at different distances and frequencies for different drivers. Indeed, using the spreadsheet, indicates that at 10kHz, for a 1 inch tweeter on a .2 meter wide baffle that is 1 meter tall and centered and 1/4 of the way from the top (longest dimension: 0.76m), you would still have a better than 3% error at 30 meters. It seems to me that we are living with errored measurement but I have a tough time believing that we are living with so much error; I wonder if the baffle does not contribute to the response as much as the mobile portion of the diaphragm and therefore the use of the longest dimension might not be an accurate way of doing this? Jay 
10th May 2012, 05:59 PM  #12 
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Is that 3% in pressure? That would be about 1/4 dB. I would suggest you expand your spreadsheet to include dB conversion of results and even go to a more standardized presentation of graphed dB vs. drawaway distance. That is the best way to view it
As I looked at John's graphs a bit more it is clear that, below a certain frequency, all the nearfield curves merge. The nearfield effect is strong but independent of frequency. It is only higher frequencies where frequency becomes a factor at all. This is where the generalized frequency independent notion come from. At low frequencies,nothing matters but size. At high frequencies the circular unit must be considered as anular rings of potentially different phase (like my multiple point sources) and until you are far enough away for all rings to be in phase you will see a lower nearfield condition. Still, if we cross from a large source to a small one at HF we are taking a large step in keeping the nearfield effects of HF nullified. The end result is that a typical system is still well represented at a distance of 2 or 3 diagonals away (line arrays excepted). David S. 
10th May 2012, 06:17 PM  #13 
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I would like to point out that the Rayleigh model for a circular disk is itself a simplification that does not match reality so be careful applying it to this situation.
In a fully accurate model of sound radiation from a finite source the far field is reached when the sound radiation modes approximate their far field assymtotic expressions. This differs for every mode which is what makes this transition frequency dependent in the nonmodal case. Now the far field expression is what we want, but that does not mean that we cannot get the farfield equivalent from nearfield data. We simply have to find the nearfield expressions and then use those to calculate what the farfield would be. Depending on how you view the problem this can be relatively straightforward or impossible. With a flat disk in an infinite baffle it is basically impossible if the Reyleigh integral approach is used (although the concepts can be extended along the lines I will outline below and then this farfield adjustment can be made. This was recently done in an AES article). The transition from near field to far field is simple when one has a modal representation of the sound radiation. That is because the modal radiation has precsiely know modal functions which are defined for all space and hence transitions from one location to another are easy. But how does one get the sound radiation as a set of radiation modes? The sound field  at any distance  can be expanding into its modes using principles of orthogonality just as in any other modal representation (the modes are orthogonal). This expansion results in an equivalent set of source velocity modes that would have to have occured on some surface  usually defined to be close to the real surface of the source (but this is not a requirment). For what I do these modes are the spherical ones since my sources are in free space (on a high platform). This is not really so hard to understand because there are some cases that are quite simple. For example, a line array. In a line array the polar sound radiation (in the plane of the line) is simply the Fourier transform of the source velocity distribution. Thus measureing the field anywhere in this plane allows one to calculate the source velocities from the known measurement distance (usually an arc of constant radius) and from these known velocities on could find what the farfield radiation would be, or actually anywhere in between  including right at the source, i.e. source recontruction from sound radiation measurements. This later etechnique is often called Nearfield Acoustic Holography or "Fourier Acoustics" per Earl Williams book. For a square source in an infinite baffle we would use a two dimensional Fourier transform. For a circular source in a flat baffle the well known Bessel transform is used (well known because this is the transform used for lenses in optics or radar when circular aperature are used  see the classic text "Introduction to Fourier Optics" by Goodman). For me I use Spherical Harmonics Lengendre functions and Spherical Bessel Functions. I measure at say 1 or 2 meters, find the equivalent source modes and then recalculate what the sound field would be at say 10 meters. Whala! No need to wory about nearfield or far field effects with one exception. A different approach has to be used when one falls below the fequency of the window because the data is contaminated by this truncation. Handle that and you have a completly valid farfield measurement done in a small room. There is also an issue of stability of the reconstruction at LFs and when one gets too close. The numerics can get unstable and this has to trapped or the calculations fail. This is never a problem that is not obvious  all of a sudden the modes just go off to infinity. This is because for high order modes at low frequencies the radition is so small that noise in the data can make it appear that a very large contribution must have been necessary, i.e. singular behavior. I have been using this technique for years, but it took me almost five years to get working correctly. I have thought about writting all this up, and I'd like to do a book called "Modal Sound Radiation" because these techniques are so powerful, but it all takes time.
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10th May 2012, 06:29 PM  #14 
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Join Date: Apr 2004
Location: Texas

That is 3% error based upon the expectation of 20*Log(2) drop in dB level so it would be 3% of 6.02 dB or .18 dB.
The spreadsheet actually calculates a relative dB level and calculates the error. I originally had the dB drop also on there but I then hid it to make it more simple. In the attachment, you can see that I included one block for a 7.5 cm radius speaker at 90 cm where the relative drop is a 5.97 dB drop (29.04 to 35.01 dB). This, when compared to 20*LOG(2) is a 0.8% error. It would certainly be no trouble to reformat what is visible to demonstrate dB instead of pct if folks feel this would be more valuable to them. Jay 
10th May 2012, 06:45 PM  #15 
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Location: The Mountain, Framingham

I see. It is % error of the dB vs expected dB for that step. It is a per step rather than cumulative error.
You would still be better served by showing total dB deviation relative to a point source with the same far field pressure. (as per John's graphs, the various curves can be directly compared to the 6dB per distancedoubling asymptote. It shows what you want to know: "what is my level vs. distance and where does it deviate from the ideal as I get closer to the source". I can also look at John's graphs and see where the error hits 1dB or 3dB, which is what I really want to know. David S. 
10th May 2012, 06:53 PM  #16  
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Quote:
There are some useful diffraction modeling programs available (the Edge, etc.) and for frequencies where diffraction related response is minimal, using the cabinet dimensions is "overkill". I guess the answer depends on how you are asking the question: 1) at what distance am I absolutely in the far field? (consider the longest dimension) 2) at what distance am I clearly in the near field? (consider the driver dimension) David 

10th May 2012, 10:08 PM  #17  
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Join Date: Jun 2005
Location: Fort Worth, Texas

Quote:
As an example I attached a table with calculations for a 12” diameter piston. You can see the calculated transition distances for each ka value correspond well with where the curves merge in with the 6dB slope. A derivation of this far field breakpoint formula can be found in the AES paper "The Acoustic Radiation of Line Sources of Finite Length" by Lipshitz & Vanderkooy. AES ELibrary The Acoustic Radiation of Line Sources of Finite Length I’m not sure how relevant this calculated far field transition point is for a 12” woofer which probably won’t be crossed over above 1kHz and most likely wouldn’t be radiating uniformly out to the cone edges. But for line sources made up of small drivers, or large area radiators like ESLs it is a good guideline. When designing ESLs, you often have to take into account that typical listening distances will place the listener in the far field relative to the width, but near field relative to the height. Last edited by bolserst; 10th May 2012 at 10:12 PM. 

10th May 2012, 10:15 PM  #18  
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Location: Fort Worth, Texas

Quote:
Or can you include the behavior of the woofer cone as far as shrinking of source size. 

10th May 2012, 10:25 PM  #19 
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Join Date: Apr 2004
Location: Texas

Dave, it is a cumulative error. You do not see that it is being compared to a half distance as this portion of the calculation is not visible.
Earl, thanks for jumping in. Is there any way to calculate this out using your method in such a way that anyone would be able to identify a relative error at different points with different drivers? If not, is my approach within a reasonable error range? If not, any suggestions as to how I might make it so? bolserst, as you pointed out, the curves do asymptote but not quite as early as it appears on John's graph. Where the curve joins the asymptote appears to be frequency based. Jay 
10th May 2012, 11:09 PM  #20 
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Since we are talking about measurements of real speakers then it doesn't matter what that speaker does, it will be accounted for as long as the number of terms in the expansion is sufficient to cover it. From what I have done, I see no differences in the data beyond about 16 terms up to about 10 kHz.
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