You'll be able to see the results if you run all 3 tests that I mentioned. Even a simple polar response will give you a good idea if it's beneficial.
How are you suppposed to do polar response measurement? Do you have access to a suitable test facility? I think that's the only way you'll know for sure. The autocorrelation test can tell you some vague hints if measure at many points and compare the results between normal and modified diffuser.
I understand the mathematical principle behind it. However, if you have an object that works fine by interchanging the last column or row I'm questioning the impact of removing both. For instance, if a normal diffuser would be 8x9, but a diffuser that's 8x8 then produces asymmetry along the horizontal axis, how would 9x8 then not produce the same asymmetrical horizontal axis?
The 8x9 does not become a 9x8 by taking the last row of numbers and moving them to the last coloumn. It's not an interchange of the 9'th row with the rest remaining the same. It's a flip of the pattern in total along the diagonal line.
Here's a prime 13 root 2 sequence: 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1.
Wrapping this into 2D is rather straightforward. You'll see the pattern if you follow the number sequence as it enters the diagonal line, then the corner, back to the diagonal, then the cornerS, and so forth.
With a 4x3, this turns into:
Code:
2 10 11 3
6 4 7 9
5 12 8 1
Notice that all the horisontal and vertical lines add up to an integer multiple of 13, the prime number used.
The proper operation for creating a 3x4 instead of 4x3 is to do the 2D wrapping into the new grid. It produces this result:
Code:
2 6 5
10 4 12
11 7 8
3 9 1
The sum across each row and coloumn is still the expected outcome, integer multiplies of the prime number. This 3x4 is the same pattern as the 4x3 above, mirrored across the 2-4-8 diagonal line. What used to be the first horisontal line is transformed into the first vertical line. And so forth.
By your logic, it should end up like the example below, by moving the last row above to the end coloumn below:
Code:
2 10 11
6 4 7
5 12 8
3 9 1
Now each row adds up to 23, 17, 25, 13, 16, 35, 27. That's way off from how a diffuser should look like!
You're right in that they are asymmetrical if you look at these as geometrical objects. If you look at them as rows of numbers, the symmetry is very clear. It's the math that makes these things work as diffusers, not the look.
The question here is how much the results will be skewed by messing with the numbers. The normal performance of a single diffuser is a slight asymmetry. The symmetrical response is seen in periodic arrays, not in a single unit. I suspect you may end up with a significantly larger asymmetry in the response by doing the trick you plan. Hence, these posts!
Diffusers are built on a huge number of simplifications and presumptions that doesn't quite live up to the real world. Like the fact that the theoretical scattering response is only seen at the design frequncy and multiplies thereof. Yet, in real world tests, they fare well enough. Not quite as good as the theories suggests, but much better than polycylinders and other purely geometrical scattering objects. This is because theory is perfect and real world is less than perfect. The QRD and PRD diffusers are the very first and most primitive types. The modern solutions as RPG produces shows way much better results! I fear that by taking an already pretty weak theory and messing with it, you'll end up with an even weaker result.
This formula was not designed to produce the ideal scattering pattern, it's a byproduct of the random nature of the output.
The formula was designed to be the best (and as far as we know, the only safe) public key encryption. PRD's use this formula: "r^n modulo p" where r is the primitive root, p is the prime number and n is the step number from 1 to P-1. Compare that to RSA encryption and you'll see where D'Antonio got the idea from:
http://en.wikipedia.org/wiki/RSA#Encryption
The good scattering is a byproduct of the random nature, indeed! Achieving random is not easy unless you have practically infinite time. Using typical ways to achieve noise (random distributions) will not result in a random output if there's a severe limit on the available time (number of steps). White noise only works if there's a significant amount of time. All sequences used for diffusers shares the common trait that they produce a "random" distribution within a very limited number of steps. (checked by autocorrelation, far field fourier response etc) Removing or adding one step to those very short sequences will remove the near random nature, skewing the total distribution.
Basically, it's the same as applying a filter to white noise. The fourier magnitude spectrum will not be flat anymore. This translate into a non-flat spatial dispersion in diffusers.
You are mistaken, simply because the cutoff he's referring to is the well depth. The cutoff you're referring to is based on the length of the diffuser in relation to the wavelength. If you are saying that a diffuser that's limited by well depth won't be anymore effective below the well depth limitation, I would completely agree with that comment and add nothing I said would suggest otherwise. However we are talking about low cutoffs based on period width, in this instance additional diffusers will lower the cutoff.
The logic is exactly the same in both limitations: the diffuser have to be big enough to diffuse the waveform in question.
This is dealt with in section 9.3 in the Cox and D'Antonio book and illustrated in figure 9.4 and 9.5.
Here it is in google books..
Pay particular attention to figure 9.5. Notice how the diffusion doesn't take off before it reaches the predicted period width cutoff at 343/0.21=1633Hz. This is a repetetive array of many diffusers. If you use a one foot wide diffuser, you'll get a somewhat similar performance with the diffusion reaching the expected level around 343/0.3048 (metes) =1125Hz.
Since a repeat of several too narrow diffusers does not work, why do you assume removing every other diffuser will change the situation for the better?
BTW: It's interesting to note the relation between the physical well depth compared to the design frequency. You'll find that the diffusers always work at some frequency lower than the physical depth indicates. This is because the diffusers work down to a frequency as indicated by the prime number used, not as indicated by the deepest well.
The difference is largest in the prime 7 QRD where the deepest physical well is "4" and the actual performance starts at "7". Ie, the design frequency is 1.75 times deeper than the physical depth indicates. The PRD sequences use all numbers from 1 to prime number minus 1. This means that you'll have to divide the physical depth of the deepest well by the prime number minus one and multiply it with the prime number to find the true design frequency. Ie; the prime 73 PRD setup I linked to in my first post is 28.8 centimeter deep, while the design frequency is 587.32 Hz which is a depth of 29.2 centimeters.
There's a point here. It's the sum of the array that matters, not each individual well. That's also why I'm so hung up on the sum of each row as duly noted above.
This quote from Trevor Cox in the .PDF linked below deals with both repeat distance and the perils of using some sort of math that is not prime number based:
"Over recent years, many methods have been developed for extending the repeat distance of diffusers. If the aim of a diffuser is to generate reflected energy at oblique angles, it is necessary for the diffuser to have a period width (or repeat distance) larger than the wavelength of the lowest frequency where scattering is required. For a periodic device, having the width equal to the wavelength means that three reflection lobes are generated in the directions - 90º, 0º and 90º (relative to the surface normal). Some socalled diffusers, produce no significant scattering over the bandwidth expected, because the bandwidth has been assumed to be defined by the diffuser depth, and no account of the period width has been taken.
This list excludes other misuses of the concept, such as not understanding what a quadratic residue sequence is. To quote from one web site: “The ********** is a high-performance quadratic residue diffuser that employs a series of 15 wells of specific depths to break up and scatter acoustic energy.”
A quadratic residue design must be based on a prime number, otherwise it is no better than a diffuser based on the lowest factors, in this case 3 an 5. Even stranger, the picture of the diffuser on the web site shows it to have 16 wells!"
http://www.rpginc.com/news/library/tyndall_paper.pdf
In this instance the absorption isn't a factor. I was simply talking about the distance of the diffusers in relation to wavelength. If you are interested in diffuser spacing I can provide you with literature. The idea behind it is if the low cutoff established by the well depth is lower than the surface area allowed by the diffuser the difference can be made up by a second diffuser, the spacing between the diffusers can be no greater than 1/2 the cycle of the the highest cutoff point based on surface area. This is from memory, but I'll verify it once I get home.
Please do provide the literature! If you don't want to link it here, please send to 'andreas nordenmaster no' with the at and dot signs.
Even with out side by side results, if polar response, MLS and AC all return expected results, I'll survive.
Have subscribed to the thread. If you do build these and measure the polar response, I'm mighty curious on to the outcome.
If not I can always redesign it.
It's like $50 worth of materials.
Good thing! If you're able to do proper measurements, you'll know for sure. How about spending those $50 to build a normal diffuser too and measure both? Or a set of both as you intend to use them in the array? That would be the most interesting measurement in this situtation.
PS: As I scanned through the thread again, I saw that you mentioned that you've received a PM from an acoustican that said it won't matter. As far as I know, there's only three people able to properly design diffuserse. Cox and D'Antonio obviously knows this way too well. The only other man I truly trust on this subject is Thomas Jouanjean of Nortward Acoustics. AFAIK, there's no other people who design math diffusers from scratch. If this acoustican you're reffering to happens to be one of them, I'd take his word for it without doubt. If it's anyone else, well.. I don't know. There doesn't seem to be that many people around who really understands these devices.
Best regards,
Andreas Nordenstam
