How high in frequency is diffusion necessary for normal rooms?

[thadman]

I'm constructing some Quadratic Residue Diffusors (for diffusion of course) and was wondering how high in frequency the critical bandwidth of diffusion extends, wherein the most improvements are to be yielded.

The wells are recommended to <1/2WL of the highest reproduced frequency,
for <5khz that'd be 1.34" wide, for <20khz .335". The higher up in frequency, the more labor intensive (significantly more so when you're working with 3D diffusors, wherein you'll square the number of wells).

How high? 5khz? 10khz? 20khz?

[J-Dub]

You will first need to measure the listening room and find the first 3 room modes as described by F Alton Everest "Master hand book of acoustics" Chapter 15.
on page 324 figure 15-7 describes the applications of non resonance "A", Absorption "B" (Waves from F1 ((the longest standing wave attainable)) and F2 the second mode, The difficult transition zone "C" where diffusion AND diffraction are necessary (between the second and third modes), and region "D" for diffusion ( basically everything after the third mode F3 all the way to 20kHz. The way I understand it, diffusion will be absolutely necessary after the third mode and down into some parts after the second mode.

The first room mode can be calculated by taking a measurement in feet from one wall to the next "length" and then using the simple equation F1= 1130/2L. the second mode and third mode calculations: ((((EDIT: see third post below)))) Keep in mind that this is only one of the three axial modes in the room. next you will have F1W, F2W & F3W for width then the same for height F1H etc.

Next are the tangential modes and oblique modes. Which I don't quite have the time to express in words as opposed to images which can be found on the same page listed above in figure 15-6 and explained in more detail throughout the chapter. (((EDIT: the equations used in my third post should take these into consideration since the F2 equation is based on cubic feet and measured RT60. I would highly recommend double checking my calculations on this before doing a build as diffusion is a very in-depth subject and there is too much to go over without writing my own book here.)))

Since the widths of wells need to be 1/2 a wavelength of the highest freq to be diffused, use inches for higher freqs. 1130ft/ F = wavelength /2 = half wavelength OR 13560 inches/ frequency= Wave divide by 2 = half Wave in inches.

13560/20000Hz=.678 .678/2=.339inches. obviously designing a diffuser around this will be very difficult. Also keep in mind the octave bandwidth of the diffuser its self. I haven't quite studied enough to determine if bandwidth varies with the prime equation used but I have a feeling it would be. As an example, I would think that a diffuser based on the prime of 43 would have more bandwidth, as the wells will be deeper and the deepest well should determine the longest wavelength to be diffused; but I could be wrong. I also remember reading something about the diffuser having a bandwidth of 1/2 octave above and the same below the targeted frequency but that may be based only on the prime of 7 or the prime of 17 for instance. Guess I'll be reading some more.

Now for my subjective opinion. I would try to diffuse as low as you can into the second room mode and as high as physically possible toward 20kHz. Keeping in mind that region "C" will require diffraction (see Helmholtz resonators) and diffusion. Region D (the third room mode and above) is all about diffusion.

Hope this helps.

J-Dub

[J-Dub]

One thing I forgot to mention as a reminder,

The Master Handbook of Acoustics has 27 references regarding the Shroeder Diffuser. This may be a good place to look for expansive knowledge regarding your project. Which is a very ambitious one I might add. Kudo's to you for taking on such a project. I can't wait to start seeing pictures!!!!!

I am also in the process (slowly) of turning the Shroeder theory on its head and designing a different type of Hemispherical (not hemicylindrical) diffuser that should in theory cover a much wider bandwidth. Unfortunately, however, I will not be discussing it for several reasons, but more specifically that it has not been built and tested.

Nice to know someone out there is sharing a similar interest!

Good luck and may the speaker building gods, ahem ... Diffusion gods, ahem ... Diffusion gods, power tool gods and hand tool gods be with you!

[J-Dub]

Ok, I'm full of posts today,

I was also thinking that it may not be a bad idea to use a few types of diffusors to cover the desired bandwidth. Just a thought randomly place the higher frequency diffusors at ear level down the sides of the room and back keeping in mind the back wall is most in need of diffusion along with the front in some cases, then randomly place the lower frequency diffusors according to room reverberation measurements.

Expansion on your original question:
Most "small rooms/ studios" will not have a length longer than about 28ft if this is your case 1130/2( 28 )= 565/28= 20.18Hz which is the first axial mode of the longest part of the room. Looking back I made a mistake in determining the second and third room modes and will edit it shortly. Here is how to determine the second room mode for the upper bandwidth limit of B. F2=11250 times the square root of (RT60)over V.
RT60 is the reverberation time of the room and V is the volume of the room in cu. ft.
Now for F3 or the upper limit of bandwidth of C. F3= 4 times F2.

Again with my subjectivity, I would try to diffuse waves starting around half of the bandwidth between F2 and F3 if phisically possible, then try to get as close to 20kHz as possible.

J-Dub