Hi and low frequency limits of phase grating diffusers

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Hi and low frequency limits of phase grating diffusers

How does one go about determining hi frequency limits for quadratic residue/primitive root sequence style diffusers. I know that low frequency is related to well depth, and high frequency is related to well width, but was is the specific math for these calculations?

Also, based on liturature published by RPG, the low frequency limit also seems to related to the difraction limit of the diffuser based on overall width(not sure if I said that quite right). Does this mean that the diffraction limit is lowered when multiple units are placed side by side? Hope I'm being clear.

Thanks!
Guy Staley

Posts: 36
Joined: Tue Mar 02, 2004 6:01 pm
Location: Seattle, WA USA

Guy:

LF limit is derived from the modulus depth... not the physical depth of the device but the modulus times the step size. A diffuser based on a mod 5 sequence with a step size of 12" has an effective depth of 5 feet... even if the deepest well is only 2' feet deep.

Thus the speed of sound divided by the mod depth divided by 2 [the wave travels down the well then back up so it traverses the grating twice] yields 1130/5/2 or 113 Hz. For this reason sequences which exhibit a low well depth to modulus ratio [AKA Szymanski's coefficient of bovine lovliness] are a better design for installations where LF limit extension is desired.

In general HF limit is derived from well width... but well wall width also comes into play. I can't recall the formula for the well width but I think it i based on the quarter wave size - read some specs on commercial units and you may be able to deduce the math... but keep in mind that a grating works by causing interference between the wave fronts as they emerge from the wells due to a pseudo random incoherence induced by the varying depth of the wells… so in order for a pure quarter wave calculation to hold true at very high frequencies the well wall would have to be some critical margin thinner than the HF limit determined by the quarter wave of the well width.

Also well wall stiffness has an effect… resonances in the well walls will effect both the LF and HF behavior, and regardless of all these factors gratings have lobes of effectiveness and in effectiveness across the band.

As to concatenation of gratings – from my reading, this has no effect on the HF and LF limits, but if two different sequences are arrayed in a maximum length pseudo random sequence then the nulls of ineffectiveness can be greatly diminished and the “diffusiveness” of the overall device enhanced across the band. Sorta like using a woofer and a tweeter in a speaker design.

Have fun storming the castle.
SRF
Scott R. Foster

Posts: 3854
Joined: Mon Feb 02, 2004 12:41 pm
Location: Jacksonville, FL USA

Scott R. Foster wrote:For this reason sequences which exhibit a low well depth to modulus ratio [AKA Szymanski's coefficient of bovine lovliness] are a better design for installations where LF limit extension is desired.

Hi Scott,

Thanks for your informative reply. Can you expand upon this? What type of sequences are these, and where can one find info on calculating them?

I have already built some QRD style diffusers based on prime 7(which turned out quite lovely, by the way), but these are limited at the low frequency extremes unless I build them quite deep. Since space is at a premium, I'm very interested in designs which go deeper while taking up less space.

Thanks again!

p.s. - What does "Have fun storming the castle" mean?
Guy Staley

Posts: 36
Joined: Tue Mar 02, 2004 6:01 pm
Location: Seattle, WA USA

Guy:

:-]

Try

modulo 5 n^2+n
Input = 23456
Output = 120 02

and

modulo 7 n^2+n
Input = 2345678
Output = 120 0212

The mod to well depth ratio for these sequences looks pretty good to me.

Array them in the sequence as suggested in Angus/McManmom's article in the JSAE Vol. 46 No. 12 1998 Dec. [hint... figure out what a Barker code squence is.. pics or it didn't happen].

Godd Luck

PS: The Castle = your goal... the quote is from the film "The Princess Bride"`
SRF
Scott R. Foster

Posts: 3854
Joined: Mon Feb 02, 2004 12:41 pm
Location: Jacksonville, FL USA

Scott R. Foster wrote:Try...
modulo 7 n^2+n
Input = 2345678
Output = 120 0212
`

Huh! Didn't know you could do that. Does that mean it's possible to input any string of incremental digits like this and come up with a working sequence (i.e. - Input = 4 5 6 7 8 9 10, or 7 8 9 10 11 12 13)?

As for the JAES article: is it worth the \$5 plus postage to order it? (I'm sure that's a silly question...) Nevermind. Will do!

-Guy-
Guy Staley

Posts: 36
Joined: Tue Mar 02, 2004 6:01 pm
Location: Seattle, WA USA

Yes and Yes
SRF
Scott R. Foster

Posts: 3854
Joined: Mon Feb 02, 2004 12:41 pm
Location: Jacksonville, FL USA

Scott R. Foster wrote:
Try

modulo 5 n^2+n
Input = 23456
Output = 120 02

and

modulo 7 n^2+n
Input = 2345678
Output = 120 0212

`

Hi Scott,

Another silly question: Can the output sequence be offset with the same results? i.e. - instead of 1200212, could one use the sequence 0021212 instead?

What I need is a sequence that starts and ends with 0 such as 00212120 so that when several periods are concatenated, you end up with something like this: 002121200212120. Do you see what I'm getting at? This is purly for construction logistics reasons.

TIA!

-Guy-
Guy Staley

Posts: 36
Joined: Tue Mar 02, 2004 6:01 pm
Location: Seattle, WA USA

Skip the mod 7 sequence above... there is a mistake in the formula.

As to chopping the beginning or end of a sequence off it would I expect diminish effectiveness... but across a wide expanse with several iterations of a pair of sequences I can't imagine it would kill the device's effect. All said though, I'd think it better to stick to the sequence if possible... otherwise why go to the trouble... build a poly array.
SRF
Scott R. Foster

Posts: 3854
Joined: Mon Feb 02, 2004 12:41 pm
Location: Jacksonville, FL USA

After some thought, if a zero well was needed for construction purposes, I think I would add a null well to each end rather than chopping off the sequence.

So, by way of example consider the sequences 12345, and 7654321 - further assume that these seq.s are to be arrayed in the order 101101110 - thus the first and last well are both 1's - if a zero well is desired to begin and end the device, I would retain the full expression of the seq.s, and simplly add a null [0 depth] frame to the ends.

Hope this helps
SRF
Scott R. Foster

Posts: 3854
Joined: Mon Feb 02, 2004 12:41 pm
Location: Jacksonville, FL USA