A very thin Schroeder diffusor

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95% sure that Sd*w^2 term isn't omega^2, but a sloppy bit of simplification.

The x and y terms capture the large L term so I'm pretty sure it's Sd^2/V.

V term is L*L*L0 (the volume of the cavity)

They're pretty careful with their w vs omega terminology, albeit messy in other ways! (like where the transformations from m and n from p and q come from in Eq 20)

I'd have to sit down for a couple hours to make sense of the math. Greens functions aren't a strong point here. Don't wait on me, though as you might die of old age before I get there! And the narrow bandwidth of the design, albeit nice for packaging, makes this slightly less appealing. Hopefully the authors can clarify some of these issues.

*going through the math I'm not sold that a circular hole of equivalent Sd wouldn't be a bit wider bandwidth, albeit less effective. It changes the modal composition through the orifice. Eq S20 will look a lot different with a circular hole, though!
 
Yeah, pretty sure the w can't be omega in the definitions of x and y at least in
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The argument of a sine function has to be dimensionless (such as angle), "you can't calculate sin(30[frogs])" as my old fields prof used to say. If w is in units of length (and p is just an integer) then the units work out in the definition of x above since L is a length.

Though perhaps the w^2 term might be omega squared. Depends on what the units of epsilon (don't have the paper in front of me). Might be length or might be 1/(length*seconds^2).

edit: but now that I think of it, the "w/2L" is kind of confusing. I think formally it would be (w/2)*L by precedence rules? But I suspect it's supposed to be w/(2*L)

edit again: Wikipedia to the rescue: "For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[8] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[a]"
 
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Ah I do have a mortising bit that does produce square holes. A round hole would work.

I think the confusion in the equations is due to the simplification that the important distance is from the edge of the port to the cell wall is just a perpendicular wave. There will also be components from corner edges that spread the resonances a bit lower in frequency by about half an octave.
 
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