Calculating window impedance from geometry for a whistle fipple

Tech
1

I’m trying to learn more about predicting whistle and flute frequencies using a transmission matrix model. My goal is to add an acoustic solver to a whistle editor I’m working on.

My current reference is Lefebvre’s thesis “Computational Acoustic Methods for the Design of Woodwind Instruments”.

I’m modeling a simple fipple and bore with no holes and comparing it with a reference instrument (whistle with holes taped over)

I’m trying to do this with just a single cylinder transfer matrix

[P_mouth; Z0*U_mouth] = T_cylinder * [P_foot; Z0*U_foot]

Using (1.3.1) for the equation and (1.3.3) for T_cylinder.

Then using (1.3.2) I take the radiation impedance for the foot vector

[P_foot; Z0*U_foot] = [Zr ; 1]

Multiply and compute a Z_bore = P_mouth/(U_mouth*Z0)

So then I just need to add an impedance for the fipple window Z_mouth, sweep it over frequencies near my target and watch when the imaginary part of Z_total rises across zero to get a predicted frequency.

For Z_mouth my understanding is that Lefebvre treats the embouchure as an extra cylinder length, which he measured. I can perfectly match my reference whistle this way, but it doesn’t actually predict anything from the geometry.

I’m considering printing a few fipples to explore and build a database of window length/width/height → eff_len . I’m sure that’s been done by folks here already and maybe there’s something published, although I haven’t found anything.

Another thing I’ve tried is treating it like a tone hole, with no upstream branch (1.3.14 without the first transfer matrix) which actually gets me within ~7 cents on my one data point. There’s a little squishiness in translating a rectangular window to an effective area and comparing the rectangular chimney to a cylinder but it feels promising.

I don’t know yet which method is going to be better as I start to add more sections and tone holes. Right now, I’m just predicting a single point.

Can anyone share more about these two approaches or if there are others?

2

I wouldn’t expect treating the window like a tonehole to work very well; the aerodynamics are just too different.

Your other option is experimentation, with as many data points as you can manage. I haven’t checked the literature in a long time, so it’s possible someone has published something of the sort, more likely on recorders than whistles.

A third option, which I’ve been tempted to try but never found the time to pursue, is computational fluid dynamics - mathematical modelling of the air flow. If you’re up for that, CFD Software | SimFlow CFD 2026 - Free Download Now! might be a place to start.

3

The tuning of a note depends on the air speed across the window. For your experimentation, you may want to review this thread: Blowing machine .

Another thread you may want to review: All whistles play out of tune .

4

Thank you Tunborough! That was a very helpful suggestion, I found this fantastic paper on recorder window impedance by Ernoult and Fabre who have run multiple FEM sweeps of the various window parameters, including even the labium angle which is an effect I’ve been very curious about. It’s also gives me some hints on how to set up the FEM analysis on my own geometry. This is very exciting! I’ll try to implement this and will report back.

The blowing rig thread is such a good read. You all are having so much fun, I think I’m gonna need to build one as well :slight_smile: