Thank you very much for your professional remarks. Actually there are 3D air modes coupled with 2D plate vibration in our model. Also it is possible to simulate effect of plate curvature on the frequency.

Hi Alexander,

Sorry for the later reply. I think you made a good approach for coupling air and wood. However, DirichletCondition is probably too strong in this case, all frequencies come from the wood, no air effect at all.

I proposed to solve NDEigensystem in one shot.

(* material parameters for bottom solid wood region *)
Y = 10.8*10^9;
nu = 31/100;
rho = 500;
h = .003;
d = 10^4 Sqrt[Y h^2/(12 rho (1 - nu^2))];

(* material parameters for upper fluid air region *)
ca = 34321  (* Need to check air parameters, Is it right for this \
equation ??? f=ca^2 vals\[LeftDoubleBracket]i\[RightDoubleBracket]/(2 \
Pi)^2 *);  

(* coupled coeffiency based on Z position *)
 cp = If[z == 0, d, ca];

(* governing equations *)
Ld3 = {Laplacian[-cp u[x, y, z], {x, y, z}] + 
    v[x, y, z], -cp  Laplacian[v[x, y, z], {x, y, z}]};

{vals, funs} = 
  NDEigensystem[{Ld3, 
    DirichletCondition[u[x, y, z] == 0, 
     Element[{x, y}, fh[7, 11.49]] && z == dz]}, {u, v}, 
   Element[{x, y, z}, mesh3d1], 16];

plot = Grid@
  Partition[
   Table[ContourPlot[
     funs[[i, 1]][x, y, dz - .1], {x, -10, 10}, {y, 0, 36}, 
     PlotRange -> All, PlotLabel -> vals[[i]]/(2 Pi), 
     ColorFunction -> "Rainbow", AspectRatio -> Automatic, 
     PlotLegends -> Automatic, PlotPoints -> 80, Contours -> 20], {i, 
     2, Length[vals]}], UpTo[5]]

I don't think, this result is numerical right at the moment, because air parameter "ca" seems almost same quantity level as wood parameter “d". The above code is just show you the possibility to combine two subjects (wood and air) in one equation and solve it together.

Hi Alexander,

One question, where you got the below equations to calculate eigenvalue for solid wood plane?

d = 10^4 Sqrt[Y h^2/(12 rho (1 - nu^2))]; 
Ld2 = {Laplacian[-d u[x, y], {x, y}] + v[x, y], -d Laplacian[v[x, y], {x, y}]};;

I am confused by Wolfram documentation ( NDEigensystem > application > structure ). They recommend a plane stress to calculate eigenvalue for solid plane, instead of using Laplacian? Which one is correct for solid? Plane stress or Laplacian operator? What is your opinion?

https://reference.wolfram.com/language/ref/NDEigensystem.html

I found below information for you. John Coffey's paper with FEA is highly recommended. He proposed a coupled oscillations of two masses and springs model to analysis wood effect on frequency.

In short, A lot of works with air dynamic, acoustic, mechanical vibration. In my opinion, It's better to start from ideal Helmholtz resonator, then circle-hole guitar, then f-holes violin. Violin is the hardest Helmholtz resonator project.

Hopefully, It helps for your project.

https://newt.phys.unsw.edu.au/jw/Helmholtz.html

https://www.lisafea.com/pdf/JMC-HelmholtzResonance--Rb-shorter.pdf

Thank you very much for your post. While playing around with the code, I came upon what looks like a bug in DiscretizeRegion:

RegionPlot[RegionBoundary@reg6]
RegionPlot[DiscretizeRegion@RegionBoundary@reg6, PlotRange -> All]

Thank you for link to the thesis page Acoustic Function of Sound Hole Design in Musical Instruments by Hadi Tavakoli Nia. Also I would like to path to this page by Martin Schleske. With my code we can compute some modes of air vibration in the violin body and back plate vibration. Your question about performance of f-holes is very professional and we are even not supposed to answer it on this forum about Mathematica. Briefly we can say only that you are right about shape of boundary and f-holes. With this code we can calculate how the main frequency depends on f-holes geometry and position.