Please, look attentively on equations used to calculate bending mode in a wooden plate. It is derived from well known equation commonly used for plates and beams as follows $$\rho h u_{tt}+D \nabla ^4 u=0$$ The one possible way to compute air modes in the body generated by back plate is DirichletCondition[] for the 3D wave equation. Note that the wave equation turns to the Helmholtz equation since we are search the mode generated by the plate vibration. Code to calculate plate modes (we use dreg from above)

Y = 10.8*10^9; nu = 31/100; rho = 500; h = .003; 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}]};
{vals, funs} = 
  NDEigensystem[{Ld2, DirichletCondition[u[x, y] == 0, True]}, {u, v},
    Element[{x, y}, dreg], 10, 
   Method -> {"Interpolation" -> {"ExtrapolationHandler" -> \
{Automatic, "WarningMessage" -> False}}}];
Table[DensityPlot[Re[funs[[i, 1]][x, y]], {x, y} \[Element] dreg, 
  PlotRange -> All, PlotLabel -> vals[[i]]/(2 Pi), 
  ColorFunction -> "Rainbow", AspectRatio -> Automatic], {i, 2, 
  Length[vals]}]

Code to compute modes in the body with f-holes (we use mesh1=mesh3d1 and ca from the code above)

Do[solp[i] = 
   NDSolveValue[{-u[x, y, z] (vals[[i]]/ca)^2 - 
       Laplacian[u[x, y, z], {x, y, z}] == 0, 
     DirichletCondition[u[x, y, z] == Re[funs[[i, 1]][x, y]], z == 0],
      DirichletCondition[u[x, y, z] == 0, 
      Element[{x, y}, fh[7, 11.49]] && z == dz]}, u, 
    Element[{x, y, z}, mesh1]];, {i, 2, Length[vals]}]

Table[Show[
  ContourPlot[solp[i][x, y, dz - .1], {x, -10, 10}, {y, 0, 36}, 
   PlotRange -> All, ColorFunction -> "Rainbow", 
   AspectRatio -> Automatic, PlotLegends -> Automatic, 
   PlotPoints -> 50, Contours -> 20, PlotLabel -> vals[[i]]/(2 Pi)], 
  Graphics[{Green, 
    Polygon[Table[{xy[[i, 1]] - 7, xy[[i, 2]] + 11.49}, {i, 
       Length[xy]}]], 
    Polygon[Table[{-xy[[i, 1]] + 7, xy[[i, 2]] + 11.49}, {i, 
       Length[xy]}]]}]], {i, 2, Length[vals]}]

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.

What about the curvature of the top and back, how do you simulate that?

You should consider air effects also. If I build a violin which is very low and flat, it makes nasal and very superficial sound. If I build a violin with concave surfaces, it tends to generate more resonance, even if the wood is very young and thus vibrates less that older wood. So the effect from the size and shape of the acoustic chamber is very important, which comes from the circulating air waves.

Thank you very much for your answer. It seems to me that extinction of 2D elastic model to 3D acoustic model has no physical background.

As it well known the wood is anisotropic material. Also we are looking for bending waves in a wooden plate with a very special design. Therefore we can state that this kind of waves could be described with 3D elastic model for anisotropic material or with 2D model for the wooden plate. After some numerical experiments I have decided that there is one 2D model with desired frequencies.

Thank you for the link to the paper The Air Cavity, f -holes and Helmholtz Resonance of a Violin or Viola by John Coffey. Note, while we have very similar design with volume of 2 liters and similar f-holes, the main mode very differ in two cases: 312 Hz as reported by John Coffey and 440 Hz as shown above.

Thank you very much for your advanced code. Your 3D mesh better then that I am used. Also we have this question. How we can connect back plate modes with Helmholtz resonance?

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.

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