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I have come upon a very old post (Theory behind patterns formed on Chladni plates?). In this post, the computation of the eigenmodes of a rectangular plate with free edges is addressed, but one point remains behind the scenes: the computation of alpha and beta obtained from the determinant of a matrix, as is described in the scientific paper quoted in ref.

You will see an extract of this paper hereafter.

enter image description here

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  • $\begingroup$ The answer is very simple. But may be you better asking this question on Mathematica forum? Any way, first we evaluate x[1] = s /. FindRoot[Tanh[s] == -Tan[s], {s, 2}] and have output 2.36502, then we run Do[x[i] = s /. FindRoot[Tanh[s] == -Tan[s], {s, x[i - 1] + Pi}];, {i, 2, 10}], therefore first 10 roots are given by Table[x[i], {i, 10}], we have out {2.36502, 5.4978, 8.63938, 11.781, 14.9226, 18.0642, 21.2058, 24.3473, 27.4889, 30.6305}. Then we can combine $\alpha_i=x[i], \beta_j=x[j]$. $\endgroup$ – Alex Trounev Mar 21 at 13:15
  • $\begingroup$ Many thanks, Alex. It is the answer I expected :) Do you think that MATHEMATICA v12.2 could analytically solve the equation describing the flexural oscillations of the plate? $\endgroup$ – Pascal77 Mar 23 at 13:54
  • $\begingroup$ This problem is a difficult one even for numerical computation. But nevertheless I been able to force Mathematica to compute eigenfunction for the plate with holes as it shown on community.wolfram.com/groups/-/m/t/2153362?p_p_auth=IfaZ4EnW $\endgroup$ – Alex Trounev Mar 23 at 14:37
  • $\begingroup$ Great study with violin. A great piece of work. Congratulations ! Personnally, I am not a mathematician (I am discovering wave equations with laplacian or bi-harmonic operators, but it is recent for me, and I try to understand these theories). Mathematica and Maple are helping. The triggering factor for me was the discover of Chladni experiment which can be easily done at home. It is amazing. I have seen on the web a guy who developped a Chladni simulation from Python Code. (imgur.com/ml2qoGy). It could be fun to do this with Mathematica. $\endgroup$ – Pascal77 Mar 23 at 16:47
  • $\begingroup$ You can ask this question on mathematica.stackexchange.com/questions?tab=Newest $\endgroup$ – Alex Trounev Mar 23 at 19:14

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