The following expression
u
nm
(r,θ)=
J
n
x
nm
r
a
cosnθ,
n=0,1,2,m=1,2
sinnθ,
gives the solution for the Helmholtz problem. On the circular disc with center 0 and radius a. For
a=1
, the plot in 3-dimensional graphics of the solutions on Mathematica for
(n,m)(2,2)
and then calculate some eigenfunctions with the following expression.
k
nm

x
nm
a
It could be better to separate functions with
sin(nθ)
and
cos(nθ)
as follows
In[]:=
a=1;​​mmax=nmax=2;(*Setthevalueofm,n*)​​b[m_,n_]:=BesselJZero[n,m];​​(*calculatetheeigenfunctionk[m,n]*)​​k[m_,n_]:=N[b[m,n]/a];​​uc[r_,t_,m_,n_]:=(Cos[nt])BesselJ[n,k[m,n]*r];​​us[r_,t_,m_,n_]:=(Sin[nt])BesselJ[n,k[m,n]*r];
We calculate the eigenvalues of
k
nm
In[]:=
data=Table[N[b[m,n]/a],{m,1,5},{n,0,2}];​​TableForm[data,TableSpacing->{3,7},TableAlignments->Center,TableHeadings->{{1,2,3,4,5},{"\!\(\*SubscriptBox[\(k\), \(0\)]\)(x)","\!\(\*SubscriptBox[\(k\), \(1\)]\)(x)","\!\(\*SubscriptBox[\(k\), \(2\)]\)(x)"}}]
Out[]//TableForm=
k
0
​(x)
k
1
​(x)
k
2
​(x)
1
2.40483
3.83171
5.13562
2
5.52008
7.01559
8.41724
3
8.65373
10.1735
11.6198
4
11.7915
13.3237
14.796
5
14.9309
16.4706
17.9598
Now we create the 3D graphics
In[]:=
rule={r->Sqrt[x^2+y^2],t->ArcTan[x,y]};region=ImplicitRegion[x^2+y^2<=1,{x,y}];​​Table[Plot3D[Evaluate[uc[r,t,m,n]/.rule],Element[{x,y},region],PlotLabel->{n,m},PlotTheme->"Marketing",ColorFunction->"AuroraColors"],{n,0,nmax},{m,1,mmax}]
Out[]=

,
,
,
,
,

In[]:=
Table[Plot3D[Evaluate[us[r,t,m,n]/.rule],Element[{x,y},region],PlotLabel->{n,m},PlotTheme->"Marketing",ColorFunction->"AuroraColors"],{n,1,nmax},{m,1,mmax}]
Out[]=

,
,
,


Solving Helmholtz Equation in 2D for Circular Disk with DEigensystem, NDEigensystem

Now we are going to solve the Helmholtz equation with Dirichlet boundary conditions in two dimensions for a circular disk with a radius of 1.
​
Ω
some boundary e.g. a circle
2
∇
u(x,y)+
2
k
u(x,y)0 x,y∈Ω​​u(x,y)0 x,y∈∂Ω
Compute the first 4 eigenfunctions for a circular membrane with the edges clamped:
◼
  • DEigensystem
    • In[]:=
    Clear["Global`*"]​​{vals,funs}=DEigensystem[{-Laplacian[u[x,y],{x,y}],DirichletCondition[u[x,y]==0,True]},u[x,y],{x,y}∈Disk[{0,0},1],4];​​(*Calculatetheeigenvalues*)​​vals//N​​(*isostaticeigenfunctioncurves*)​​Table[Plot3D[funs[[i]]//N//Evaluate,{x,y}∈Disk[{0,0},1],PlotRange->All,PlotLabel->vals[[i]],PlotTheme->"Minimal"],{i,Length[vals]}]
    Out[]=
    {5.78319,14.682,14.682,26.3746}
    Out[]=
    
    ,
    ,
    ,
    
    ◼
  • NDEigensystem
    • In[]:=
    Clear["Global`*"]​​{vals,funs}=NDEigensystem[{-Laplacian[u[x,y],{x,y}],DirichletCondition[u[x,y]==0,True]},u[x,y],{x,y}∈Disk[{0,0},1],4];​​(*Calculatetheeigenvalues*)​​vals​​(*isostaticeigenfunctioncurves*)​​Table[Plot3D[funs[[i]],{x,y}∈Disk[{0,0},1],PlotRange->All,PlotLabel->vals[[i]],PlotTheme->"Minimal"],{i,Length[vals]}]
    Out[]=
    {5.78323,14.6827,14.6827,26.3784}
    Out[]=
    
    ,
    ,
    ,
    
    We found the eigenvalues but they do not match the
    k
    nm
    of the above table. So we have to notice the following that the eigenvalues
    λ
    i
    are defined as the solution of equation
    -
    2
    ∇
    u(x,y)
    λ
    i
    u(x,y).
    In a case of
    ucos(nθ)
    J
    n
    (
    k
    mn
    r)
    we have
    In[]:=
    -Laplacian[Cos[nθ]BesselJ[n,kmnr],{r,θ},"Polar"]//FullSimplify
    Out[]=
    2
    kmn
    BesselJ[n,kmnr]Cos[nθ]
    Therefore,
    λ
    mn
    
    2
    k
    mn
    . So the first eigenvalues could be calculated as follows
    In[]:=
    b[m_,n_]:=BesselJZero[n,m];​​k[m_,n_]:=N[b[m,n]];Table[{m,n,k[m,n]^2},{m,1,3},{n,0,2}]
    Out[]=
    {{{1,0,5.78319},{1,1,14.682},{1,2,26.3746}},{{2,0,30.4713},{2,1,49.2185},{2,2,70.85}},{{3,0,74.887},{3,1,103.499},{3,2,135.021}}}
    Greetings: I would like to thank Alex Trounev from Mathematica Stack Exchange for his help and for solving my questions

    CITE THIS NOTEBOOK

    Helmholtz problem in circular disk​
    by Athanasios Paraskevopoulos​
    Wolfram Community, STAFF PICKS, April 19 2023
    https://community.wolfram.com/groups/-/m/t/2901086