Obtain a Clamped Triangular Membrane's Symbolic Eigenfunctions
Specify a Laplacian operator.
In[1]:=
![Click for copyable input](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/In_119.png)
\[ScriptCapitalL] = -Laplacian[u[x, y], {x, y}];
Specify homogeneous Dirichlet boundary conditions.
In[2]:=
![Click for copyable input](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/In_120.png)
\[ScriptCapitalB] = DirichletCondition[u[x, y] == 0, True];
Find the four smallest eigenvalues and eigenfunctions of the operator in a triangle.
In[3]:=
![Click for copyable input](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/In_121.png)
{vals, funs} =
DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x, y], {x, y} \[Element] Triangle[], 4];
In[4]:=
![Click for copyable input](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/In_122.png)
vals
Out[4]=
![](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/O_59.png)
Visualize the eigenfunctions.
In[5]:=
![Click for copyable input](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/In_123.png)
Table[Plot3D[funs[[i]], {x, y} \[Element] Triangle[], Boxed -> False,
Axes -> False, Method -> {"ShrinkWrap" -> True}], {i, 4}]
Out[5]=
![](assets.en/obtain-a-clamped-triangular-membranes-symbolic-eig/O_60.png)