Obtain a Clamped Triangular Membrane's Symbolic Eigenfunctions

Specify a Laplacian operator.

In[1]:=

\[ScriptCapitalL] = -Laplacian[u[x, y], {x, y}];

Specify homogeneous Dirichlet boundary conditions.

In[2]:=

\[ScriptCapitalB] = DirichletCondition[u[x, y] == 0, True];

Find the four smallest eigenvalues and eigenfunctions of the operator in a triangle.

In[3]:=

{vals, funs} = DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x, y], {x, y} \[Element] Triangle[], 4];

In[4]:=

vals

Out[4]=

Visualize the eigenfunctions.

In[5]:=

Table[Plot3D[funs[[i]], {x, y} \[Element] Triangle[], Boxed -> False, Axes -> False, Method -> {"ShrinkWrap" -> True}], {i, 4}]

Out[5]=