Nodal set of the Dirichlet Laplacian on the square
The eigenmodes od the Dirichlet Laplacian on the square [0,π]x[0,π] are explicit and given by
u(j,k)(x,y)=sin(j x) sin(k y), λ(j,k)=j2+k2
Consequently we have λ(1,r)=λ(r,1) and any linear combination of u(1,r) and u(r,1) is still an eigenfunction. Let us look at the nodal sets of some combination.