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)(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.
r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8 r=9 r=10
u(1,r)+u(r,1)
   
r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8 r=9 r=10
u(1,r)+0.97 u(r,1)
   
r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8 r=9 r=10
u(1,r)-0.3 u(r,1)
   
r=11 r=11 r=11 r=11
u(1,r)+ u(r,1) u(1,r)- u(r,1) u(1,r)-0.97 u(r,1) u(1,r)-0.1 u(r,1)
   
r=12 r=12
u(1,r)+ u(r,1) u(1,r)+0.97 u(r,1)