I'm solving a Helmholtz equation uxx+uyy+lambda*u=0 in a rectangle: 0<=x<=L, 0<=y<=H with the following boundary conditions:(adsbygoogle = window.adsbygoogle || []).push({});

u(x,0)=u(x,H)=0 and ux(0,y)=ux(L,y)=0

I found the eigenvalues to be:

lambda(nm)=(n Pi/L)^2+(m Pi/H)^2

and the eigenfunctions to be:

u(nm)=Cos(n Pi x/L)*Sin(m Pi y/H)

Now the question I'm stuck on is to show that if L=H (a square) then most eigenvalues have more than one eigenfunction

and, Are any two eigenfunctions of this eigenvalue problem orthogonal in a two-dimensional sense?

Any help would be greatly appreciated.

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# Helmholtz Equation in a Square

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