Norm independent solution for partial differential eqution

Question

We are looking for raidal (norm independent) solutions for $u_{xx}=u_{yy}$. I have solved this equation and get that: u(x,y)=$f(x+y)+g(x-y)$. Now since the solution should be norm independent translate the solution into polar coordinates $u(x,y)=f(r(\cos \phi + \sin \phi))+g(r(\cos \phi - \sin \phi))$ since right hand side should be independent on $\phi$ we can put $\phi=\pi/4$. We finally get $u(x,y)=f(\sqrt{x^2+y^2})$ we can ommit the constant $g(0)$. The problem is I can't determine the functions $f$ that actually makes it a solution (tried use that $f(x+y)=f(\sqrt{x^2+y^2})$) and what happends at $0$? I will be gratefoul for your help.