I have the problem $$u_{tt}=C^2(u_{xx}+u_{yy}+u_{zz})$$ with initial condition $$u(x,y,z,0)=x-y+1$$ Therefore, I wish to find all functions $F(s)$ such that $$F(ax+by+cz-t)=u(x,y,z,t)$$ is a solution of my pde and $$F(ax+by+cz)=x-y+1 \:\:\:\:\:(*)$$ where $a^2+b^2+c^2=\frac{1}{C^2}$.
I thought about first finding a general form of the functions $F$ satisfying $(*)$ and see where I can go from there. Any suggestions on how to proceed?
P.S.- I don't know how to solve a PDE using Fourier series