Im trying to solve the 1D Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \exp(-100(x-0.5)^2), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary condition $$u(x,t) = u(x+1,t)$$ From what I know the general solution to this type of wave equation is $$u(x,t) = \frac{1}{2}(g(x+ct) + g(x-ct))$$ where $g(x) = u(x,0)$, which in this case is $$u(x,t) = \frac{1}{2} (\exp(-100(x-t-0.5)^2) + \exp(-100(x+t-0.5)^2)$$ however this does not satisfy the above wave equation.
Does anyone know what the correct closed form analytical solution is of this wave equation?
Best Answer
Because you have periodic boundary conditions the solution is different from what you stated. With this starting condition you cannot have a solution from the basic fact that it is not periodic.
In order to get the general solution you will need to do separation of variables and account for the periodic boundary conditions like follows:
Let's assume $u(x,t)=X(x)T(t)$ you can see that the equation becomes: $$X_{xx}=-k^2X, T_{tt}=-k^2T$$ (The minus and the square is just for simplicity and because k can be complex doesn't eliminate potential solutions)
The general solutions for these equations are: $$X(x)=Ae^{ikx}+Be^{-ikx}$$ From periodic boundary conditions you can get the value of k and see it is $2\pi$.