Show that the general solution of the wave equation

$$\frac{1}{c^{2}} \frac{\partial^{2} y}{\partial t^{2}}-\frac{\partial^{2} y}{\partial x^{2}}=0$$

can be written in the form

$$y(x, t)=f(c t-x)+g(c t+x) .$$

For the boundary conditions

$$y(0, t)=y(L, t)=0, \quad t>0,$$

find the relation between $f$ and $g$ and show that they are $2 L$-periodic. Hence show that

$$E(t)=\frac{1}{2} \int_{0}^{L}\left(\frac{1}{c^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right) d x$$

is independent of $t$.