Show how the general solution of the wave equation for $y(x, t)$,

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

can be expressed as

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

Show that the boundary conditions $y(0, t)=y(L, t)=0$ relate the functions $f$ and $g$ and require them to be periodic with period $2 L$.

Show that, with these boundary conditions,

$$\frac{1}{2} \int_{0}^{L}\left(\frac{1}{c^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right) \mathrm{d} x=\int_{-L}^{L} g^{\prime}(c t+x)^{2} \mathrm{~d} x$$

and that this is a constant independent of $t$.