Write down the wave equation for the displacement $y(x, t)$ of a stretched string with constant mass density and tension. Obtain the general solution in the form

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

where $c$ is the wave velocity. For a solution in the region $0 \leqslant x<\infty$, with $y(0, t)=0$ and $y \rightarrow 0$ as $x \rightarrow \infty$, show that

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

is constant in time. Express $E$ in terms of the general solution in this case.