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

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

Hence derive the solution $y(x, t)$ subject to the initial conditions

$$y(x, 0)=0, \quad \frac{\partial y}{\partial t}(x, 0)=\psi(x)$$