Wave equation in 1D

The wave equation for the scalar $u$ in the one dimensional case reads

$$\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}.$$ (2.2)

The general solution of Eq. (2.2) was first derived by Jean le Rond d'Alembert. Let us introduce new coordinates $(\xi, \eta)$ by use of the transformation

$$\xi=x-ct,\qquad \eta=x+ct.$$ (2.3)

In the new coordinate system Eq. (2.2) becomes

$$\frac{\partial^2 u}{\partial \xi\partial \eta}=0.$$ (2.4)

This equation means that the function $u$ remains constant along the curves (2.3), i.e., (2.3) are characteristic curves of the wave equation (2.2). Moreover, one can see that the derivative $\partial u/\partial \xi$ does not depends on $\eta$ , i.e.,

$$\frac{\partial}{\partial\eta}\biggl(\frac{\partial u}{\partial\xi}\biggr)=0\,\Leftrightarrow\, \frac{\partial u}{\partial\xi}=f(\xi).$$ (2.5)

After integration with respect to $\xi$ one obtains

$$u(\xi,\eta)=F(\xi)+G(\eta),$$

where $F$ is the primitive function of $f$ and $G$ is the ''constant`` of integration, in general the function of $\eta$ . Turning back to the coordinates $(x,t)$ one obtains the general solution of Eq. (2.2)

$$\boxed{u(x,t)=F(x-ct)+G(x+ct).}$$ (2.6)