Solution of one-dimensional wave equations by canonical transformation

Consider a one dimensional wave equation

$$u_{xx}-c^2u_{tt}=0.$$

The discriminant of this equation is $\Delta=4c^2>0.$ Which means given equation is hyperbolic. The characteristic equation are given by solution of

$$u(x,t) =\frac{-b\pm \sqrt{b^2-4ac}}{2a},$$

$$=\mp c,$$

$$\implies~x\pm ct = .$$

The canonical variables are

$$\xi=x-ct~$$and$$~\eta=x+ct,$$

$$u_x =u_{\xi}+u_{\eta},$$

$$u_t =-cu_{\xi}+cu_{\eta},$$

$$u_{xx}=(u_{\xi}+u_{\eta})_x=u_{\xi \xi}+2u_{\xi \eta}+u_{\eta \eta},$$

$$u_{tt}=(-cu_{\xi}+cu_{\eta})_t=c^2u_{\xi \xi}-2u_{\xi \eta}+u_{\eta \eta}.$$

Putting it in given equation, one gets

$$u_{\xi \eta} = 0,$$

$$u_{\xi} =f(\xi),$$

$$u =\int f(\xi) d\xi+G(\eta),$$

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

$$u(x,t)=F(x-ct)+G(x+ct),$$ (3.49)

where $F$ and $G$ are arbitrary functions.

   Think it?$$~\begin{cases} F(x-ct) & \text{wave travelling to right}\\ G(x+ct) & \text{wave travelling to left} \end{cases}$$

To understand motion

1. Let $u_1(x,t)=F(x-ct).$ The initial wave profile is given by $t=0$ i.e. $u_1(x,0)=F(x).$ Also shown in Fig. .

   Figure:

   
2. Let us understand how this profile evolves as time goes. The trick is to use 'appropriate' time instants.
3. At $t=\frac{1}{c},~u_1(x,\frac{1}{c})=F(x-1).$ Let $\Tilde{x}=x-1,$ so $F(x-1)=F(\Tilde{x}).$ This shows that the initial wave profile is mentioned, even if the origin is shifted by one unit to the right along the $x-$axis i.e. the graph of $u_1(x,\frac{1}{c})$ is same as the graph of the initial wave profile translated one unit to the right. Also shown in Fig. .

   Figure:

   
4. At $t=\frac{2}{c}$, the profile will be shifted by two units to the right of the $x-$axis. Which implies that $F(x-ct)$ is a wave propagating to right.

Now we use analytic solution Eq. () to solve initial value problem for wave equation.

D'Alembert: Historically, D'Alembert was the first to formulate the equation $u_x=v_y, u_y=-v_x$ during his hydrodynamical investigation. But later, these equations were came to known as C-R equations due to work of Cauchy Riemann to emphasis the equations role w.r.t. compare analytic functions.