Examples

Initial shape no velocity, divides into left-moving and right-moving halves

Initial shape no velocity, divides into left-moving and right-moving of same shape half size

Problem: Suppose the initial condition is given by $y(x,0)=f(x)$ and $y_t(x,0)=0$. How does the wave evolve?

Solution: Write the solution in the format $y(x,t)=\ell (x+vt)+r(x-vt)$. The initial conditions determine two equations:

$$\begin{array}{rl}y(x,0)& =\ell (x)+r(x)=f(x)\\ y_t(x,0)& =v{\ell }^{\prime }(x)-v{r}^{\prime }(x)=0.\end{array}$$

The first equation implies ${\ell }^{\prime }(x)+r^{\prime }(x)=f^{\prime }(x)$, which we solve for $r^{\prime }$ and plug into the second:

$$\begin{array}{rl}v{\ell }^{\prime }-v(f^{\prime }-{\ell }^{\prime })& =0\\ 2v{\ell }^{\prime }& =v{f}^{\prime }\\ 2{\ell }^{\prime }& =f^{\prime }\\ \ell & =\frac{1}{2}f+c.\end{array}$$

Now plug this in for $\ell$ in the first equation to obtain $r=\frac{1}{2}f-c$, and therefore:

$$y(x,t)=\frac{1}{2}(f(x+vt)+f(x-vt)).$$

This is the equation of a $\frac{1}{2}$ wave with shape $f$ moving to the left superposed on a $\frac{1}{2}$ wave with shape $f$ moving to the right.