wave question with piecewise initial conditions

Question

My problem is regarding the wave equation of the form: $$u_{tt}=u_{xx}$$ in the domain $$D = \{ (x,t) \mid - \infty<x< \infty\ ,t>0\} $$ subject to the initial conditions:

$$u(x,0)=\left\{ \begin{array}{c l} x^3-x, & |x|\leq1\\ 0, & |x|\geq1 \end{array}\right.$$

$$u_t(x,0)=\left\{ \begin{array}{c l} 1-x^2, & |x|\leq1\\ 0, & |x|\geq1 \end{array}\right.$$

I'm pretty sure this can be solved using D'Alambert's solution for a general wave equation, however I'm slightly unsure as to how to do this question when there are piecewise initial conditions. I know the initial conditions are continuous. How do I tackle this? Any help would be greatly appreciated!

Answer 1

Try going over the derivation of D'Lambert's solution again (i.e use the change of variables $r=x+t$ and $s=x-t$ to show that the PDE really says $\frac{\partial ^2u}{\partial r \partial s} = 0$, then integrate with respect to r and s, plug in the boundary conditions).

You should see that not much changes when you have piecewise initial conditions, you just need to take care evaluating the integral in D'Lambert's solution, be careful about where the function (speed) $$u_t(x,0) \begin{cases} 1-x^2 &|x|\leq 1 \\ \\ 0 &|x|\geq 1 \end{cases}$$ is zero!.

Answer 2

In fact the real result should be $u(x,t)=\dfrac{f(x+t)+f(x-t)+g(x+t)-g(x-t)}{2}$ , where $f(x)=\begin{cases}x^3-x&\text{for}~|x|\leq1\\0&\text{for}~|x|\geq1\end{cases}$ and $g(x)=\begin{cases}-\dfrac{2}{3}&\text{for}~x\leq-1\\x-\dfrac{x^3}{3}&\text{for}~|x|\leq1\\\dfrac{2}{3}&\text{for}~x\geq1\end{cases}$