# wave question with piecewise initial conditions [duplicate]

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!

• This looks like a duplicate of this question.
– robjohn
Commented Mar 13, 2014 at 9:50

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!.

• I have worked out D'Alambert's solution to be $x^3+3xt^2-x+t-tx^2-\frac{t^3}{3}$ which seems to satisfy the initial and boundary conditions. Is this undefined in some places? I feel like I've been too casual with the piecewise conditions... How would I sketch the wave on a graph?
– Lucy
Commented Mar 12, 2014 at 21:01

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 $g(x)=\begin{cases}-\dfrac{2}{3}&\text{for}~x\leq-1\\x-\dfrac{x^3}{3}&\text{for}~|x|\leq1\\\dfrac{2}{3}&\text{for}~x\geq1\end{cases}$

• helpful when you just give an answer with no details as to how you arrived at it.
– user100463
Commented Jan 6, 2017 at 17:54