Wave equation on infinite line with piecewise $2\pi$-periodic i.c.

Question

I need to solve the following p.d.e.:

$\begin{align} u_{tt}(x,t) = a^{2}u_{xx}(x,t) & \,\,\,\, -\infty < x < \infty, \,\, t>0 \\ u(x,0)=0, \,\,\,\,\,\,\,& u_{t}(x,0) = g(x), \,\,\,\, -\infty < x < \infty \end{align}$ $\text{where}\, g(x) \, \text{denotes the}\, 2\pi-\text{periodic function satisfying}\, g(x) = \begin{cases} -1, & \pi

Now, by D'Alembert's formula, $\displaystyle u(x,t) = \frac{1}{2}\left[f(x+at) + f(x-at) \right] + \frac{1}{2a}\int_{x-at}^{x+at}g(z)dz $.

Since here, $f(x)=0$, we have that $\displaystyle u(x,t)=\frac{1}{2a}\int_{x+at}^{x-at}g(z)dz $.

Normally, what I would do next is evaluate the integral, but I'm not sure how to "plug" $g(z)$ in here. I don't think I need to make a periodic extension, because it's already given to be $2\pi-$periodic on the real line, so how am I supposed to deal with this kind of an initial condition? There are no examples like this in my book and I have been unable to find any similar examples on the internet, so I do not know how to proceed with this problem. Please help me finish!

Thanks.

---

Just to be clear, the fact that the function $g$ is periodic is throwing me off as to how I even integrate it. I'm probably overthinking - I guarantee you it's that, and not that I don't know how to find integrals.

Answer 1

So far you have $u(x,t)=\frac{1}{2a}(G(x+at)-G(x-at)$ where $G$ is an antiderivative of $g$. A periodic function whose integral over its period is $0$ has a periodic antiderivative. In this case, it's $|x|$ restricted to $[-\pi,\pi]$ and extended periodically. There are various ways to express this function in a formula, for example $G(x) = \operatorname{dist}(x, 2\pi \mathbb{Z})$ which means the distance from $x$ to the nearest element of the set $2\pi \mathbb{Z}$.