Consider a bar of length $\pi$ with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$ :

$$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$$

for $x \in(0, \pi)$ and $t>0$ with boundary conditions:

$$\frac{\partial y}{\partial x}(0, t)=\frac{\partial y}{\partial x}(\pi, t)=0$$

for $t>0$. The bar is initially at rest so that

$$\frac{\partial y}{\partial t}(x, 0)=0$$

for $x \in(0, \pi)$, with a spatially varying initial longitudinal displacement given by

$$y(x, 0)=b x$$

for $x \in(0, \pi)$, where $b$ is a real constant.

(a) Using separation of variables, show that

$$y(x, t)=\frac{b \pi}{2}-\frac{4 b}{\pi} \sum_{n=1}^{\infty} \frac{\cos [(2 n-1) x] \cos [(2 n-1) c t]}{(2 n-1)^{2}}$$

(b) Determine a periodic function $P(x)$ such that this solution may be expressed as

$$y(x, t)=\frac{1}{2}[P(x+c t)+P(x-c t)]$$