A rod of length \(L\), with ends at \((x=0, L)\), has an initial displacement function \(\eta(0, x)=b\) for \(0 \leq x \leq L / 2\) and \(\eta(0, x)=-b\) for \(L / 2 \leq x \leq L\), where \(b\) is a positive constant. At time \(t=0\) the derivative of \(\eta(t, x)\) is \(\left.\partial \eta(t, x) \partial t\right|_{0}=-v_{0}\) for \(0 \leq x \leq L / 2\) and equal to \(+v_{0}\) for \(L / 2 \leq x \leq L\), where \(v_{0}\) is a positive constant. Find a Fourier-series representation of the solution of the wave equation at all future times using the doubling trick.