For a plucked string obeying \(u_{xx} - \ddot{u}/c^2 =0\), \(b_n = 0\) and* \begin{align}\label{eq:recover:pluck:a} a_n = \frac{8}{n^2\pi^2} \sin\frac{n\pi}{2} \end{align}
The analytical (gray) and numerical (red) initial conditions and spectra are compared below:
*Blackstock 2000, p. 224